A237048 Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists 1's interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0

Offset

1

Comments

The sum of row n gives A001227(n), the number of odd divisors of n.

Row n has length A003056(n), hence column k starts in row A000217(k).

If n = 2^j then the only positive integer in row n is T(n,1) = 1.

If n is an odd prime then the only two positive integers in row n are T(n,1) = 1 and T(n,2) = 1.

The partial sums of column k give the column k of A235791.

The connection with A196020 is as follows: A235791 --> A236104 --> A196020.

The connection with the symmetric representation of sigma is as follows: A235791 --> A237591 --> A237593 --> A239660 --> A237270.

From Hartmut F. W. Hoft, Oct 23 2014: (Start)

Property: Let n = 2^m*s*t with m >= 0 and 1 <= s, t odd, and r(n) = floor(sqrt(8*n+1) - 1)/2) = A003056(n). T(n, k) = 1 precisely when k is odd and k|n, or k = 2^(m+1)*s when 1 <= s < 2^(m+1)*s <= r(n) < t. Thus each odd divisor greater than r(n) is matched by a unique even index less than or equal to r(n).

For further connections with the symmetric representation of sigma see also A249223. (End)

From Omar E. Pol, Jan 21 2017: (Start)

Conjecture 1: alternating sum of row n gives A067742(n), the number of middle divisors of n.

The sum of row n also gives the number of subparts in the symmetric representation of sigma(n), equaling A001227(n), the number of odd divisors of n. For more information see A279387. (End)

From Omar E. Pol, Feb 08 2017, Feb 22 2017: (Start)

Conjecture 2: Alternating sum of row n also gives the number of central subparts in the symmetric representation of sigma(n), equaling the width of the terrace at the n-th level in the main diagonal of the pyramid described in A245092.

Conjecture 3: The sum of the odd-indexed terms in row n gives A082647(n): the number of odd divisors of n less than sqrt(2*n), also the number of partitions of n into an odd number of consecutive parts.

Conjecture 4: The sum of the even-indexed terms in row n gives A131576(n): the number of odd divisors of n greater than sqrt(2*n), also the number of partitions of n into an even number of consecutive parts.

Conjecture 5: The sum of the even-indexed terms in row n also gives the number of pairs of equidistant subparts in the symmetric representation of sigma(n). (End)

Conjecture 6: T(n,k) is also the number of partitions of n into exactly k consecutive parts. - Omar E. Pol, Apr 28 2017

The number of zeros in the n-th row equals A238005(n). - Omar E. Pol, Sep 11 2021

This triangle is a member of an infinite family of irregular triangles read by rows in which column k lists 1's interleaved with k-1 zeros, and the first element of column k is where the row number equals the k-th (m+2)-gonal number, with n >= 1, k >= 1, m >= 0. T(n,k) is also the number of partitions of n into k consecutive parts that differ by m. This is the case for m = 1. For other values of m see the cross-references. - Omar E. Pol, Sep 29 2021

The indices of the rows where the number of 1's increases to a record give A053624. - Omar E. Pol, Mar 04 2023

Links

G. C. Greubel, Table of n, a(n) for the first 150 rows, flattened

Formula

For n >= 1 and k = 1, ..., A003056(n): if k is odd then T(n, k) = 1 if k|n, otherwise 0, and if k is even then T(n, k) = 1 if k|(n-k/2), otherwise 0. - Hartmut F. W. Hoft, Oct 23 2014

a(n) = A057427(A196020(n)) = A057427(A261699(n)). - Omar E. Pol, Nov 14 2016

A000203(n) = Sum_{k=1..A003056(n)} (-1)^(k-1) * ((Sum_{j=k*(k+1)/2..n} T(j,k))^2 - (Sum_{j=k*(k+1)/2..n} T(j-1,k))^2), assuming that T(k*(k+1)/2-1,k) = 0. - Omar E. Pol, Oct 10 2018

T(n,k) = A285914(n,k)/k. - Omar E. Pol, Sep 29 2021

From Hartmut F. W. Hoft, Apr 30 2024: (Start)

Another way of expressing the formula above, using S(n,k) for entries in the triangle of A235791, is:

T(n,k) = S(n,k) - S(n-1,k), for all n >=1 and 1 <= k <= A003056(n), noting that for triangular numbers n(n+1)/2, S(n(n+1)/2 - 1, A003056(n(n+1)/2) = S(n(n+1)/2 - 1, n) = 0.

Also, T(n,k) = 1 if n - k(k+1)/2 (mod k) = 0, and 0 otherwise. (End)

Example

Triangle begins (rows 1..28):
1;
1;
1,  1;
1,  0;
1,  1;
1,  0,  1;
1,  1,  0;
1,  0,  0;
1,  1,  1;
1,  0,  0,  1;
1,  1,  0,  0;
1,  0,  1,  0;
1,  1,  0,  0;
1,  0,  0,  1;
1,  1,  1,  0,  1;
1,  0,  0,  0,  0;
1,  1,  0,  0,  0;
1,  0,  1,  1,  0;
1,  1,  0,  0,  0;
1,  0,  0,  0,  1;
1,  1,  1,  0,  0,  1;
1,  0,  0,  1,  0,  0;
1,  1,  0,  0,  0,  0;
1,  0,  1,  0,  0,  0;
1,  1,  0,  0,  1,  0;
1,  0,  0,  1,  0,  0;
1,  1,  1,  0,  0,  1;
1,  0,  0,  0,  0,  0,  1;
...
For n = 20 the divisors of 20 are 1, 2, 4, 5, 10, 20.
There are two odd divisors: 1 and 5. On the other hand the 20th row of triangle is [1, 0, 0, 0, 1] and the row sum is 2, equaling the number of odd divisors of 20.
From Hartmut F. W. Hoft, Oct 23 2014: (Start)
For n = 18 the divisors are 1, 2, 3, 6, 9, 18.
There are three odd divisors: 1 and 3 are in their respective columns, but 9 is accounted for in column 4 = 2^2*1 since 18 = 2^1*1*9 and 9>5, the number of columns in row 18. (End)
From Omar E. Pol, Dec 17 2016: (Start)
Illustration of initial terms:
Row                                                         _
1                                                         _|1|
2                                                       _|1 _|
3                                                     _|1  |1|
4                                                   _|1   _|0|
5                                                 _|1    |1 _|
6                                               _|1     _|0|1|
7                                             _|1      |1  |0|
8                                           _|1       _|0 _|0|
9                                         _|1        |1  |1 _|
10                                      _|1         _|0  |0|1|
11                                    _|1          |1   _|0|0|
12                                  _|1           _|0  |1  |0|
13                                _|1            |1    |0 _|0|
14                              _|1             _|0   _|0|1 _|
15                            _|1              |1    |1  |0|1|
16                          _|1               _|0    |0  |0|0|
17                        _|1                |1     _|0 _|0|0|
18                      _|1                 _|0    |1  |1  |0|
19                    _|1                  |1      |0  |0 _|0|
20                  _|1                   _|0     _|0  |0|1 _|
21                _|1                    |1      |1   _|0|0|1|
22              _|1                     _|0      |0  |1  |0|0|
23            _|1                      |1       _|0  |0  |0|0|
24          _|1                       _|0      |1    |0 _|0|0|
25        _|1                        |1        |0   _|0|1  |0|
26      _|1                         _|0       _|0  |1  |0 _|0|
27    _|1                          |1        |1    |0  |0|1 _|
28   |1                            |0        |0    |0  |0|0|1|
...
Note that the 1's are placed exactly below the horizontal line segments.
Also the above structure represents the left hand part of the front view of the pyramid described in A245092. For more information about the pyramid and the symmetric representation of sigma see A237593. (End)

Maple

r := proc(n) floor((sqrt(1+8*n)-1)/2) ; end proc: # A003056
A237048:=proc(n, k) local i; global r;
if n<(k-1)*k/2 or k>r(n) then return(0); fi;
if (k mod 2)=1 and (n mod k)=0 then return(1); fi;
if (k mod 2)=0 and ((n-k/2) mod k) = 0 then return(1); fi;
return(0);
end;
for n from 1 to 12 do lprint([seq(A237048(n, k), k=1..r(n))]); od; # N. J. A. Sloane, Jan 15 2021

Mathematica

cd[n_, k_] := If[Divisible[n, k], 1, 0]
row[n_] := Floor[(Sqrt[8n+1] - 1)/2]
a237048[n_, k_] := If[OddQ[k], cd[n, k], cd[n - k/2, k]]
a237048[n_] := Map[a237048[n, #]&, Range[row[n]]]
Flatten[Map[a237048, Range[24]]] (* data: 24 rows of triangle *)
(* Hartmut F. W. Hoft, Oct 23 2014 *)

Prog

(PARI) t(n, k) = if (k % 2, (n % k) == 0, ((n - k/2) % k) == 0);
tabf(nn) = {for (n=1, nn, for (k=1, floor((sqrt(1+8*n)-1)/2), print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Sep 20 2015
(Python)
from sympy import sqrt
import math
def T(n, k): return (n%k == 0)*1 if k%2 == 1 else (((n - k/2)%k) == 0)*1
for n in range(1, 21): print([T(n, k) for k in range(1, int(math.floor((sqrt(8*n + 1) - 1)/2)) + 1)]) # Indranil Ghosh, Apr 21 2017

Crossrefs

Indices of 1's are also the indices of nonzero terms in A196020, A211343, A236106, A239662, A261699, A272026, A280850, A285891, A285914, A286013, A339275.

The MMA code here is also used in A262045.

Triangles of the same family related to partitions into consecutive parts that differ by m are: A051731 (m=0), this sequence (m=1), A303300 (m=2), A330887 (m=3), A334460 (m=4), A334465 (m=5).

Cf. A000203, A001227, A003056, A053624, A057427, A067742, A082647, A131576, A236104, A235791, A237270, A237271, A237591, A237593, A238005, A239657, A245092, A249351, A262611, A262626, A279387, A279693, A285898, A334466.

Keyword

nonn,easy,tabf

Author

Omar E. Pol, Mar 01 2014

Status

approved

A030190 Binary Champernowne sequence (or word): write the numbers 0,1,2,3,4,... in base 2 and juxtapose.

0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0

Offset

0,1

Comments

a(A003607(n)) = 0 and for n > 0: a(A030303(n)) = 1. - Reinhard Zumkeller, Dec 11 2011

An irregular table in which the n-th row lists the bits of n (see the example section). - Jason Kimberley, Dec 07 2012

The binary Champernowne constant: it is normal in base 2. - Jason Kimberley, Dec 07 2012

This is the characteristic function of A030303, which gives the indices of 1's in this sequence and has first differences given by A066099. - M. F. Hasler, Oct 12 2020

References

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Jean Berstel, Home Page (in case the following link should be broken)

Jean Berstel and Juhani Karhumäki, Combinatorics on words-a tutorial. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, # 79, pp. 178-228, 2003.

S. Ferenczi, Complexity of sequences and dynamical systems, Discrete Math., 206 (1999), 145-154.

Eric Weisstein's World of Mathematics, Binary Champernowne Constant

Example

As an array, this begins:
0,
1,
1, 0,
1, 1,
1, 0, 0,
1, 0, 1,
1, 1, 0,
1, 1, 1,
1, 0, 0, 0,
1, 0, 0, 1,
1, 0, 1, 0,
1, 0, 1, 1,
1, 1, 0, 0,
1, 1, 0, 1,
1, 1, 1, 0,
1, 1, 1, 1,
1, 0, 0, 0, 0,
1, 0, 0, 0, 1,
...

Mathematica

Flatten[ Table[ IntegerDigits[n, 2], {n, 0, 26}]] (* Robert G. Wilson v, Mar 08 2005 *)
First[RealDigits[ChampernowneNumber[2], 2, 100, 0]] (* Paolo Xausa, Jun 16 2024 *)

Prog

(Haskell)
import Data.List (unfoldr)
a030190 n = a030190_list !! n
a030190_list = concatMap reverse a030308_tabf
-- Reinhard Zumkeller, Jun 16 2012, Dec 11 2011
(Magma) [0]cat &cat[Reverse(IntegerToSequence(n, 2)):n in[1..31]]; // Jason Kimberley, Dec 07 2012
(PARI) A030190_row(n)=if(n, binary(n), [0]) \\ M. F. Hasler, Oct 12 2020
(Python)
from itertools import count, islice
def A030190_gen(): return (int(d) for m in count(0) for d in bin(m)[2:])
A030190_list = list(islice(A030190_gen(), 30)) # Chai Wah Wu, Jan 07 2022

Crossrefs

Cf. A007376, A003137, A030308. Same as and more fundamental than A030302, but I have left A030302 in the OEIS because there are several sequences that are based on it (A030303 etc.). - N. J. A. Sloane.

a(n) = T(A030530(n), A083652(A030530(n))-n-1), T as defined in A083651, a(A083652(k))=1.

Tables in which the n-th row lists the base b digits of n: this sequence and A030302 (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

A076478 is a similar sequence.

For run lengths see A056062; see also A318924.

See also A066099 for (run lengths of 0s) + 1 = first difference of positions of 1s given by A030303.

Keyword

nonn,base,cons,easy,tabf

Author

Clark Kimberling

Status

approved

A000161 Number of partitions of n into 2 squares.

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0

Offset

0,26

Comments

Number of ways of writing n as a sum of 2 (possibly zero) squares when order does not matter.

Number of similar sublattices of square lattice with index n.

Let Pk = the number of partitions of n into k nonzero squares. Then we have A000161 = P0 + P1 + P2, A002635 = P0 + P1 + P2 + P3 + P4, A010052 = P1, A025426 = P2, A025427 = P3, A025428 = P4. - Charles R Greathouse IV, Mar 08 2010, amended by M. F. Hasler, Jan 25 2013

a(A022544(n))=0; a(A001481(n))>0; a(A125022(n))=1; a(A118882(n))>1. - Reinhard Zumkeller, Aug 16 2011

References

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 339

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

Henry Bottomley, Illustration of initial terms

R. T. Bumby, Sums of four squares, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.

J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).

Michael Gilleland, Some Self-Similar Integer Sequences

E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 84.

M. D. Hirschhorn, Some formulas for partitions into squares, Discrete Math, 211 (2000), pp. 225-228. [From Ant King, Oct 05 2010]

Index entries for sequences related to sublattices

Index entries for sequences related to sums of squares

Index entries for "core" sequences

Formula

a(n) = card { { a,b } c N | a^2+b^2 = n }. - M. F. Hasler, Nov 23 2007

Let f(n)= the number of divisors of n that are congruent to 1 modulo 4 minus the number of its divisors that are congruent to 3 modulo 4, and define delta(n) to be 1 if n is a perfect square and 0 otherwise. Then a(n)=1/2 (f(n)+delta(n)+delta(1/2 n)). - Ant King, Oct 05 2010

Example

25 = 3^2+4^2 = 5^2, so a(25) = 2.

Maple

A000161 := proc(n) local i, j, ans; ans := 0; for i from 0 to n do for j from i to n do if i^2+j^2=n then ans := ans+1 fi od od; RETURN(ans); end; [ seq(A000161(i), i=0..50) ];
A000161 := n -> nops( numtheory[sum2sqr](n) ); # M. F. Hasler, Nov 23 2007

Mathematica

Length[PowersRepresentations[ #, 2, 2]] &/@Range[0, 150] (* Ant King, Oct 05 2010 *)

Prog

(PARI) a(n)=sum(i=0, n, sum(j=0, i, if(i^2+j^2-n, 0, 1))) \\ for illustrative purpose
(PARI) A000161(n)=sum(k=sqrtint((n-1)\2)+1, sqrtint(n), issquare(n-k^2)) \\ Charles R Greathouse IV, Mar 21 2014, improves earlier code by M. F. Hasler, Nov 23 2007
(PARI) A000161(n)=#sum2sqr(n) \\ See A133388 for sum2sqr(). - M. F. Hasler, May 13 2018
(Haskell)
a000161 n =
   sum $ map (a010052 . (n -)) $ takeWhile (<= n `div` 2) a000290_list
a000161_list = map a000161 [0..]
-- Reinhard Zumkeller, Aug 16 2011
(Python)
from math import prod
from sympy import factorint
def A000161(n):
    f = factorint(n)
    return int(not any(e&1 for e in f.values())) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 1 # Chai Wah Wu, Sep 08 2022

Crossrefs

Cf. A002654, A001481, A025427, A025428, A063725, A025426, A000290.

Cf. A000925, A247367.

Equivalent sequences for other numbers of squares: A010052 (1), A000164 (3), A002635 (4), A000174 (5).

Keyword

nonn,core,easy,nice

Author

N. J. A. Sloane

Status

approved

A030302 Write n in base 2 and juxtapose; irregular table in which row n lists the binary expansion of n.

1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1

Offset

1,1

Comments

The binary Champernowne constant: it is normal in base 2. - Jason Kimberley, Dec 07 2012

A word that is recurrent, but neither morphic nor uniformly recurrent. - N. J. A. Sloane, Jul 14 2018

See A030303 for the indices of 1's (so this is the characteristic function of A030303), with first differences (i.e., run lengths of 0's, increased by 1, with two consecutive 1's delimiting a run of zero 0's) given by A066099. - M. F. Hasler, Oct 12 2020

References

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Links

Table of n, a(n) for n=1..90.

Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.

Formula

a(n) = (floor(2^(((n + 2^i - 2) mod i) - i + 1) * ceiling((n + 2^i - 1)/i - 1))) mod 2 where i = ceiling( W(log(2)/2 (n - 1))/log(2) + 1 ) and W denotes the principal branch of the Lambert W function. See also Mathematica code. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007

Maple

A030302 := proc(n) local i, t1, t2; t1:=convert(n, base, 2); t2:=nops(t1); [seq(t1[t2+1-i], i=1..t2)]; end; # N. J. A. Sloane, Apr 08 2021

Mathematica

i[n_] := Ceiling[FullSimplify[ProductLog[Log[2]/2 (n - 1)]/Log[2] + 1]]; a[n_] := Mod[Floor[2^(Mod[n + 2^i[n] - 2, i[n]] - i[n] + 1) Ceiling[(n + 2^i[n] - 1)/i[n] - 1]], 2]; (* David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007 *)
Join @@ Table[ IntegerDigits[i, 2], {i, 1, 40}] (* Olivier Gérard, Mar 28 2011 *)
Flatten@ IntegerDigits[ Range@ 25, 2] (* or *)
almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ almostNatural[#, 2] &, 105] (* Robert G. Wilson v, Jun 29 2014 *)

Prog

(Magma) &cat[Reverse(IntegerToSequence(n, 2)): n in [1..31]]; // Jason Kimberley, Mar 02 2012
(Python)
from itertools import count, islice
def A030302_gen(): # generator of terms
    return (int(d) for n in count(1) for d in bin(n)[2:])
A030302_list = list(islice(A030302_gen(), 30)) # Chai Wah Wu, Feb 18 2022

Crossrefs

Essentially the same as A007088 and A030190. Cf. A030303, A007088.

Tables in which the n-th row lists the base b digits of n: A030190 and this sequence (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). [Jason Kimberley, Dec 06 2012]

Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

Keyword

nonn,base,cons,easy,tabf

Author

Clark Kimberling

Status

approved

A051023 Middle column of rule-30 1-D cellular automaton, from a lone 1 cell.

1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1

Offset

0,1

Comments

A092539(n) gives the value of prefix of length n+1, seen as a binary number. - Reinhard Zumkeller, Jun 08 2013

Also middle column of rule 86 1-D cellular automaton, from a lone 1 cell, as rule 86 is the mirror image of rule 30. - Antti Karttunen, Oct 03 2019

Links

Antti Karttunen, Table of n, a(n) for n = 0..100000 (terms 0..10000 from Reinhard Zumkeller)

Pedro Hecht, PQC: R-Propping a Chaotic Cellular Automata, Univ. of Buenos Aires (Argentina, 2021).

Erica Jen, Global properties of cellular automata, Journal of Statistical Physics 43 (1986), pp 219-242.

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.

Eric Weisstein's World of Mathematics, Rule 30

Eric Weisstein's World of Mathematics, Random Number

Xiangdong Wen, A Billion Bits of the Center Column of the Rule 30 Cellular Automaton, 2019

Stephen Wolfram, Announcing the Rule 30 Prizes, 2019

Stephen Wolfram, A Million Bits of the Center Column of the Rule 30 Cellular Automaton, 2017

Wolfram Mathematica, Cellular automata: Mathematica notebooks

Index entries for sequences related to cellular automata

Index entries for characteristic functions

Formula

a(n) = A070950(n,n). - Reinhard Zumkeller, Jun 06 2013

a(n) = 1 - A226474(n). - Reinhard Zumkeller, Jun 08 2013

From Antti Karttunen, Oct 04 2019: (Start)

a(n) = A000035(floor(A110240(n) / 2^n)).

For n>= 2, a(n) = (A328100(n) OR A328101(n)) XOR A328101(1+n). ["Sideways evaluation"]

(End)

Mathematica

CellularAutomaton[30, {{1}, 0}, 101, {All, {0}}]//Flatten

Prog

(Haskell)
a051023 n = a070950 n n  -- Reinhard Zumkeller, Jun 06 2013
(PARI)
A051023(n) = ((A110240(n)>>n)%2);
\\ Or for fast creation of b-files:
A051023write(up_to) = { my(s=1, n=0); for(n=0, up_to, write("b051023.txt", n, " ", ((s>>n)%2)); s = A269160(s)); }; \\ Antti Karttunen, Oct 03 2019

Crossrefs

Cf. A070950, A269160 (rule 30), A071032, A269161 (rule 86).

Cf. A000035, A092539, A226474.

Cf. A327974 (adjacent bits xored), A327982 (partial sums), A327983 (run lengths).

Characteristic function of A327984 (gives the positions of ones in this sequence), A327985 (positions of zeros).

Cf. also A328100, A328101, A328102 (neighbor columns).

Cf. A365254 (converted to base 10).

Keyword

nonn

Author

Eric W. Weisstein

Extensions

Corrected from 64th term by Daniel B. Cristofani (cristofd(AT)hevanet.com), Jan 07 2004

Status

approved

A011751 Expansion of (1 + x^4)/(1 + x + x^3 + x^4 + x^5) mod 2.

1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0

Offset

0,1

Links

Muniru A Asiru, Table of n, a(n) for n = 0..1000

Michael Gilleland, Some Self-Similar Integer Sequences

R. Gold, Characteristic linear sequences and their coset functions, J. SIAM Applied. Math., 14 (1966), 980-985.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1), i.e., 31-periodic.

Formula

a(n+31) = a(n) for all n. - M. F. Hasler, Feb 17 2018

Maple

series((1+x^4)/(1+x+x^3+x^4+x^5), x, 100) mod 2;
[seq(coeff(series((1+x^4)/(1+x+x^3+x^4+x^5), x, 100) mod 2, x, n), n=0..80)]; # Muniru A Asiru, Feb 18 2018
A011751 := n -> coeftayl((1+x^4)/(1+x+x^3+x^4+x^5), x=0, n) mod 2 # M. F. Hasler, Feb 18 2018

Mathematica

Mod[ CoefficientList[ Series[(1 + x^4)/(1 + x + x^3 + x^4 + x^5), {x, 0, 105}], x], 2] (* Robert G. Wilson v, Feb 19 2018 *)

Prog

(PARI) a(n)=bittest(1826728215, n%31) \\ M. F. Hasler, Feb 17 2018

Crossrefs

Cf. A011655..A011750 for other binary m-sequences.

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved

A024316 a(n) = s(1)*s(n) + s(2)*s(n-1) + ... + s(k)*s(n+1-k), where k = floor((n+1)/2), s = A023531.

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1

Offset

1,28

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{j=1..floor((n+1)/2)} A023531(j)*A023531(n-j+1). - G. C. Greubel, Jan 17 2022

Mathematica

A023531[n_]:= SquaresR[1, 8n+9]/2;
a[n_]:= a[n]= Sum[A023531[j]*A023531[n-j+1], {j, Floor[(n+1)/2]}];
Table[a[n], {n, 110}] (* G. C. Greubel, Jan 17 2022 *)

Prog

(Haskell)
a024316 n = sum $ take (div (n + 1) 2) $ zipWith (*) zs $ reverse zs
            where zs = take n $ tail a023531_list
-- Reinhard Zumkeller, Feb 14 2015
(Magma)
A023531:= func< n | IsIntegral( (Sqrt(8*n+9) - 3)/2 ) select 1 else 0 >;
[ (&+[A023531(j)*A023531(n-j+1): j in [1..Floor((n+1)/2)]]) : n in [1..110]]; // G. C. Greubel, Jan 17 2022
(Sage)
def A023531(n):
    if ((sqrt(8*n+9) -3)/2).is_integer(): return 1
    else: return 0
[sum( A023531(j)*A023531(n-j+1) for j in (1..floor((n+1)/2)) ) for n in (1..110)] # G. C. Greubel, Jan 17 2022

Crossrefs

Cf. A024312, A024313, A024314, A024315, A024317, A024318, A024319, A024320, A024321, A024322, A024323, A024324, A024325, A024326, A024327.

Cf. A023531.

Keyword

nonn

Author

Clark Kimberling

Status

approved

A189920 Zeckendorf representation of natural numbers.

1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1

Offset

1

Comments

The row lengths sequence of this array is A072649(n), n >= 1.

Note that the Fibonacci numbers F(0)=0 and F(1)=1 are not used in this unique representation of n >= 1. No neighboring Fibonacci numbers are allowed (no 1,1, subsequence in any row n).

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., 1994, Addison-Wesley, Reading MA, pp. 295-296.

E. Zeckendorf, Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41.3-4 (1972) 179-182 (with the proof from 1939).

Links

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Formula

n = Sum_{m=1..rl(n)} a(n,m)*F(rl(n) + 2 - m), n >= 1, with rl(n):=A072649(n)(row length) and F(n):=A000045(n) (Fibonacci numbers).

T(n,k) = A213676(n, A072649(n, k)-1) for k = 1..A072649(k). - Reinhard Zumkeller, Mar 10 2013

Example

n=1:  1;
n=2:  1, 0;
n=3:  1, 0, 0;
n=4:  1, 0, 1;
n=5:  1, 0, 0, 0;
n=6:  1, 0, 0, 1;
n=7:  1, 0, 1, 0;
n=8:  1, 0, 0, 0, 0;
n=9:  1, 0, 0, 0, 1;
n=10: 1, 0, 0, 1, 0;
n=11: 1, 0, 1, 0, 0;
n=12: 1, 0, 1, 0, 1;
n=13: 1, 0, 0, 0, 0, 0;
...
1 = F(2),
6 = F(5) + F(2),
11 = F(6) + F(4).

Mathematica

f[n_] := (k = 1; ff = {}; While[(fi = Fibonacci[k]) <= n, AppendTo[ff, fi]; k++]; Drop[ff, 1]); a[n_] := (fn = f[n]; xx = Array[x, Length[fn]]; r = xx /. {ToRules[ Reduce[ And @@ (0 <= # <= 1 & ) /@ xx && fn . xx == n, xx, Integers]]}; Reverse[ First[ Select[ r, FreeQ[ Split[#], {1, 1, ___}] & ]]]); Flatten[ Table[ a[n], {n, 1, 25}]] (* Jean-François Alcover, Sep 29 2011 *)

Prog

(Haskell)
a189920 n k = a189920_row n !! k
a189920_row n = z n $ reverse $ takeWhile (<= n) $ tail a000045_list where
   z x (f:fs'@(_:fs)) | f == 1 = if x == 1 then [1] else []
                      | f == x = 1 : replicate (length fs) 0
                      | f < x  = 1 : 0 : z (x - f) fs
                      | f > x  = 0 : z x fs'
a189920_tabf = map a189920_row [1..]
-- Reinhard Zumkeller, Mar 10 2013

Crossrefs

Cf. A035517, A014417.

Keyword

nonn,easy,tabf,base

Author

Wolfdieter Lang, Jun 12 2011

Status

approved

A129405 Expansion of L(3, chi3) in base 2, where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0

Offset

0,1

Comments

Contributed to OEIS on Apr 15 2007 -- the 300th anniversary of the birth of Leonhard Euler.

References

Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292

Links

Table of n, a(n) for n=0..103.

Formula

chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.

Series: L(3, chi3) = sum_{k >= 1} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...

Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).

Example

L(3, chi3) = A129404 = (0.111000100100111101100010011100000101101...)_2

Mathematica

nmax = 1000; First[ RealDigits[4 Pi^3/(81 Sqrt[3]) - (1/2) * 2^(-nmax), 2, nmax] ]

Crossrefs

Cf. A129404, A129406, A129407, A129408, A129409, A129410, A129411.

Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664, A129665

Keyword

nonn,base,cons,easy

Author

Stuart Clary, Apr 15 2007

Extensions

Offset corrected by R. J. Mathar, Feb 05 2009

Status

approved

A142724 Irregular triangle read by rows: row n gives coefficients in expansion of Product_{k=1..n} (1 + x^(2*k + 1)) for n >= 0.

1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2

Offset

0,43

Comments

For n >= 1, row n is the Poincaré polynomial for the Lie group A_n.

Row sums are powers of 2.

References

Borel, A. and Chevalley, C., The Betti numbers of the exceptional groups, Mem. Amer. Math. Soc. 1955, no. 14, pp 1-9.

Samuel I. Goldberg, Curvature and Homology, Dover, New York, 1998, page 144

Links

Alois P. Heinz, Rows n = 0..40, flattened

Formula

p(x,n) = Product[(1 + x^(2*k + 1)), {k, 1, n}]; t(n,m)=coefficients(p(x,n)).

Example

Triangle begins:
{1} (the empty product)
{1, 0, 0, 1},
{1, 0, 0, 1, 0, 1, 0, 0, 1},
{1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1},
{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1},
{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 0, 2, 1, 2, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1},
...

Maple

A:=n->mul(1+x^(2*r+1), r=1..n);
for n from 1 to 12 do lprint(seriestolist(series(A(n), x, 10000))); od:

Mathematica

Clear[p, x, n, m]; p[x_, n_] = Product[(1 + x^(2*k + 1)), {k, 1, n}]; Table[CoefficientList[p[x, n], x], {n, 1, 10}]; Flatten[%]

Crossrefs

Cf. A178600, A178602, A178580, A178651, A178665, A178666.

Rows: A178686. A178687, A178691, A178692, A178695, A178696, A178697.

Keyword

nonn,tabf

Author

Roger L. Bagula and Gary W. Adamson, Sep 26 2008

Extensions

Edited by N. J. A. Sloane, Dec 25 2010

Status

approved