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A001316 Gould's sequence: a(n) = Sum_{k=0..n} (C(n,k) mod 2); number of odd entries in row n of Pascal's triangle (A007318); 2^A000120(n).
(Formerly M0297 N0109)
133
1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 4, 8, 8, 16, 8, 16, 16, 32, 8, 16, 16, 32, 16, 32, 32, 64, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 4, 8, 8, 16, 8, 16, 16, 32 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also called Dress's sequence.

All terms are powers of 2. The first occurrence of 2^k is when n = 2^k - 1: e.g. the first occurrence of 16 is at n = 15 - Robert G. Wilson v, Dec 06 2000

a(n) is the highest power of 2 dividing C(2n,n)=A000984(n) - Benoit Cloitre, Jan 23 2002

Also number of 1's in n-th row of triangle in A070886. - Hans Havermann, May 26 2002. Equivalently, number of live cells in generation n of a one-dimensional cellular automaton, Rule 90, starting with a single live cell. - Ben Branman, Feb 28 2009. Ditto for Rule 18. - N. J. A. Sloane, Aug 09 2014

Also number of numbers k, 0<=k<=n, such that (k OR n) = n (bitwise logical OR): a(n) = #{k : T(n,k)=n, 0<=k<=n}, where T is defined as in A080098. - Reinhard Zumkeller, Jan 28 2003

To construct the sequence, start with 1 and use the rule: If k>=0 and a(0),a(1),...,a(2^k-1) are the 2^k first terms, then the next 2^k terms are 2*a(0),2*a(1),...,2*a(2^k-1). - Benoit Cloitre, Jan 30 2003

Also, numerator((2^k)/k!). - Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 03 2004

The odd entries in Pascal's triangle form the Sierpinski Gasket (a fractal). - Amarnath Murthy, Nov 20 2004.

Row sums of Sierpinski’s Gasket A047999. - Johannes W. Meijer, Jun 05 2011

Fixed point of the morphism "1" -> "1,2", "2" -> "2,4", "4" -> "4,8", ..., "2^k" -> "2^k,2^(k+1)", ... starting with a(0) = 1; 1 -> 12 -> 1224 -> = 12242448 -> 122424482448488(16) -> . . . - Philippe Deléham, Jun 18 2005

a(n) = number of 1's of stage n of the one-dimensional cellular automaton with Rule 90. - Andras Erszegi (erszegi.andras(AT)chello.hu), Apr 01 2006

a[33..63]=A117973[1..31] - Stephen Crowley, Mar 21 2007

Or the number of solutions of the equation: A000120(x)+A000120(n-x)=A000120(n). - Vladimir Shevelev, Jul 19 2009

For positive n, a(n) equals the denominator of the permanent of the n X n matrix consisting entirely of (1/2)'s. - John M. Campbell, May 26, 2011

Companions to A001316 are A048896, A105321, A117973, A151930 and A191488. They all have the same structure. We observe that for all these sequences a((2*n+1)*2^p-1) = C(p)*A001316(n), p>=0. If C(p) = 2^p then a(n) = A001316(n), if C(p) = 1 then a(n) = A048896(n), if C(p) = 2^p+2 then a(n) = A105321(n+1), if C(p) = 2^(p+1) then a(n) = A117973(n), if C(p) = 2^p-2 then a(n) = (-1)*A151930(n) and if C(p) = 2^(p+1)+2 then a(n) = A191488(n). Furthermore for all a(2^p - 1) = C(p). - Johannes W. Meijer, Jun 05 2011

a(n) = number of zeros in n-th row of A219463 = number of ones in n-th row of A047999. - Reinhard Zumkeller, Nov 30 2012

a(n) = A226078(n,1). - Reinhard Zumkeller, May 25 2013

This is the Run Length Transform of S(n) = {1,2,4,8,16,...} (cf. A000079}. The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product). - N. J. A. Sloane, Sep 05 2014

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 75ff.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.

H. W. Gould, Exponential Binomial Coefficient Series. Tech. Rep. 4, Math. Dept., West Virginia Univ., Morgantown, WV, Sep 1961.

Sam Northshield, "Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,...", Amer. Math. Month., Vol. 117 (7), pp. 581-598, 2010.

D. G. Poole, The towers and triangles of Professor Claus (or, Pascal knows Hanoi), Math. Mag., 67 (1994), 323-344.

M. R. Schroeder, Fractals, Chaos, Power Laws, W.H. Freeman, NY, 1991, page 383.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Wuensche, Andrew. Exploring Discrete Dynamics. Luniver Press, 2011. See Fig. 2.3.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.

E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

Philippe Dumas, Diviser pour regner Comportement asymptotique (has many references)

S. R. Finch, Stolarsky-Harborth Constant

Michael Gilleland, Some Self-Similar Integer Sequences

T. Pisanski and T. W. Tucker, Growth in Repeated Truncations of Maps, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), suppl., 167-176.

N. J. A. Sloane, Illustration of first 20 generations of Rule 90

R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Index entries for sequences related to cellular automata

FORMULA

a(n) = 2^A000120(n).

a(0) = 1; for n>0, write n = 2^i + j where 0 <= j < 2^i; then a(n) = 2*a(j).

a(n) = 2a(n-1)/A006519(n) = A000079(n)*A049606(n)/A000142(n)

a(n) = A038573(n) + 1

G.f.: Prod_{k>=0} (1+2*z^(2^k)). - Ralf Stephan, Apr 06 2003.

a(n)=sum(i=0, 2*n, (binomial(2*n, i) (mod 2))*(-1)^i) - Benoit Cloitre, Nov 16 2003

a(n) {mod 3}=A001285(n) - Benoit Cloitre, May 09 2004

2^n-2*sum{k=0..n, floor(C(n, k)/2)} - Paul Barry, Dec 24 2004

a(n)=product{k=0..log_2(n), 2^b(n, k)}, b(n, k)=coefficient of 2^k in binary expansion of n. Formula from Paul D. Hanna.

Sum_{k<n} a(k) = A006046(n).

a(n)=(n/2)+(1/2)+(sum(-(-1)^binomial(n,k),k=0..n)/2) - Stephen Crowley, Mar 21 2007

G.f.: (1/2)*z^(1/2)*sinh(2*z^(1/2)) - Johannes W. Meijer, Feb 20 2009

Equals infinite convolution product of [1,2,0,0,0,0,0,0,0] aerated A000079 - 1 times, i.e. [1,2,0,0,0,0,0,0,0] * [1,0,2,0,0,0,0,0,0] * [1,0,0,0,2,0,0,0,0]. - Mats Granvik, Gary W. Adamson, Oct 02 2009

a(n) = f(n, 1) with f(x, y) = if x = 0 then y else f(floor(x/2), y*(1 + x mod 2)). - Reinhard Zumkeller, Nov 21 2009

a(n) = 2 ^ (number of 1's in binary form of (n-1)) - Gabriel C. Benamy, Dec 08 2009

a((2*n+1)*2^p-1) = (2^p)*a(n), p>=0. - Johannes W. Meijer, Jun 05 2011

a(n) = A000120(A001317(n)). - Reinhard Zumkeller, Nov 24 2012

EXAMPLE

Has a natural structure as a triangle:

.1,

.2,

.2,4,

.2,4,4,8,

.2,4,4,8,4,8,8,16,

.2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,

.2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32,64,

....

The rows converge to A117973.

Contribution from Omar E. Pol, Jun 07 2009: (Start)

Also, triangle begins:

.1;

.2,2;

.4,2,4,4;

.8,2,4,4,8,4,8,8;

16,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16;

32,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,32;

64,2,4,4,8,4,8,8,16,4,8,8,16,8,16,16,32,4,8,8,16,8,16,16,32,8,16,16,32,16,32,...

(End)

MAPLE

A001316 := proc(n) local k; add(binomial(n, k) mod 2, k=0..n); end;

S:=[1]; S:=[op(S), op(2*s)]; # repeat ad infinitum!

a := n -> 2^add(i, i=convert(n, base, 2)); # Peter Luschny, Mar 11 2009

MATHEMATICA

Table[ Sum[ Mod[ Binomial[n, k], 2], {k, 0, n} ], {n, 0, 100} ]

Nest[ Join[#, 2#] &, {1}, 7] (* Robert G. Wilson v, Jan 24 2006 and modified Jul 27 2014 *)

Map[Function[Apply[Plus, Flatten[ #1]]], CellularAutomaton[90, {{1}, 0}, 100]] (* Produces counts of ON cells. N. J. A. Sloane, Aug 10 2009 *)

ArrayPlot[CellularAutomaton[90, {{1}, 0}, 20]] (* Illustration of first 20 generations. - N. J. A. Sloane, Aug 14 2014 *)

Table[2^(RealDigits[n - 1, 2][[1]] // Total), {n, 1, 100}] (* Gabriel C. Benamy, Dec 08 2009 *)

CoefficientList[Series[Exp[2*x], {x, 0, 100}], x] // Numerator (* Jean-François Alcover, Oct 25 2013 *)

PROG

(PARI) a(n)=if(n<0, 0, numerator(2^n/n!))

(PARI) A001316(n)=1<<norml2(binary(n)) \\ M. F. Hasler, May 03 2009

(PARI) a(n)=2^hammingweight(n) \\ Charles R Greathouse IV, Jan 04 2013

(Haskell)

a001316 = sum . a047999_row  -- Reinhard Zumkeller, Nov 24 2012

a001316_list = 1 : gould [1] where

   gould (x:xs) = zs ++ gould zs where

                  zs = (2*x) : xs ++ map (* 2) (x:xs)

-- Reinhard Zumkeller, Sep 16 2011

(Sage)

def A001316(n):

    if n <= 1: return Integer(n+1)

    return A001316(n//2) << n%2

[A001316(n) for n in range(88)]  # Peter Luschny, Nov 19 2012

CROSSREFS

Equals left border of triangle A166548. - Gary W. Adamson, Oct 16 2009

For generating functions Prod_{k>=0} (1+a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.

For partial sums see A006046. For first differences see A151930.

This is the numerator of 2^n/n!, while A049606 gives the denominator.

Cf. A051638, A048967, A007318, A094959, A048896, A117973, A008977, A139541, A048883, A102376.

Cf. A156769 = Gould's sequence appears in the numerators. - Johannes W. Meijer, Feb 20 2009

Cf. A038573, A159913, A000079, A166548, A000079.

Cf. A006047, A089898

Sequence in context: A094269 A157227 A054536 * A161831 A096865 A116466

Adjacent sequences:  A001313 A001314 A001315 * A001317 A001318 A001319

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Additional comments from Henry Bottomley, Mar 12 2001

Additional comments from N. J. A. Sloane, May 30 2009

STATUS

approved

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Last modified October 24 05:24 EDT 2014. Contains 248500 sequences.