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A004737 Concatenation of sequences (1,2,..,n-1,n,n-1,..,1) for n >= 1. 27
1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Also called the Smarandache Crescendo Pyramidal sequence.

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003

a(A002061(n)) = n; a(A000290(n)) = a(A002522(n)) = 1. - Reinhard Zumkeller, Mar 10 2006

The ordinal transform of a sequence b_0, b_1, b_2, ... is the sequence a_0, a_1, a_2, ... where a_n is the number of times b_n has occurred in {b_0 ... b_n}.

From Artur Jasinski, Mar 07 2010: (Start)

Zeta[2,k/p]+Zeta[2,(p-k)/p]=(Pi/Sin[(Pi*a(n))/p])2, where p=2,3,4, k=1..p-1.

This sequence is even subset of A003983 for odd p=2,4,6,8,....

For odd subset of A003983 see A004739. (End)

From Gary W. Adamson, Mar 30 2010: (Start)

Given the triangle rows: (1; 1,2,1; 1,2,3,2,1;...) as polcoeff with offset 0:

q = (1 + 2x + x^2), r = (1 + 2x + 3x^2 + 2x^3 +x^4),...etc; then

(1 + 2x + 3x^2 + ...) = q(x) * q(x^2) * q(x^4) * q(^8) * ...

..................... = r(x) * r(x^3) * r(x^9) * r(x^27) * ...

..................... = s(x) * x(x^4) * s(x^16)* s(x^64) * ...

... (End)

From L. Edson Jeffery, Jan 13 2012: (Start)

Let U_1(t)=1, U_2(t)=2*t, and U_r(t)=2*t*U_(r-1)(t)-U(r-2)(t), r>2, be Chebyshev polynomials of the second kind. For q>1 an integer, let N=2*q and x_k=cos((2*k-1)*Pi/N), and define the ordered column vectors V_k=[U_k(x_1), U_k(x_2), ..., U_k(x_q)]^T, k=1,...,q, where A^T denotes the transpose of matrix A. Let E_N=[V_1, V_2, ..., V_q] be the q X q matrix formed from the ordered components of the V_k. E_N contains the joint spectra of the Danzer basis (see [Jeffery]) associated with N. Let M_N=(1/q)*[E_N]^T*E_N. For the trivial case q=1, let M_2=[1]. CONJECTURE: E_N and M_N are always integral and symmetric, with M_N having diagonal entries {1,2,...} beginning at entries 1,j (j odd) in the first row and i,1 (i odd) in the first column and with zeros elsewhere. If N is allowed to increase without bound, and assuming the conjecture is true, then triangle A004737 emerges in its entirety from the successive antidiagonals containing those entries [M_N]_(i,j) such that i+j=2*v, for each v in {1,2,...,floor((q+1)/2)}. For example, for N=18 and q=9 (omitting the zeros for clarity),

M_18=[

(1   1   1   1   1);

(  2   2   2   2  );

(1   3   3   3   3);

(  2   4   4   4  );

(1   3   5   5   5);

(  2   4   6   6  );

(1   3   5   7   7);

(  2   4   6   8  );

(1   3   5   7   9)],

from which the first five rows of the sequence can be read off in succession. (End)

T(n,k) =  min(n,k). The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

Expanded form of T(2,k) k=0,1,...2m for ascending m-nomial triangles. - Bob Selcoe, Feb 07 2014

Terms in the first nine rows of the triangle can be duplicated by performing (111...)^2 with <= nine ones. By way of example, (11111)^2 = 123454321. - Gary W. Adamson, Mar 27 2015

REFERENCES

Miklós Laczkovich, Conjecture and Proof, TypoTex, Budapest, 1998. See Chapter 10.

F. Smarandache, "Numerical Sequences", University of Craiova, 1975.

LINKS

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

Jerry Brown et al., Problem 4619, School Science and Mathematics, USA, Vol. 97 (4), 1997, pp. 221-222.

L. E. Jeffery, Danzer matrices (definitions)

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

F. Smarandache, Collected Papers, Vol. II, Tempus Publ. Hse., Bucharest, 1996.

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems, Hexis, Phoenix, 2006.

Eric Weisstein's World of Mathematics, Smarandache Sequences.

FORMULA

a(n) = if n<3 then 1 else (if a(n-1)=1 then 1 + 0^(a(n-2)-1) else a(n-1) - 0^X + (a(n-1)-a(n-2))*(1 - 0^X)), where X = A003059(n-1)-a(n-1). - Reinhard Zumkeller, Mar 10 2006

Let b(n) = floor(sqrt(n-1)). Then a(n) = min(n - b(n)^2, (b(n)+1)^2 - n + 1). - Franklin T. Adams-Watters, Jun 09 2006

Ordinal transform of A004741. - Franklin T. Adams-Watters, Aug 28 2006

If the sequence is read as a triangular array, beginning [1]; [1,2,1]; [1,2,3,2,1]; ..., then the o.g.f. is (1+qx)/((1-x)(1-qx)(1-q^2x)) = 1 + x(1 + 2q + q^2) + x^2(1 + 2q + 3q^2 + 2q^3 +q^4) + .... The row polynomials for this triangle are (1 + q + ... + q^n)^2 =[n,2]_q + q[n-1,2]_q, where [n,2]_q are Gaussian polynomials (see A008967). - Peter Bala, Sep 23 2007

a(n+1)=a(n)+{floor[sqrt(n)]-floor[sqrt(n-1)]-1}*(-1)^{floor[(n-1)/floor[(1/2)*(1+sqrt(4*n-3))]]-floor[(1/2)*(1+sqrt(4*n-3))]+1}, with a(1)=1. - Paolo P. Lava, Nov 13 2008

See Mathematica code. - Artur Jasinski, Mar 07 2010

a(n) = floor(sqrt(n-1)) - |n - floor(sqrt(n-1))^2 - floor(sqrt(n-1)) - 1| + 1. - Boris Putievskiy, Jan 13 2013

Read as a triangular array, then T(n,k) = n-|n-k-1|; T(n,0) = 1; T(n,n-1) = n. - Juan Pablo Herrera P., Oct 17 2016

EXAMPLE

From Boris Putievskiy, Jan 13 2013: (Start)

The start of the sequence as table:

1..1..1..1..1..1...

1..2..2..2..2..2...

1..2..3..3..3..3...

1..2..3..4..4..4...

1..2..3..4..5..5...

1..2..3..4..5..6...

. . .

The start of the sequence as an irregular triangle array read by rows:

1;

1,2,1;

1,2,3,2,1;

1,2,3,4,3,2,1;

1,2,3,4,5,4,3,2,1;

1,2,3,4,5,6,5,4,3,2,1;

. . .

Row number k contains 2*k-1 numbers: 1,2,..,k-1,k,k-1,..,1. (End)

The sequence of fractions A196199/A004737 = 0/1, -1/1, 0/2, 1/1, -2/1, -1/2, 0/3, 1/2, 2/1, -3/1, -2/2, -1/3, 0/4, 1/3, 2/2, 3/1, -4/4. -3/2, ... contains every rational number (infinitely ofter) [Laczkovich]. - N. J. A. Sloane, Oct 09 2013

MAPLE

P:=proc(i) local a, b, c, n; a:=1; print(a); for n from 1 by 1 to i do a:=a+(floor(sqrt(n))-floor(sqrt(n-1))-1)*(-1)^(floor((n-1)/floor(1/2*(1+sqrt(4*n-3))))-floor(1/2*(1+sqrt(4*n-3)))+1); print(a); od; end: P(100); # Paolo P. Lava, Nov 13 2008

MATHEMATICA

aa = {}; Do[Do[AppendTo[aa, (p/Pi) ArcSin[Sqrt[1/((1/Pi2) (Zeta[2, k/p] + Zeta[2, (p - k)/p]))]]], {k, 1, p - 1}], {p, 2, 50, 2}]; Round[N[aa, 50]] (* Artur Jasinski, Mar 07 2010 *)

Table[Min[n - #^2, (# + 1)^2 - n + 1] &@ Floor[Sqrt[n - 1]], {n, 105}] (* or *)

Table[Floor@ # - Abs[n - Floor[#]^2 - Floor@ # - 1] + 1 &@ Sqrt[n - 1], {n, 105}] (* Michael De Vlieger, Oct 21 2016 *)

PROG

(Haskell)

import Data.List (inits)

a004737 n = a004737_list !! (n-1)

a004737_list = concatMap f $ tail $ inits [1..]

   where f xs = xs ++ tail (reverse xs)

-- Reinhard Zumkeller, May 11 2014, Mar 26 2011

(PARI) a(n) = n--; my(m=sqrtint(n)); m+1-abs(n-m^2-m) \\ David A. Corneth, Oct 18 2016

CROSSREFS

Cf. A004738, A008967, A003983, A196199.

Cf. A242357.

Sequence in context: A234503 A236325 A080345 * A255616 A014600 A165475

Adjacent sequences:  A004734 A004735 A004736 * A004738 A004739 A004740

KEYWORD

nonn,frac,easy,tabf

AUTHOR

R. Muller

EXTENSIONS

More terms from Patrick De Geest, Jun 15 1998

STATUS

approved

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Last modified March 29 06:34 EDT 2017. Contains 284250 sequences.