A131205 a(n) = a(n-1) + a(floor(n/2)) + a(ceiling(n/2)).

1, 3, 7, 13, 23, 37, 57, 83, 119, 165, 225, 299, 393, 507, 647, 813, 1015, 1253, 1537, 1867, 2257, 2707, 3231, 3829, 4521, 5307, 6207, 7221, 8375, 9669, 11129, 12755, 14583, 16613, 18881, 21387, 24177, 27251, 30655, 34389, 38513, 43027, 47991

Offset: 1

Author

Reinhard Zumkeller, Oct 22 2007

Keywords

• nonn

Comments

From Gary W. Adamson, Dec 16 2009: (Start)

Let M = an infinite lower triangular matrix with (1, 3, 4, 4, 4, ...) in every column shifted down twice, with the rest zeros:

1;

3, 0;

4, 1, 0;

4, 3, 0, 0;

4, 4, 1, 0, 0;

4, 4, 3, 0, 0, 0;

...

A131205 = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. (End)

The subsequence of primes in this sequence begins with 5 in a row: 3, 7, 13, 23, 37, 83, 647, 1867, 2707, 88873, 388837, 655121, 754903, 928621, 1062443. - Jonathan Vos Post, Apr 25 2010

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Cristina Ballantine, George Beck, and Mircea Merca, Partitions and elementary symmetric polynomials -- an experimental approach, arXiv:2408.13346 [math.CO], 2024. See p. 13.
• Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, Elementary symmetric partitions, arXiv:2409.11268 [math.CO], 2024. See pp. 5, 7.

Crossrefs

Cf. A000123, A008619. Bisection of A033485.

Programs

Haskell

a131205 n = a131205_list !! (n-1)
a131205_list = scanl1 (+) a000123_list -- Reinhard Zumkeller, Oct 10 2013

Maple

A[1]:= 1:
for n from 2 to 100 do A[n]:= A[n-1] + A[floor(n/2)] + A[ceil(n/2)] od:
seq(A[n],n=1..100); # Robert Israel, Sep 06 2016

Mathematica

Nest[Append[#1, #1[[-1]] + #1[[Floor@ #3]] + #[[Ceiling@ #3]] ] & @@ {#1, #2, #2/2} & @@ {#, Length@ # + 1} &, {1}, 42] (* Michael De Vlieger, Jan 16 2020 *)

Formula

Partial sums of A000123. - Gary W. Adamson, Oct 26 2007

G.f.: r(x) * r(x^2) * r(x^4) * r(x^8) * ... where r(x) = (1 + 3x + 4x^2 + 4x^3 + 4x^4 + ...) is the g.f. of A113311. - Gary W. Adamson, Sep 01 2016

G.f.: (x/(1 - x))*Product_{k>=0} (1 + x^(2^k))/(1 - x^(2^k)). - Ilya Gutkovskiy, Jun 05 2017

a(n) = A033485(2n-1). - Jean-Paul Allouche, Aug 11 2021