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URL: https://oeis.org/A131205

⇱ A131205 - OEIS

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A131205
a(n) = a(n-1) + a(floor(n/2)) + a(ceiling(n/2)).
7
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
1,2
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
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.
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
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 *)
PROG
(Haskell)
a131205 n = a131205_list !! (n-1)
a131205_list = scanl1 (+) a000123_list -- Reinhard Zumkeller, Oct 10 2013
CROSSREFS
Cf. A000123, A008619. Bisection of A033485.
Sequence in context: A075321 A258030 A164787 * A256309 A058682 A081995
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Oct 22 2007
STATUS
approved