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A131205 a(n) = a(n-1) + a(floor(n/2)) + a(ceiling(n/2)). 4
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 (list; graph; refs; listen; history; text; internal format)
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->inf.} 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

FORMULA

Partial sums of A000123: (1, 2, 4, 6, 10, 14, 20, 26,...). - 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 + ...). - 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

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. A033485, A000123, A008619.

Sequence in context: A075321 A258030 A164787 * A256309 A058682 A081995

Adjacent sequences:  A131202 A131203 A131204 * A131206 A131207 A131208

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Oct 22 2007

STATUS

approved

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Last modified September 26 05:00 EDT 2020. Contains 337346 sequences. (Running on oeis4.)