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%I #40 Aug 11 2021 08:49:29
%S 1,3,7,13,23,37,57,83,119,165,225,299,393,507,647,813,1015,1253,1537,
%T 1867,2257,2707,3231,3829,4521,5307,6207,7221,8375,9669,11129,12755,
%U 14583,16613,18881,21387,24177,27251,30655,34389,38513,43027,47991
%N a(n) = a(n-1) + a(floor(n/2)) + a(ceiling(n/2)).
%C From _Gary W. Adamson_, Dec 16 2009: (Start)
%C Let M = an infinite lower triangular matrix with (1, 3, 4, 4, 4, ...) in every column shifted down twice, with the rest zeros:
%C 1;
%C 3, 0;
%C 4, 1, 0;
%C 4, 3, 0, 0;
%C 4, 4, 1, 0, 0;
%C 4, 4, 3, 0, 0, 0;
%C ...
%C A131205 = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. (End)
%C 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
%H T. D. Noe, <a href="/A131205/b131205.txt">Table of n, a(n) for n = 1..1000</a>
%F Partial sums of A000123. - _Gary W. Adamson_, Oct 26 2007
%F 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
%F G.f.: (x/(1 - x))*Product_{k>=0} (1 + x^(2^k))/(1 - x^(2^k)). - _Ilya Gutkovskiy_, Jun 05 2017
%F a(n) = A033485(2n-1). - _Jean-Paul Allouche_, Aug 11 2021
%p A[1]:= 1:
%p for n from 2 to 100 do A[n]:= A[n-1] + A[floor(n/2)] + A[ceil(n/2)] od:
%p seq(A[n],n=1..100); # _Robert Israel_, Sep 06 2016
%t Nest[Append[#1, #1[[-1]] + #1[[Floor@ #3]] + #[[Ceiling@ #3]] ] & @@ {#1, #2, #2/2} & @@ {#, Length@ # + 1} &, {1}, 42] (* _Michael De Vlieger_, Jan 16 2020 *)
%o (Haskell)
%o a131205 n = a131205_list !! (n-1)
%o a131205_list = scanl1 (+) a000123_list -- _Reinhard Zumkeller_, Oct 10 2013
%Y Cf. A000123, A008619. Bisection of A033485.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Oct 22 2007
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