A129095 - OEIS

%I #25 Sep 16 2024 12:52:00

%S 1,1,3,1,5,3,11,1,13,5,23,3,29,11,51,1,53,13,79,5,89,23,135,3,141,29,

%T 199,11,221,51,323,1,325,53,431,13,457,79,615,5,625,89,803,23,849,135,

%U 1119,3,1125,141,1407,29,1465,199,1863,11,1885,221,2327,51,2429,323

%N Semi-Pell numbers: a(n) = a(n/2) (n even), a(n) = 2*a(n-1) + a(n-2) (n odd >1), with a(1) = 1.

%C Bisection A129096 is monotonically increasing.

%H Paul D. Hanna, <a href="/A129095/b129095.txt">Table of n, a(n) for n = 1..1024</a>

%H Cristina Ballantine and George Beck, <a href="https://arxiv.org/abs/2303.11493">Partitions enumerated by self-similar sequences</a>, arXiv:2303.11493 [math.CO], 2023.

%H William J. Keith and Augustine O. Munagi, <a href="https://arxiv.org/abs/1912.11148">Binary compositions and semi-Pell compositions</a>, arXiv:1912.11148 [math.CO], 2019.

%H William J. Keith and Augustine O. Munagi, <a href="https://hosted.math.rochester.edu/ojac/vol18/276.pdf">Power compositions and semi-Pell compositions</a>, Univ. Rochester, Online J. Analytic Comb. (2023) Issue 18, Art No. 2. See p. 2.

%e Terms may be arranged into an irregular-shaped triangle:

%e   1;

%e   1, 3;

%e   1, 5, 3, 11;

%e   1, 13, 5, 23, 3, 29, 11, 51;

%e   1, 53, 13, 79, 5, 89, 23, 135, 3, 141, 29, 199, 11, 221, 51, 323;

%e   ...

%e where final terms of rows form A129097,

%e central terms are given by A129098,

%e and row sums are equal to A129099.

%t Nest[Append[#1, If[EvenQ[#2], #1[[#2/2]], 2 #1[[-1]] + #1[[-2]] ] ] & @@ {#, Length@ # + 1} &, {1}, 61] (* _Michael De Vlieger_, Mar 10 2020 *)

%o (PARI) a(n)=if(n==1 || n==0,1,if(n%2==0,a(n/2),2*a(n-1)+a(n-2)))

%Y Cf. A129096, A129097, A129098, A129099.

%Y Cf. A030067 (semi-Fibonacci), A074364 (semi-tribonacci).

%K easy,nonn

%O 1,3

%A _Paul D. Hanna_, Apr 11 2007