A129095 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.

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, 199, 11, 221, 51, 323, 1, 325, 53, 431, 13, 457, 79, 615, 5, 625, 89, 803, 23, 849, 135, 1119, 3, 1125, 141, 1407, 29, 1465, 199, 1863, 11, 1885, 221, 2327, 51, 2429, 323

Offset

1,3

Comments

Bisection A129096 is monotonically increasing.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1024

Cristina Ballantine and George Beck, Partitions enumerated by self-similar sequences, arXiv:2303.11493 [math.CO], 2023.

William J. Keith and Augustine O. Munagi, Binary compositions and semi-Pell compositions, arXiv:1912.11148 [math.CO], 2019.

William J. Keith and Augustine O. Munagi, Power compositions and semi-Pell compositions, Univ. Rochester, Online J. Analytic Comb. (2023) Issue 18, Art No. 2. See p. 2.

Example

Terms may be arranged into an irregular-shaped triangle:
  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, 199, 11, 221, 51, 323;
  ...
where final terms of rows form A129097,
central terms are given by A129098,
and row sums are equal to A129099.

Mathematica

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

Prog

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

Crossrefs

Cf. A129096, A129097, A129098, A129099.

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

Sequence in context: A212641 A195835 A077881 * A105604 A117576 A112447

Adjacent sequences: A129092 A129093 A129094 * A129096 A129097 A129098

Keyword

easy,nonn

Author

Paul D. Hanna, Apr 11 2007

Status

approved