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
Examples
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.
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.
Crossrefs
Programs
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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 *) -
PARI
a(n)=if(n==1 || n==0,1,if(n%2==0,a(n/2),2*a(n-1)+a(n-2)))
Comments