A176205 Coefficients of S(x) which satisfies S(x) = (1 + 3*x + 5*x^2)*S(x^2).

1, 3, 8, 9, 23, 24, 49, 27, 68, 69, 139, 72, 169, 147, 272, 81, 203, 204, 409, 207, 484, 417, 767, 216, 529, 507, 992, 441, 1007, 816, 1441, 243, 608, 609, 1219, 612, 1429, 1227, 2252, 621, 1519, 1452, 2837, 1251, 2852, 2301, 4051, 648, 1609, 1587, 3152, 1521, 3527, 2976, 5401, 1323, 3212, 3021, 5851, 2448

Offset

0,2

Comments

Has an apparent fractal structure with the following properties:

a(n) for n = 0, 1, 3, 7, 15, ... are 1, 3, 9, 27, ....

Odd n-th terms are divisible by 3 (starting with n=1) creating the same sequence.

Then the result is relabeled with n=0,1,2,...; with the odds again divisible by 3, getting (1, 3, 8, 9, 23, ...); and so on.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

Given M = an infinite triangular matrix with (1, 3, 5, ...) in each column; shifted down twice for columns > 0. Then A176205 = lim_{n->infinity} M^n, the left shifted vector considered as a sequence.

a(2*n) = a(n) + 5*a(n-1) and a(2*n+1) = 3*a(n), with a(0) = 1. - G. C. Greubel, Mar 13 2020

Maple

a:= proc(n) option remember;
      if n=0 then 1
    elif `mod`(n, 2)=0 then a(n/2) + 5*a(n/2 -1)
    else 3*a((n-1)/2)
      fi; end:
seq( a(n), n=0..60); # G. C. Greubel, Mar 13 2020

Mathematica

a[n_]:= If[n==0, 1, If[EvenQ[n], a[n/2] +5*a[n/2 -1], 3*a[(n-1)/2]]]; Table[a[n], {n, 0, 60}] (* G. C. Greubel, Mar 13 2020 *)

Prog

(Sage)
@CachedFunction
def a(n):
    if (n==0): return 1
    elif (n%2==0): return a(n/2) + 5*a(n/2 -1)
    else: return 3*a((n-1)/2)
[a(n) for n in (0..60)] # G. C. Greubel, Mar 13 2020

Crossrefs

Sequence in context: A295289 A212849 A191487 * A099256 A167344 A025615

Adjacent sequences: A176202 A176203 A176204 * A176206 A176207 A176208

Keyword

nonn

Author

Gary W. Adamson, Apr 11 2010

Extensions

Terms a(25) onward added by G. C. Greubel, Mar 13 2020

Status

approved