A176205 - OEIS

%I #14 Feb 16 2022 23:22:52

%S 1,3,8,9,23,24,49,27,68,69,139,72,169,147,272,81,203,204,409,207,484,

%T 417,767,216,529,507,992,441,1007,816,1441,243,608,609,1219,612,1429,

%U 1227,2252,621,1519,1452,2837,1251,2852,2301,4051,648,1609,1587,3152,1521,3527,2976,5401,1323,3212,3021,5851,2448

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

%C Has an apparent fractal structure with the following properties:

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

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

%C 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.

%H G. C. Greubel, <a href="/A176205/b176205.txt">Table of n, a(n) for n = 0..1000</a>

%F 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.

%F 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

%p a:= proc(n) option remember;

%p       if n=0 then 1

%p     elif `mod`(n,2)=0 then a(n/2) + 5*a(n/2 -1)

%p     else 3*a((n-1)/2)

%p       fi; end:

%p seq( a(n), n=0..60); # _G. C. Greubel_, Mar 13 2020

%t 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 *)

%o (Sage)

%o @CachedFunction

%o def a(n):

%o     if (n==0): return 1

%o     elif (n%2==0): return a(n/2) + 5*a(n/2 -1)

%o     else: return 3*a((n-1)/2)

%o [a(n) for n in (0..60)] # _G. C. Greubel_, Mar 13 2020

%K nonn

%O 0,2

%A _Gary W. Adamson_, Apr 11 2010

%E Terms a(25) onward added by _G. C. Greubel_, Mar 13 2020