A296470 - OEIS

A296470

Decimal expansion of limiting power-ratio for A295862; see Comments.

6, 5, 1, 4, 7, 1, 0, 0, 7, 5, 0, 5, 5, 2, 7, 2, 7, 7, 3, 3, 5, 8, 8, 6, 8, 5, 7, 5, 3, 4, 3, 9, 6, 9, 4, 2, 5, 3, 7, 3, 9, 6, 8, 8, 7, 6, 0, 9, 1, 3, 1, 7, 9, 5, 6, 8, 7, 8, 8, 5, 9, 6, 7, 9, 6, 3, 6, 5, 5, 1, 8, 0, 8, 3, 2, 6, 4, 8, 9, 8, 3, 8, 2, 4, 4, 0

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Suppose that A = (a(n)), for n >=0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A295862 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

LINKS

Table of n, a(n) for n=1..86.

FORMULA

limiting power-ratio = 6.514710075055272773358868575343969425373...

MATHEMATICA

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;

a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];

j = 1; While[j < 12, k = a[j] - j - 1;

While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

Table[a[n], {n, 0, 15}] (* A295862 *)

z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];

StringJoin[StringTake[ToString[h[[z]]], 41], "..."]