A296498 - OEIS

%I #7 Jan 04 2018 21:23:33

%S 5,4,1,6,4,3,7,4,3,0,1,3,8,4,8,6,3,1,3,8,4,8,4,9,4,9,5,0,8,8,0,8,8,8,

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

%U 1,1,5,9,4,1,6,6,6,0,3,7,2,9,1,4,5,2

%N Decimal expansion of limiting power-ratio for A294541; see Comments.

%C 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.

%C For A = A294541, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%e limiting power-ratio = 5.416437430138486313848494950880888008486...

%t a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;

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

%t j = 1; While[j < 13, k = a[j] - j - 1;

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

%t Table[a[n], {n, 0, k}]; (* A294541 *)

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

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

%t Take[RealDigits[Last[h], 10][[1]], 120]   (* A296498 *)

%Y Cf. A001622, A294541, A296284, A296497.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Dec 20 2017