%I #7 Jan 04 2018 21:23:39

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

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

%U 4,4,6,9,0,6,2,9,1,5,1,4,8,0,9,6,3,4

%N Decimal expansion of limiting power-ratio for A294546; 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. For A = A294546, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%e limiting power-ratio = 7.920208071319294432128690529988578214476...

%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] + n;

%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}]; (* A294546 *)

%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] (* A296500 *)

%Y Cf. A001622, A294546, A296284, A296499.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Dec 20 2017