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A296570 Decimal expansion of limiting power-ratio for A294558; see Comments.

%I A296570 #4 Jan 07 2018 21:38:03
%S A296570 4,0,9,0,3,7,2,6,6,9,7,6,7,6,0,7,8,3,9,0,9,7,9,0,9,0,1,2,7,4,5,9,5,6,
%T A296570 5,0,1,9,4,2,5,4,5,6,6,2,3,2,1,8,6,8,0,7,8,1,2,1,0,8,9,7,5,6,5,4,9,6,
%U A296570 7,0,3,7,7,2,7,0,0,1,0,2,0,2,7,8,2,0
%N A296570 Decimal expansion of limiting power-ratio for A294558; see Comments.
%C A296570 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 = A294558, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296570 limiting power-ratio = 4.090372669767607839097909012745956501942...
%t A296570 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A296570 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - n;
%t A296570 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296570  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296570 Table[a[n], {n, 0, k}]; (* A294558 *)
%t A296570 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296570 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296570 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296570 *)
%Y A296570 Cf. A001622, A294558, A296469, A296569.
%K A296570 nonn,easy,cons
%O A296570 1,1
%A A296570 _Clark Kimberling_, Jan 07 2018