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A296845 Decimal expansion of limiting power-ratio for A296843; see Comments.

%I A296845 #4 Jan 12 2018 23:43:45
%S A296845 6,8,5,7,7,3,9,7,3,7,0,4,2,3,0,5,2,4,8,6,4,0,8,0,9,7,8,2,0,2,2,1,3,6,
%T A296845 6,2,8,4,1,1,8,6,5,7,9,6,7,3,7,9,4,5,7,5,9,2,7,6,6,9,4,2,2,7,4,3,0,3,
%U A296845 4,8,8,2,6,9,2,0,1,5,5,0,4,7,6,4,5,3
%N A296845 Decimal expansion of limiting power-ratio for A296843; see Comments.
%C A296845 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 = A296843, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296845 limiting power-ratio = 6.136385518666220790955343949152636124460...
%t A296845 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; b[3] = 6;
%t A296845 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n + 1];
%t A296845 j = 1; While[j < 16, k = a[j] - j - 1;
%t A296845  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296845 Table[a[n], {n, 0, k}];  (* A296843 *)
%t A296845 z = 1700; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296845 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296845 Take[RealDigits[Last[h], 10][[1]], 120] (* A296845 *)
%Y A296845 Cf. A001622, A296469, A296843, A296844.
%K A296845 nonn,easy,cons
%O A296845 1,1
%A A296845 _Clark Kimberling_, Jan 12 2018