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%I #4 Apr 17 2018 18:32:15
%S 6,1,3,6,3,8,5,5,1,8,7,0,7,8,2,2,2,2,1,7,6,8,4,1,2,7,9,0,4,4,0,0,8,9,
%T 9,6,0,0,3,7,0,4,0,0,3,1,5,5,9,4,5,8,4,1,4,2,0,8,2,2,8,2,1,3,1,8,0,0,
%U 7,5,4,2,2,5,0,2,5,0,8,5,9,0,3,1,0,1
%N Decimal expansion of limiting power-ratio for A294553; 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 = A294553, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e limiting power-ratio = 6.136385518707822221768412790440089960037...
%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - 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}]; (* A294553 *)
%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] (* A296504 *)
%Y Cf. A001622, A294553, A296469, A296503.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Apr 14 2018
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