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A296493 Decimal expansion of ratio-sum for A296555; see Comments. 7

%I

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

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

%U 1,7,2,8,6,1,3,8,5,5,7,5,0,7,2,4,7,8

%N Decimal expansion of ratio-sum for A296555; 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 ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + . . . , assuming that this series converges. For A = A296555, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%e 5.204091649313251611130187115558413050194...

%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;

%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 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}]; (* A296555 *)

%t g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]

%t Take[RealDigits[s, 10][[1]], 100] (* A296493 *)

%Y Cf. A001622, A296284, A296494, A296555.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Dec 19 2017

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Last modified July 14 10:29 EDT 2020. Contains 335721 sequences. (Running on oeis4.)