A296499 - OEIS

%I #7 Jul 25 2021 20:58:04

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

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

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

%N Decimal expansion of ratio-sum 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 ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A294546, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%e 4.737057998792217365428375621295759326159...

%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] + 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}]; (* A294549 *)

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

%t Take[RealDigits[s, 10][[1]], 100]  (* A296499 *)

%Y Cf. A001622, A294546, A296284, A296500.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Dec 20 2017