A296487 - OEIS

%I #7 Jul 25 2021 20:57:36

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

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

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

%N Decimal expansion of ratio-sum for A293076; 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 = A293076, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%e ratio-sum = 3.092262285773106330183534655202716162425...

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

%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2];

%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}]; (* A293076 *)

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

%t Take[RealDigits[s, 10][[1]], 100]  (* A296487 *)

%Y Cf. A001622, A293076, A296284, A296488.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Dec 19 2017