%I #7 Jul 25 2021 20:58:27

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

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

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

%N Decimal expansion of ratio-sum for A294557; 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 = A294557, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%e 2.574408888082831272283741429619184772661...

%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 - 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}]; (* A294557 *)

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

%t Take[RealDigits[s, 10][[1]], 100] (* A296567 *)

%Y Cf. A001622, A294557, A296469, A296568.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Dec 20 2017