|
%I #7 Jul 25 2021 20:58:15
%S 6,3,5,8,6,8,5,3,1,1,1,1,1,1,9,2,3,2,1,0,0,9,7,0,1,9,7,0,0,5,7,1,5,6,
%T 4,8,3,9,3,9,4,1,4,1,1,9,4,3,6,9,0,9,1,9,3,2,6,3,4,6,3,6,9,2,2,9,5,5,
%U 1,6,1,6,1,2,7,7,9,4,6,7,8,2,7,3,1,8
%N Decimal expansion of ratio-sum for A294552; 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 = A294552, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e 6.358685311111192321009701970057156483939...
%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}]; (* A294552 *)
%t g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t Take[RealDigits[s, 10][[1]], 100] (* A296501 *)
%Y Cf. A001622, A294552, A296469.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Dec 20 2017
|
