A296483 - OEIS

%I #9 Feb 09 2018 17:36:00

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

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

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

%N Decimal expansion of ratio-sum for A295953; 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 = A295953, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%e ratio-sum = 4.194678679056371758991840818123954420964...

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

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

%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}]; (* A295953 *)

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

%t Take[RealDigits[s, 10][[1]], 100]  (* A296483 *)

%Y Cf. A001622, A295953, A296284, A296848.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Jan 06 2018