A296471 - OEIS

%I #7 Jul 25 2021 22:19:28

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

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

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

%N Decimal expansion of ratio-sum for A295947; 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 = A295947, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%e ratio-sum = 2.427179488056039424423653103831452251757...

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

%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[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}]; (* A295947 *)

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

%t Take[RealDigits[s, 10][[1]], 100]  (* A296471 *)

%Y Cf. A001622, A295947, A296284, A296472.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Dec 18 2017