A296479 - OEIS

%I #7 Jul 25 2021 20:53:35

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

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

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

%N Decimal expansion of ratio-sum for A295951; 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 = A295951, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

%e ratio-sum = 2.571971369307578245340672489672087659876...

%t a[0] = 2; a[1] = 3; b[0] = 1; b[1 ] = 4; 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}]; (* A295951 *)

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

%t Take[RealDigits[s, 10][[1]], 100]  (* A296479 *)

%Y Cf. A001622, A295951, A296284, A296480.

%K nonn,easy,cons

%O 1,1

%A _Clark Kimberling_, Jan 05 2018