A296849 -id:A296849 - cp's OEIS frontend

A296850 Decimal expansion of ratio-sum for A296849; see Comments.

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

Offset: 1

Clark Kimberling, Jan 12 2018

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 = A296849, we have g = 1 + sqrt(2). See A296425..A296434 for related ratio-sums and A296452..A296461 for related limiting power-ratios.

			ratio-sum = 2.898430373605188679364554049974531941517...

Cf. A296849.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = 2*a[n - 1] + a[n - 2] + b[n];
j = 1; While[j < 8, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A296849 *)
r = 1 + Sqrt[2]; s = N[Sum[-r + a[n]/a[n - 1], {n, 1, 1000}], 200];
StringJoin[StringTake[ToString[s], 41], "..."]
Take[RealDigits[s, 10][[1]], 100] (* A296850 *)

A296851 Decimal expansion of limiting power-ratio for A296849; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 13 2018

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 limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A296849, we have g = 1+ sqrt(2). See the guide at A296469 for related sequences.

			limiting power-ratio = 2.283378414042995122799666600398678749089...

Cf. A296849, A296850.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + b[n];
j = 1; While[j < 8, k = a[j] - j - 1;
 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A296849 *)
z = 1700; r = 1 + Sqrt[2]; h = Table[N[a[n]/r^n, z], {n, 0, z}];
StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
Take[RealDigits[Last[h], 10][[1]], 120] (* A296851 *)

cp's OEIS Frontend

A296850 Decimal expansion of ratio-sum for A296849; see Comments.

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