A296462 -id:A296462 - cp's OEIS frontend

A298171 Decimal expansion of ratio-sum for A296776; see Comments.

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

Offset: 1

Clark Kimberling, Feb 09 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 = A298171, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296462 for related sequences.

			ratio-sum = 5.611562057656225477032443456092579480982...

Cf. A001622, A296462, A296776.

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 2 n;
j = 1; While[j < 16, 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}];  (* A296776 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A298171 *)

A298172 Decimal expansion of limiting power-ratio for A296776; see Comments.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Feb 09 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 = A296776, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296462 for related sequences.

			limiting power-ratio = 11.65837748505646668170868260629394947392...

Cf. A001622, A296462, A296776.

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 2 n;
j = 1; While[j < 16, 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}];  (* A296776 *)
z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
Take[RealDigits[Last[h], 10][[1]], 120]   (* A298172 *)

A296463 Expansion of e.g.f. arcsinh(x)*arctanh(x) (even powers only).

Original entry on oeis.org

0, 2, 4, 158, 3624, 427482, 29665260, 6948032310, 991515848400, 383952670412850, 93532380775766100, 53913667654307868750, 20087427376748637675000, 16096655588343149442026250, 8531309209053208518037597500, 9057367559484733295974741323750, 6486329752640392315697926589700000

Offset: 0

Ilya Gutkovskiy, Dec 13 2017

nonn

			arcsinh(x)*arctanh(x) = 2*x^2/2! + 4*x^4/4! + 158*x^6/6! + 3624*x^8/8! + 427482*x^10/10! + ..

Cf. A001818, A009744, A009747, A010050, A012752, A296462.

nmax = 16; Table[(CoefficientList[Series[ArcSinh[x] ArcTanh[x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 16; Table[(CoefficientList[Series[(Log[1 + x] - Log[1 - x]) Log[x + Sqrt[1 + x^2]]/2, {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

E.g.f.: arcsin(x)*arctan(x) (even powers only, absolute values).
E.g.f.: (log(1 + x) - log(1 - x))*log(x + sqrt(1 + x^2))/2 (even powers only).
a(n) ~ (2*n-1)! * log(1+sqrt(2)) * (1 - (-1)^n * sqrt(Pi) / (4 * log(1+sqrt(2)) * sqrt(n))). - Vaclav Kotesovec, Dec 13 2017

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A298171 Decimal expansion of ratio-sum for A296776; see Comments.

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A298172 Decimal expansion of limiting power-ratio for A296776; see Comments.

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A296463 Expansion of e.g.f. arcsinh(x)*arctanh(x) (even powers only).

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