A298520 -id:A298520 - cp's OEIS frontend

A298512 Decimal expansion of lim_ {n->oo} ((n + 1)*g - s(0) - s(1) - ... - s(n)), where g = (1 + sqrt (5))/2, s(n) = (s(n - 1) + 1)^(1/2), s(0) = 1.

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

Offset: 0

Clark Kimberling, Feb 11 2018

Lim_{n->oo} s(n) = g = golden ratio, A001622.  In the following guide to related sequences, the sequence gives the decimal expansion for lim_{n->oo} |(n+1)*g - s(0) - s(1) - ... - s(n)|, where s(n) = (s(n-1) + d)^p, and tau = (1+sqrt(5))/2.
***
sequence   d            p       a(0)         g
A298512    1           1/2       1       (1+sqrt(5))/2
A298513    1           1/2       2       (1+sqrt(5))/2
A298514    1           1/2       3       (1+sqrt(5))/2
A298515    1/2         1/2       1       (1+sqrt(3))/2
A298516    2           1/2       1       2
A298517    3           1/2       1       (1+sqrt(13))/2
A298518    1           1/3       1       1.3247...
A298519    1           1/3       2       1.3247...
A298520    1           1/3       3       1.3247...
A298521    1           2/3       1       2.1478...
A298522    tau         1/2       1       1.8667...
A298523    tau         1/2       2       1.8667...
A298524    sqrt(2)     1/2       1       1.7900...
A298525    sqrt(2)     1/2       2       1.7900...
A298526    sqrt(3)     1/2       1       1.9078...
A298527    sqrt(3)     1/2       2       1.9078...
A298528    e           1/2       1       2.2228...
A298529    e           1/2       e       2.2228...
A298530    Pi          1/2       1       2.3416...
A298531    Pi          1/2       Pi      2.3416...
A298532    tau         1/2      tau      2.3416...

			s(n) = (1, 1.4142..., 1.5537..., 1.5980..., 1.6118..., ...) with limit g = 1.618... = (1+sqrt(5))/2.
((n + 1)*g - s(0) - s(1) - ... - s(n)) -> 0.9150498480151349148436312146030...

Cf. A001622, A298513, A298514.

s[0] = 1; d = 1; p = 1/2; s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
z = 200 ; g = GoldenRatio; s = N[(z + 1)*g - Sum[s[n], {n, 0, z}], 150 ];
RealDigits[s, 10][[1]];  (* A298512 *)

A298518 Decimal expansion of lim_ {n->oo} ((n + 1)*g - s(0) - s(1) - ... - s(n)), where g = 1.324717957..., s(n) = (s(n - 1) + 1)^(1/3), s(0) = 1.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Feb 11 2018

(lim_ {n->oo} s(n)) = g = real zero of x^3 - x - 1. See A298512 for a guide to related sequences.

			((n + 1)*g - s(0) - s(1) - ... - s(n)) -> 0.40485767742624961326290607445802...

Cf. A298512, A298519, A298520.

s[0] = 1; d = 1; p = 1/3;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[3]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[g - s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298518 *)

A298519 Decimal expansion of lim_ {n->oo} (s(0) + s(1) + ... + s(n) - (n+1)*g), where g = 1.324717957..., s(n) = (s(n - 1) + 1)^(1/3), s(0) = 2.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Feb 11 2018

(lim_ {n->oo} s(n)) = g = real zero of x^3 - x - 1. See A298512 for a guide to related sequences.

			s(0) + s(1) + ... + s(n) - (n+1)*g -> 0.819904707664104372564776035917499198052...

Cf. A298512, A298518, A298520.

s[0] = 2; d = 1; p = 1/3;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[3]]
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[-g + s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298519 *)

A298521 Decimal expansion of lim_ {n->oo} ((n + 1)*g - s(0) - s(1) - ... - s(n)), where g = 2.1478990357..., s(n) = (s(n - 1) + 1)^(2/3), s(0) = 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 12 2018

(lim_ {n->oo} s(n)) = g = real zero of x^(2/3) - x - 1. See A298512 for a guide to related sequences.

			((n + 1)*g - s(0) - s(1) - ... - s(n)) -> 2.19524431421173816419739210020867...

Cf. A298512, A298520.

s[0] = 1; d = 1; p = 2/3;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])
s[n_] := s[n] = (s[n - 1] + d)^p
N[Table[s[n], {n, 0, 30}]]
s = N[Sum[g - s[n], {n, 0, 200}], 150 ];
RealDigits[s, 10][[1]] (* A298521 *)

cp's OEIS Frontend

A298512 Decimal expansion of lim_ {n->oo} ((n + 1)*g - s(0) - s(1) - ... - s(n)), where g = (1 + sqrt (5))/2, s(n) = (s(n - 1) + 1)^(1/2), s(0) = 1.

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A298518 Decimal expansion of lim_ {n->oo} ((n + 1)*g - s(0) - s(1) - ... - s(n)), where g = 1.324717957..., s(n) = (s(n - 1) + 1)^(1/3), s(0) = 1.

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A298519 Decimal expansion of lim_ {n->oo} (s(0) + s(1) + ... + s(n) - (n+1)*g), where g = 1.324717957..., s(n) = (s(n - 1) + 1)^(1/3), s(0) = 2.

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A298521 Decimal expansion of lim_ {n->oo} ((n + 1)*g - s(0) - s(1) - ... - s(n)), where g = 2.1478990357..., s(n) = (s(n - 1) + 1)^(2/3), s(0) = 1.

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