A298523 -id:A298523 - 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 *)

A298522 Decimal expansion of lim_ {n->oo} ((n + 1)*g - s(0) - s(1) - ... - s(n)), where g = 1.86676039917386..., s(n) = (s(n - 1) + (1+sqrt(5))/2)^(1/2), s(0) = 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 12 2018

(lim_ {n->oo} s(n)) = g = real zero of x^2 - x - (1+sqrt(5))/2. See A298512 for a guide to related sequences.

			((n + 1)*g - s(0) - s(1) - ... - s(n)) -> 1.20829379722676782518591549956095278...

Cf. A298512, A298523.

s[0] = 1; d = GoldenRatio; p = 1/2;
g = (x /. NSolve[x^(1/p) - x - d == 0, x, 200])[[2]]
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]] (* A298522 *)

A298738 Decimal expansion of (1/2)(1 + sqrt(7 + 2*sqrt(5))).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Feb 13 2018

			constant = 2.1935270853310539386... = positive zero of x^2 - x - r^2, where r = golden ratio = (1+ sqrt(5))/2; see A001622.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A001622, A190157, A298522, A298523, A275828.

r = (1/2)*(1 + Sqrt[7 + 2*Sqrt[5]])
RealDigits[N[r, 100], 10][[1]];  (* A298738 *)
RealDigits[(1+Sqrt[7+2*Sqrt[5]])/2,10,120][[1]] (* Harvey P. Dale, Jun 13 2025 *)

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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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A298522 Decimal expansion of lim_ {n->oo} ((n + 1)*g - s(0) - s(1) - ... - s(n)), where g = 1.86676039917386..., s(n) = (s(n - 1) + (1+sqrt(5))/2)^(1/2), s(0) = 1.

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A298738 Decimal expansion of (1/2)(1 + sqrt(7 + 2*sqrt(5))).

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