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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. 21
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

(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...

LINKS

Table of n, a(n) for n=0..85.

EXAMPLE

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...

MATHEMATICA

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 *)

CROSSREFS

Cf. A001622, A298513, A298514.

Sequence in context: A178745 A114893 A089101 * A192930 A010168 A340004

Adjacent sequences:  A298509 A298510 A298511 * A298513 A298514 A298515

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, Feb 11 2018

STATUS

approved

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Last modified January 24 19:12 EST 2021. Contains 340411 sequences. (Running on oeis4.)