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

0,1

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: A114893 A089101 A359809 * A192930 A010168 A340004

Adjacent sequences: A298509 A298510 A298511 * A298513 A298514 A298515

Keyword

nonn,easy,cons

Author

Clark Kimberling, Feb 11 2018

Status

approved