A373021 Decimal expansion of Sum_{k>=0} sin(k*Pi/5)/2^k.

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

Offset

0,1

Comments

Guide to related sequences:

sequence summand approximation minimal polynomial

(a(n)) sin(k*Pi/5)/2^k 0.6664488708 5 - 65*x^2 + 121*x^4

A373022 sin(2k*Pi/5)/2^k 0.5053526528 5 - 265*x^2 + 961*x^4

A373023 sin(3k*Pi/5)/2^k 0.3050180080 5 - 65*x^2 + 121*x^4

A373024 sin(4k*Pi/5)/2^k 0.1427344344 5 - 265*x^2 + 961*x^4

A373025 cos(k*Pi/5)/2^k 1.3503729060 11 - 23*x + 11*x^2

A373026 cos(2k*Pi/5)/2^k 0.8985194182 19 - 49*x + 31*x^2

A373027 cos(3k*Pi/5)/2^k 0.7405361848 11 - 23*x + 11*x^2

A373028 cos(4k*Pi/5)/2^k 0.6821257430 19 - 49*x + 31*x^2

Links

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

Formula

Equals sqrt(10 - 2*sqrt*(5)) / (-8 + 2*sqrt(5)).

Equals (-1)*Sum_{k>=0} sin(9*k*Pi/5)/2^k.

Peter J. C. Moses (May 22 2024) found the following generalized summation identities for the eight sequences in Comments and many other sequences:

Sum_{k>=0} sin(h*k + Pi/m)/b^(k+r) = b^(1-r)*(b*sin(Pi/m) + sin(h - Pi/m)/(1 + b^2 - 2*b*cos*(Pi/m)).

Sum_{k>=0} cos(h*k + Pi/m)/b^(k+r) = b^(1-r)*(b*cos(Pi/m) + cos(h - Pi/m)/(1 + b^2 - 2*b*cos*(Pi/m)).

Example

0.666448870812313914861635732850178653200791742032...

Mathematica

{b, m, h} = {2, 5, 1}; s = Sum[Sin[ h  k  Pi/m]/b^k, {k, 0, Infinity}]
d = N[s, 100]
First[RealDigits[d], 100]

Crossrefs

Cf. A373021, A373022, A373023, A373024, A373025, A373026, A373027, A373028.

Sequence in context: A349187 A153509 A248093 * A361006 A292091 A080066

Adjacent sequences: A373018 A373019 A373020 * A373026 A373027 A373028

Keyword

nonn,cons

Author

Clark Kimberling, Jun 09 2024

Status

approved