login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A373021
Decimal expansion of Sum_{k>=0} sin(k*Pi/5)/2^k.
10
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
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]
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jun 09 2024
STATUS
approved