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%I #15 Jun 12 2024 17:37:27
%S 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,
%T 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,
%U 4,8,6,5,9,3,7,7,1,6,8,8,6,5,3,8,7,4
%N Decimal expansion of Sum_{k>=0} sin(k*Pi/5)/2^k.
%C Guide to related sequences:
%C sequence summand approximation minimal polynomial
%C (a(n)) sin(k*Pi/5)/2^k 0.6664488708 5 - 65*x^2 + 121*x^4
%C A373022 sin(2k*Pi/5)/2^k 0.5053526528 5 - 265*x^2 + 961*x^4
%C A373023 sin(3k*Pi/5)/2^k 0.3050180080 5 - 65*x^2 + 121*x^4
%C A373024 sin(4k*Pi/5)/2^k 0.1427344344 5 - 265*x^2 + 961*x^4
%C A373025 cos(k*Pi/5)/2^k 1.3503729060 11 - 23*x + 11*x^2
%C A373026 cos(2k*Pi/5)/2^k 0.8985194182 19 - 49*x + 31*x^2
%C A373027 cos(3k*Pi/5)/2^k 0.7405361848 11 - 23*x + 11*x^2
%C A373028 cos(4k*Pi/5)/2^k 0.6821257430 19 - 49*x + 31*x^2
%F Equals sqrt(10 - 2*sqrt*(5)) / (-8 + 2*sqrt(5)).
%F Equals (-1)*Sum_{k>=0} sin(9*k*Pi/5)/2^k.
%F _Peter J. C. Moses_ (May 22 2024) found the following generalized summation identities for the eight sequences in Comments and many other sequences:
%F 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)).
%F 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)).
%e 0.666448870812313914861635732850178653200791742032...
%t {b, m, h} = {2, 5, 1}; s = Sum[Sin[ h k Pi/m]/b^k, {k, 0, Infinity}]
%t d = N[s, 100]
%t First[RealDigits[d], 100]
%Y Cf. A373021, A373022, A373023, A373024, A373025, A373026, A373027, A373028.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Jun 09 2024
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