cp's OEIS Frontend

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

%I A373021 #20 Apr 13 2025 19:57:58
%S A373021 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 A373021 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 A373021 4,8,6,5,9,3,7,7,1,6,8,8,6,5,3,8,7,4
%N A373021 Decimal expansion of Sum_{k>=0} sin(k*Pi/5)/2^k.
%C A373021 Guide to related sequences:
%C A373021 sequence   summand              approximation     minimal polynomial
%C A373021 (a(n))     sin(k*Pi/5)/2^k      0.6664488708     5 -  65*x^2 + 121*x^4
%C A373021 A373022    sin(2*k*Pi/5)/2^k    0.5053526528     5 - 265*x^2 + 961*x^4
%C A373021 A373023    sin(3*k*Pi/5)/2^k    0.3050180080     5 -  65*x^2 + 121*x^4
%C A373021 A373024    sin(4*k*Pi/5)/2^k    0.1427344344     5 - 265*x^2 + 961*x^4
%C A373021 A373025    cos(k*Pi/5)/2^k      1.3503729060    11 -  23*x   +  11*x^2
%C A373021 A373026    cos(2*k*Pi/5)/2^k    0.8985194182    19 -  49*x   +  31*x^2
%C A373021 A373027    cos(3*k*Pi/5)/2^k    0.7405361848    11 -  23*x   +  11*x^2
%C A373021 A373028    cos(4*k*Pi/5)/2^k    0.6821257430    19 -  49*x   +  31*x^2
%F A373021 Equals sqrt(10 - 2*sqrt(5)) / (8 - 2*sqrt(5)).
%F A373021 Equals (-1)*Sum_{k>=0} sin(9*k*Pi/5)/2^k.
%F A373021 _Peter J. C. Moses_ (May 22 2024) found the following generalized summation identities for the eight sequences in Comments and many other sequences:
%F A373021 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 A373021 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 A373021 0.666448870812313914861635732850178653200791742032...
%t A373021 {b, m, h} = {2, 5, 1}; s = Sum[Sin[ h  k  Pi/m]/b^k, {k, 0, Infinity}]
%t A373021 d = N[s, 100]
%t A373021 First[RealDigits[d], 100]
%Y A373021 Cf. A373021, A373022, A373023, A373024, A373025, A373026, A373027, A373028.
%K A373021 nonn,cons
%O A373021 0,1
%A A373021 _Clark Kimberling_, Jun 09 2024