A193013 - OEIS

%I #10 Jan 18 2022 02:34:56

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

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

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

%N Decimal expansion of the constant term of the reduction of cos(x) by x^2->x+1.

%C Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.

%F From _Amiram Eldar_, Jan 18 2022: (Start)

%F Equals 1 + Sum_{k>=1} (-1)^k*Fibonacci(2*k-1)/(2*k)!.

%F Equals cos(1/2)*cos(sqrt(5)/2) + sin(1/2)*sin(sqrt(5)/2)/sqrt(5). (End)

%e 0.5767021231946339181854043477561375348200573480...

%t f[x_] := Cos[x]; r[n_] := Fibonacci[n];

%t c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]

%t u0 = N[Sum[c[n]*r[n - 1], {n, 0, 300}], 100]

%t RealDigits[u0, 10]

%Y Cf. A000045, A193010, A192232, A193014.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Jul 14 2011