A078112 Coefficients a(n) in the unique expansion sin(1) = Sum[a(n)/n!, n>=1], where a(n) satisfies 0<=a(n)

0, 1, 2, 0, 0, 5, 6, 0, 0, 9, 10, 0, 0, 13, 14, 0, 0, 17, 18, 0, 0, 21, 22, 0, 0, 25, 26, 0, 0, 29, 30, 0, 0, 33, 34, 0, 0, 37, 38, 0, 0, 41, 42, 0, 0, 45, 46, 0, 0, 49, 50, 0, 0, 53, 54, 0, 0, 57, 58, 0, 0, 61, 62, 0, 0, 65, 66, 0, 0, 69, 70, 0, 0, 73, 74, 0, 0, 77, 78, 0, 0, 81, 82, 0, 0, 85

Offset: 1

Author

John W. Layman, Dec 04 2002

Keywords

• nonn
• easy

Examples

			sum(i=1,10,a(i)/i!)=0.84147073..., sin(1)=0.841470984...
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

Crossrefs

Cf. A077814.

Programs

PARI

concat(0, Vec(x^2*(1-x^2+2*x^3)/((1-x)^2*(1+x^2)^2) + O(x^100))) \\ Colin Barker, Feb 15 2016

Formula

a(n) = floor(n!*sin(1)) - n*floor((n-1)!*sin(1)). a(n)=0 if n==0 or 1 (mod 4); a(n)=n-1 if n==2 or 3 (mod 4). - Benoit Cloitre, Dec 07 2002

From Colin Barker, Feb 15 2016: (Start)

a(n) = 2*a(n-1)-3*a(n-2)+4*a(n-3)-3*a(n-4)+2*a(n-5)-a(n-6) for n>6.

G.f.: x^2*(1-x^2+2*x^3) / ((1-x)^2*(1+x^2)^2). (End)

Extensions

More terms from Benoit Cloitre, Dec 07 2002