A077814 -id:A077814 - cp's OEIS frontend

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

John W. Layman, Dec 04 2002

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

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).

Cf. A077814.

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

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)

More terms from Benoit Cloitre, Dec 07 2002

A087620 #{0<=k<=n: k*n is divisible by 4}.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 4, 2, 9, 3, 6, 3, 13, 4, 8, 4, 17, 5, 10, 5, 21, 6, 12, 6, 25, 7, 14, 7, 29, 8, 16, 8, 33, 9, 18, 9, 37, 10, 20, 10, 41, 11, 22, 11, 45, 12, 24, 12, 49, 13, 26, 13, 53, 14, 28, 14, 57, 15, 30, 15, 61, 16, 32, 16, 65, 17, 34, 17, 69, 18, 36, 18, 73, 19, 38, 19, 77, 20

Offset: 0

Paul Barry, Sep 13 2003

With the similar remainder 1, 2 and 3 sequences provides a four-fold partition of A000027.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A026741, A077814, A087507.

I:=[1,1,2,1,5,2,4,2]; [n le 8 select I[n] else 2*Self(n-4)-Self(n-8): n in [1..80]]; // Vincenzo Librandi, May 03 2015

CoefficientList[Series[(3 x^4 + x^3 + 2 x^2 + x + 1)/((x - 1)^2 (x + 1)^2 (x^2 + 1)^2), {x, 0, 80}], x] (* Vincenzo Librandi, May 03 2015 *)

Vec((3*x^4+x^3+2*x^2+x+1)/((x-1)^2*(x+1)^2*(x^2+1)^2) + O(x^100)) \\ Colin Barker, May 03 2015

a(n) = Sum_{k=0..n} if (k*n mod 4 = 0, 1, 0).
From Colin Barker, May 03 2015: (Start)
  a(n) = (6+4*n+i^n*(-i+n)+(-i)^n*(i+n)+2*(-1)^n*(1+n))/8 where i=sqrt(-1).
  a(n) = 2*a(n-4)-a(n-8) for n>7.
  G.f.: (3*x^4+x^3+2*x^2+x+1) / ((x-1)^2*(x+1)^2*(x^2+1)^2).
(End)

A131728 a(4n) = n, a(4n+1) = 2n+1, a(4n+2) = n+1, a(4n+3) = 0.

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 2, 0, 2, 5, 3, 0, 3, 7, 4, 0, 4, 9, 5, 0, 5, 11, 6, 0, 6, 13, 7, 0, 7, 15, 8, 0, 8, 17, 9, 0, 9, 19, 10, 0, 10, 21, 11, 0, 11, 23, 12, 0, 12, 25, 13, 0, 13, 27, 14, 0, 14, 29, 15, 0, 15, 31, 16, 0, 16, 33, 17, 0, 17, 35, 18, 0, 18, 37, 19, 0, 19, 39, 20, 0, 20, 41, 21, 0, 21

Offset: 0

Paul Curtz, Sep 17 2007

Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

LinearRecurrence[{2,-3,4,-3,2,-1},{0,1,1,0,1,3},90] (* Harvey P. Dale, Aug 15 2013 *)

a(n)=A077814(n+1). - R. J. Mathar, Jun 13 2008

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A087620 #{0<=k<=n: k*n is divisible by 4}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A131728 a(4n) = n, a(4n+1) = 2n+1, a(4n+2) = n+1, a(4n+3) = 0.

Views

Author

Keywords

Links

Programs

Mathematica

Formula