A066530 Expansion of (1+x+x^3)/((1-x)*(1-x^4)).

1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 14, 15, 16, 17, 17, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 41, 41, 42, 43, 44, 44, 45, 46, 47, 47, 48, 49, 50, 50, 51, 52, 53, 53, 54, 55

Offset: 0

Author

Robert G. Wilson v, Jan 06 2002

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A037915.

Programs

Magma

[Floor((3*n + 5)/4): n in [0..100]]; // Wesley Ivan Hurt, Jan 02 2017

Maple

A066530:=n->floor((3*n+5)/4): seq(A066530(n), n=0..100); # Wesley Ivan Hurt, Jan 02 2017

Mathematica

CoefficientList[ Series[ (1 + x + x^3)/((1 - x)*(1 - x^4)), {x, 0, 75} ], x] (* or *) Table[Floor[(3 n + 5)/4], {n, 0, 75}]
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 2, 3, 4}, 80] (* Harvey P. Dale, Apr 01 2013 *)

Formula

G.f.: (1 + x + x^3)/((1 - x)*(1 - x^4)).

a(n) = floor((3*n + 5)/4).

a(n) = a(n-1) + a(n-4) - a(n-5) for n>4, with a(0)=1, a(1)=2, a(2)=2, a(3)=3, a(4)=4. - Harvey P. Dale, Apr 01 2013

a(n) = (6*n+7+2*cos(n*Pi/2)+cos((n+1)*Pi)+2*sin(n*Pi/2))/8. - Wesley Ivan Hurt, Oct 01 2017

Sum_{n>=0} (-1)^n/a(n) = log(3)/2 + Pi/(6*sqrt(3)). - Amiram Eldar, Jan 31 2023