A008748 Expansion of (1 + x^5) / ((1-x) * (1-x^2) * (1-x^3)) in powers of x.

1, 1, 2, 3, 4, 6, 8, 10, 13, 16, 19, 23, 27, 31, 36, 41, 46, 52, 58, 64, 71, 78, 85, 93, 101, 109, 118, 127, 136, 146, 156, 166, 177, 188, 199, 211, 223, 235, 248, 261, 274, 288, 302, 316, 331, 346, 361, 377, 393, 409, 426, 443, 460, 478, 496, 514, 533, 552, 571

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 10*x^7 + 13*x^8 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Peter M. Chema, Illustration of first 26 terms as corners of a double hexagon spiral from 1
• Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A001840, A008748, A061347.

Programs

GAP

List([0..60], n-> 1 + Int(n*(n+1)/6)); # G. C. Greubel, Aug 03 2019

Magma

[1 + Floor(n*(n+1)/6): n in [0..60]]; // G. C. Greubel, Aug 03 2019

Maple

A061347 := proc(n) op(1+(n mod 3),[-2,1,1]) ; end proc:
A008748 := proc(n) 1/6*n^2+1/6*n+8/9+A061347(n+2)/9 ; end proc:
seq(A008748(n),n=0..60) ; # R. J. Mathar, Mar 22 2011

Mathematica

Table[Floor[((n*(n+1)+2)/2+3)/3],{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 26 2010 *)
CoefficientList[Series[(1+x^5)/((1-x)(1-x^2)(1-x^3)), {x,0,60}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2,-1,1,-2,1}, {1,1,2,3,4}, 60] (* Harvey P. Dale, Apr 08 2019 *)

PARI

{a(n) = (n^2 + n)\6 + 1} /* Michael Somos, Sep 06 2013 */

Sage

((1 + x^5)/((1-x)*(1-x^2)*(1-x^3))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 03 2019

Formula

a(n) = 1 + floor( n(n+1)/6 ). - Michael Somos, Jun 16 1999

a(n) = 1 + A001840(n-1). - Michael Somos, Jun 16 1999

a(n) = 1 + a(n-1) + a(n-3) - a(n-4) if n>4; a(n) = n if n=1..4. - Michael Somos, Jun 16 1999

a(-1-n) = a(n). - Michael Somos, Sep 06 2013