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

0,3

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).

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

Example

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 + ...

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 *)

Prog

(PARI) {a(n) = (n^2 + n)\6 + 1} /* Michael Somos, Sep 06 2013 */
(Magma) [1 + Floor(n*(n+1)/6): n in [0..60]]; // G. C. Greubel, Aug 03 2019
(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
(GAP) List([0..60], n-> 1 + Int(n*(n+1)/6)); # G. C. Greubel, Aug 03 2019

Crossrefs

Cf. A001840, A008748, A061347.

Sequence in context: A025695 A025694 A022796 * A089649 A049700 A002984

Adjacent sequences: A008745 A008746 A008747 * A008749 A008750 A008751

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved