A008671 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^7)).

1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 13, 14, 14, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 28, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 57, 58, 60, 62, 63, 65, 67, 69, 70, 73, 74, 76, 78, 80

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of partitions of n into parts 2, 3, and 7. - Joerg Arndt, Jul 08 2013

Examples

			G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 3*x^10 + ...
		

References

• A. Adler, Hirzebruch's curves F_1, F_2, F_4, F_14, F_28 for Q(sqrt 7), pp. 221-285 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999 (see p. 262).
• L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 24).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 227
• Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1,0,1,0,-1,-1,0,1).

Crossrefs

First differences of A029001.

Cf. A008614.

Programs

GAP

a:=[1,0,1,1,1,1,2,2,2,3,3,3];; for n in [13..80] do a[n]:=a[n-2] +a[n-3] -a[n-5] +a[n-7] -a[n-9] -a[n-10] +a[n-12]; od; a; # G. C. Greubel, Sep 08 2019

Magma

R:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^2)*(1-x^3)*(1-x^7)) )); // G. C. Greubel, Sep 08 2019

Maple

seq(coeff(series(1/((1-x^2)*(1-x^3)*(1-x^7)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Sep 08 2019

Mathematica

CoefficientList[Series[1/((1-x^2)(1-x^3)(1-x^7)), {x,0,80}], x] (* Vincenzo Librandi, Jun 22 2013 *)
a[ n_] := With[{m = If[ n < 0, -12 - n, n]}, SeriesCoefficient[ 1 / ((1 - x^2) (1 - x^3) (1 - x^7)), {x, 0, m}]]; (* Michael Somos, Mar 18 2015 *)
a[ n_] := Quotient[ 3 n^2 + 36 n + If[ OddQ[n], 189, 252], 252]; (* Michael Somos, Mar 18 2015 *)
LinearRecurrence[{0,1,1,0,-1,0,1,0,-1,-1,0,1},{1,0,1,1,1,1,2,2,2,3,3,3},100] (* Harvey P. Dale, Dec 18 2023 *)

PARI

{a(n) = if( n<0, n = -12-n); polcoeff( 1 / ((1 - x^2) * (1 - x^3) * (1 - x^7)) + x * O(x^n), n)}; /* Michael Somos, Oct 11 2006 */

PARI

{a(n) = (3*n^2 + 36*n + if( n%2, 189, 252)) \ 252}; /* Michael Somos, Mar 18 2015 */

Sage

def A008671_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(1/((1-x^2)*(1-x^3)*(1-x^7))).list()
A008671_list(80) # G. C. Greubel, Sep 08 2019

Formula

Euler transform of length 7 sequence [ 0, 1, 1, 0, 0, 0, 1]. - Michael Somos, Oct 11 2006

a(n) = a(-12-n), a(n) = a(n-2) + a(n-3) - a(n-5) + a(n-7) - a(n-9) - a(n-10) + a(n-12) for all n in Z. - Michael Somos, Oct 11 2006

a(n) = floor((3*n^2+36*n+196)/252 + (-1/9)*(-2)^floor((n+2-3*floor((n+2)/3))/2)). - Tani Akinari, Jul 07 2013

a(n) ~ 1/84*n^2. - Ralf Stephan, Apr 29 2014

0 = a(n) - a(n+2) - a(n+3) + a(n+5) - (mod(n, 7) == 2) for all n in Z. - Michael Somos, Mar 18 2015

a(n) = A008614(2*n). - Michael Somos, Mar 18 2015

a(n) = floor((n^2 + 12*n + 56 + 28*[(n mod 3)=0])/84). - Hoang Xuan Thanh, Jun 24 2025