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

0,7

Comments

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

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).

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

Example

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 + ...

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 *)

Prog

(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 */
(Magma) R<x>:=PowerSeriesRing(Integers(), 80); Coefficients(R!( 1/((1-x^2)*(1-x^3)*(1-x^7)) )); // G. C. Greubel, Sep 08 2019
(SageMath)
def A008671_list(prec):
    P.<x> = 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
(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

Crossrefs

First differences of A029001.

Cf. A008614.

Sequence in context: A054404 A334415 A307152 * A199017 A189709 A025771

Adjacent sequences: A008668 A008669 A008670 * A008672 A008673 A008674

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved