cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A206799 Based on an erroneous version of A008614.

Original entry on oeis.org

4, 1, 0, 2, 4, 3, 4, 4, 4, 5, 4, 6, 8, 7, 8, 8, 8, 9, 12, 10, 12, 15, 12, 12, 16, 17, 16, 18, 20, 19, 20, 20, 24, 25, 24, 26, 28, 27, 28, 32, 32, 33, 36, 34, 36, 39, 40, 40, 44, 45, 44
Offset: 0

Views

Author

N. J. A. Sloane, Feb 21 2012

Keywords

Comments

This is based on the formula in Burnside, Section 267, at the foot of page 363. Unfortunately there is a typo in the formula - the term with numerator 21 should have denominator (1+x)(1-x^3). This produces a sequence with 4's in the denominators. Multiplying by 4 gives a sequence of integers, shown here. This is included in the OEIS in accordance with our policy of publishing incorrect sequences together with pointers to the correct versions. - N. J. A. Sloane, Feb 21 2012

References

  • W. Burnside, Theory of Groups of Finite Order, Dover, NY, 1955, section 267, page 363

Crossrefs

Cf. A008616.

Programs

  • Mathematica
    (* expansion*)
w = Exp[I*2*Pi/7];
p[x_] = FullSimplify[ExpandAll[(4/168)*(1/(1 - x)^3 + 21/((1 - x)*(1 - x^2)) + 42/((1 - x)*(1 + x^2)) + 56/(1 - x^3) + 24/((1 - w*x)*(1 - w^2*x)*(1 - w^4*x)) + 24/((1 - w^6*x)*(1 - x*w^5)*(1 - x*w^3)))]];
a = Table[SeriesCoefficient[Series[FullSimplify[ExpandAll[p[x]]], {x, 0, 50}], n], {n, 0, 50}]
(* recursion*)
b[1] = 4; b[2] = 1; b[3] = 0; b[4] = 2; b[5] = 4; b[6] = 3;
b[7] = 4; b[8] = 4; b[9] = 4; b[10] = 5; b[11] = 4;
b[n_Integer?Positive] :=
b[n] = -489 + 11 n + n^2 - b[-11 + n] - 3 b[-10 + n] - 6 b[-9 + n] -
   9 b[-8 + n] - 11 b[-7 + n] - 12 b[-6 + n] - 12 b[-5 + n] -
   11 b[-4 + n] - 9 b[-3 + n] - 6 b[-2 + n] - 3 b[-1 + n];
Table[b[n], {n, 1, Length[a]}]
  • PARI
    Vec((-4-x+2*x^3+x^4-2*x^5-2*x^6+2*x^7+3*x^8+2*x^9-3*x^11)/(-1+x^3*(1+x-x^7-x^8+x^11))+O(x^9)) \\ Charles R Greathouse IV, Feb 13 2012

Formula

A precise definition is: Take the generating function as given by Burnside, expand as a Taylor series, and multiply by 4.
Expansion of (-4 - x + 2 x^3 + x^4 - 2 x^5 - 2 x^6 + 2 x^7 + 3 x^8 + 2 x^9 - 3 x^11)/(-1 + x^3 (1 + x - x^7 - x^8 + x^11))

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

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

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
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.