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 1-3 of 3 results.

A026811 Number of partitions of n in which the greatest part is 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 84, 101, 119, 141, 164, 192, 221, 255, 291, 333, 377, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 1342, 1469, 1602, 1747, 1898, 2062, 2233, 2418, 2611, 2818, 3034, 3266, 3507, 3765
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Essentially same as A001401: five zeros followed by A001401.
Also number of partitions of n into exactly 5 parts.

References

  • D. E. Knuth, The Art of Computer Programming, vol. 4, fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.

Links

Crossrefs

Cf. A026810, A026812, A026813, A026814, A026815, A026816, A002622 (partial sums), A008667 (first differences).

Programs

  • GAP
    List([0..70],n->NrPartitions(n,5)); # Muniru A Asiru, May 17 2018
  • Mathematica
    Table[Count[IntegerPartitions[n], {5, _}], {n, 0, 55}] (* corrected by Harvey P. Dale, Oct 24 2011 *)
Table[Length[IntegerPartitions[n, {5}]], {n, 0, 55}] (* Eric Rowland, Mar 02 2017 *)
CoefficientList[Series[x^5/Product[1 - x^k, {k, 1, 5}], {x, 0, 65}], x] (* Robert A. Russell, May 13 2018 *)
Drop[LinearRecurrence[{1,1,0,0,-1,-1,-1,1,1,1,0,0,-1,-1,1}, Append[Table[0,{14}],1],110],9] (* Robert A. Russell, May 17 2018 *)
  • PARI
    a(n)=round((n^4+10*(n^3+n^2)-75*n-45*(-1)^n*n)/2880);
for(n=0,10000,print(n," ",a(n))); /* b-file format */
/* Washington Bomfim, Jul 03 2012 */
  • PARI
    x='x+O('x^99); concat(vector(5), Vec(x^5/prod(k=1, 5, 1-x^k))) \\ Altug Alkan, May 17 2018

Formula

a(n) = round( ((n^4+10*(n^3+n^2)-75*n -45*n*(-1)^n)) / 2880 ). - Washington Bomfim, Jul 03 2012
G.f.: x^5/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)). - Joerg Arndt, Jul 04 2012
a(n) = A008284(n,5). - Robert A. Russell, May 13 2018
From Gregory L. Simay, Jul 28 2019: (Start)
a(2n) = a(2n-1) + a(n+1) + a(n) - a(n-3) - a(n-4);
a(2n+1) = a(2n) + a(n+3) - a(n-5). (End)
From R. J. Mathar, Jun 23 2021: (Start)
a(n) - a(n-5) = A001400(n-5).
a(n) - a(n-4) = A008669(n-5).
a(n) - a(n-3) = A029007(n-5).
a(n) - a(n-2) = A029032(n-5).
a(n) = +a(n-1) +a(n-2) -a(n-5) -a(n-6) -a(n-7) +a(n-8) +a(n-9) +a(n-10) -a(n-13) -a(n-14) +a(n-15). (End)

Extensions

More terms from Robert G. Wilson v, Jan 11 2002
a(0)=0 inserted by Joerg Arndt, Jul 04 2012

A249020 a(n) = floor( n * (n+5) / 10) + 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 9, 11, 13, 16, 18, 21, 24, 27, 31, 34, 38, 42, 46, 51, 55, 60, 65, 70, 76, 81, 87, 93, 99, 106, 112, 119, 126, 133, 141, 148, 156, 164, 172, 181, 189, 198, 207, 216, 226, 235, 245, 255, 265, 276, 286, 297, 308, 319, 331, 342, 354, 366
Offset: 0

Views

Author

Michael Somos, Oct 19 2014

Keywords

Comments

If the s(n) are the Somos-4 polynomials, then s(n) = x^a(n-6) * y^a(n-4) * z^a(n-5) * f(n) where f(n) is an irreducible polynomial. - Michael Somos, Feb 21 2020

Examples

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 7*x^6 + 9*x^7 + ...
Somos-4 polynomial s(7) = x^1 * y^3 * z^2 * (z + 2*y*z + x*y^2 + y^2*z + x*y*z + x*y^2*z) where 1 = a(7-6), 3 = a(7-4), 2 = a(7-5). - _Michael Somos_, Feb 21 2020

Links

Crossrefs

Cf. A008669, A249013.

Programs

  • Magma
    [Floor(n*(n+5)/10) + 1: n in [0..60]]; // G. C. Greubel, Aug 04 2018
  • Mathematica
    a[ n_] := Quotient[ n (n + 5), 10] + 1;
CoefficientList[Series[(1-x+x^2)/((1-x)^2*(1-x^5)), {x,0,60}], x] (* or *) Table[Floor[n*(n+5)/10]+1, {n,0,60}] (* G. C. Greubel, Aug 04 2018 *)
  • PARI
    {a(n) = n * (n + 5) \ 10 + 1};
  • PARI
    {a(n) = if( n<0, n = -5-n); polcoeff( (1 - x + x^2) / ((1 - x)^2 * (1 - x^5)) + x * O(x^n), n)};

Formula

G.f.: (1 - x + x^2) / ((1 - x)^2 * (1 - x^5)) = (1-x+x^2)/ ( (1-x)^3*(1+x+x^2+x^3+x^4)).
Euler transform of length 6 sequence [1, 1, 1, 0, 1, -1].
a(n) = a(-5-n) for all n in Z.
a(n) = a(n-5) + n for all n in Z.
a(n) + a(n+4) = min( a(n+1) + a(n+3), a(n+2) + a(n+2)) + 1 for all n in Z.
a(n) = A249013(n+1) + 1 for all n in Z.
a(n) = A008669(n) - A008669(n-6) for all n in Z.

A028291 Expansion of 1/((1-x)^2(1-x^2)(1-x^3)(1-x^5)) in powers of x.

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 25, 35, 48, 64, 84, 108, 137, 171, 211, 258, 312, 374, 445, 525, 616, 718, 832, 959, 1100, 1256, 1428, 1617, 1824, 2050, 2297, 2565, 2856, 3171, 3511, 3878, 4273, 4697, 5152, 5639, 6160, 6716, 7309, 7940, 8611, 9324, 10080, 10881, 11729
Offset: 0

Views

Author

N. J. A. Sloane, Dec 11 1999

Keywords

Comments

Partitions of n into parts 1, 2, 3, and 5. - Joerg Arndt, Jun 05 2014

Examples

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 17*x^5 + 25*x^6 + 35*x^7 + ...

References

  • Susan Elle, Ore extensions of global dimension 5, Abstract 1110-17-204, Abstracts Amer. Math. Soc., 36 (No. 2, 2015), p. 822.

Links

Crossrefs

Cf. A001304, A008669.

Programs

  • Mathematica
    a[ n_] := Quotient[n (n + 12) (n^2 + 12 n + 52), 720] + 1; (* Michael Somos, Jun 05 2014 *)
a[ n_] := With[{m = If[ n < 0, -12 - n, n]}, SeriesCoefficient[ 1 / ((1 - x)^2*(1 - x^2)*(1 - x^3)*(1 - x^5)), {x, 0, m}]]; (* Michael Somos, Jun 05 2014 *)
Table[Round[(n + 1)*(n^3 + 23*n^2 + 173*n + 451)/720], {n, 0, 40}] (* Wesley Ivan Hurt, Jun 05 2014 *)
LinearRecurrence[{2,0,-1,-1,1,0,-1,1,1,0,-2,1},{1,2,4,7,11,17,25,35,48,64,84,108},50] (* Harvey P. Dale, Sep 06 2022 *)
  • PARI
    Vec(1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^5))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012
  • PARI
    {a(n) = n * (n+12) * (n^2 + 12*n + 52) \ 720 + 1}; /* Michael Somos, Jun 05 2014 */
  • PARI
    {a(n) = if( n<0, n = -12 - n); polcoeff( 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^3) * (1 - x^5)) + x * O(x^n), n)}; /* Michael Somos, Jun 05 2014 */

Formula

a(n) = round((n+1)*(n^3+23*n^2+173*n+451)/720). - Tani Akinari, Jun 05 2014
a(n) - 2*a(n-1) + a(n+3) + a(n+4) - 2*a(n+6) + a(n+7) = 1 if n == 3 (mod 5) else 0. - Michael Somos, Jun 05 2014
a(n) = a(-12 - n) for all n in Z. - Michael Somos, May 14 2015
a(n) - a(n-1) = A008669(n), a(n) - a(n-3) = A001304(n) for all n in Z. - Michael Somos, May 14 2015
Euler transform of length 5 sequence [ 2, 1, 1, 0, 1]. - Michael Somos, May 14 2015
Showing 1-3 of 3 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.