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.

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A133103 Number of partitions of n^3 into n nonzero squares.

Original entry on oeis.org

1, 1, 2, 1, 10, 34, 156, 734, 3599, 18956, 99893, 548373, 3078558, 17510598, 101960454, 599522778, 3565904170, 21438347021, 129905092421, 794292345434, 4890875249113, 30326545789640, 189195772457341, 1187032920371427
Offset: 1

Views

Author

Hugo Pfoertner, Sep 11 2007

Keywords

Examples

			a(2)=1 because the only way to express 2^3 = 8 as a sum of two squares is 8 = 2^2 + 2^2.
a(3)=2 because 3^3 = 27 = 1^2 + 1^2 + 5^2 = 3^2 + 3^2 + 3^2.

Links

Crossrefs

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.
Cf. A133102 (number of ways to express n^3 as a sum of n distinct nonzero squares), A133104 (number of ways to express n^4 as a sum of n nonzero squares).

Programs

  • PARI
    a(i, n, k)=local(s, j); if(k==1, if(issquare(n), return(1), return(0)), s=0; for(j=ceil(sqrt(n/k)), min(i, floor(sqrt(n-k+1))), s+=a(j, n-j^2, k-1)); return(s)) for(n=1,50, m=n^3; k=n; print1(a(m, m, k)", ") ) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

Extensions

2 more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
More terms from Robert Gerbicz, May 09 2008

A133105 Number of partitions of n^4 into n distinct nonzero squares.

Original entry on oeis.org

1, 0, 1, 0, 21, 266, 2843, 55932, 884756, 13816633, 283194588, 5375499165, 125889124371, 3202887665805, 80542392920980, 2270543992935431, 64253268814048352, 1892633465941308859, 59116753827795287519, 1886846993941912938452
Offset: 1

Views

Author

Hugo Pfoertner, Sep 12 2007

Keywords

Examples

			a(3)=1 because there is exactly one way to express 3^4 as the sum of 3 distinct nonzero squares: 81 = 1^2 + 4^2 + 8^2.

Links

Crossrefs

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.
Cf. A133104 (number of ways to express n^4 as a sum of n nonzero squares), A133102 (number of ways to express n^3 as a sum of n distinct nonzero squares).

Programs

  • PARI
    a(i, n, k)=local(s, j); if(k==1, if(issquare(n) && n

Extensions

a(10) from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
a(11) onwards from Robert Gerbicz, May 09 2008

A133104 Number of partitions of n^4 into n nonzero squares.

Original entry on oeis.org

1, 0, 3, 1, 49, 732, 9659, 190169, 3225654, 61896383, 1360483727, 30969769918, 778612992660, 20749789703573, 579672756740101, 17115189938667708, 525530773660159970, 16825686497823918869, 561044904645283065043, 19368002907483932784642
Offset: 1

Views

Author

Hugo Pfoertner, Sep 11 2007

Keywords

Examples

			a(3)=3 because there are 3 ways to express 3^4 = 81 as a sum of 3 nonzero squares: 81 = 1^2 + 4^2 + 8^2 = 3^2 + 6^2 + 6^2 = 4^2 + 4^2 + 7^2.
a(4)=1 because the only way to express 4^4 = 256 as a sum of 4 nonzero squares is 256 = 8^2 + 8^2 + 8^2 + 8^2.

Crossrefs

Cf. A000161, A000378, A000141, A005875, A000118, A000132, A008451.
Cf. A133105 (number of ways to express n^4 as a sum of n distinct nonzero squares), A133103 (number of ways to express n^3 as a sum of n nonzero squares).

Programs

  • PARI
    a(i, n, k)=local(s, j); if(k==1, if(issquare(n), return(1), return(0)), s=0; for(j=ceil(sqrt(n/k)), min(i, floor(sqrt(n-k+1))), s+=a(j, n-j^2, k-1)); return(s)) for(n=1,50, m=n^4; k=n; print1(a(m, m, k)", ") ) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007

Extensions

a(9) from Herman Jamke (hermanjamke(AT)fastmail.fm), Dec 16 2007
a(10) onwards from Robert Gerbicz, May 09 2008

A179015 Number of ways in which n^2 can be expressed as the sum of exactly five positive squares.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 5, 2, 6, 6, 9, 9, 15, 8, 25, 20, 21, 25, 39, 26, 46, 44, 57, 49, 71, 52, 102, 81, 81, 99, 145, 92, 156, 126, 164, 160, 204, 151, 247, 217, 236, 245, 326, 211, 357, 319, 381, 360, 416, 344, 518, 446, 476, 450, 670, 468, 675, 607, 661, 668, 825, 625
Offset: 1

Views

Author

Carmine Suriano, Jun 24 2010

Keywords

Comments

As n goes to infinity the ratio of a(n)/a(n) of sequence A178898 (using all different squares) tends to 5/4.

Links

Crossrefs

Cf. A000290, A178898.
Cf. A000132. - R. J. Mathar, Jun 26 2010
Cf. A065458, A065459.

Programs

  • Maple
    a(8) = 5 since 64 can be expressed in five different ways as the sum of 5 squares (order is ignored): 8^2 = 7^2+3^2+2^2+1^2+1^2 = 6^2+5^2+1^2+1^2+1^2 = 6^2+4^2+2^2+2^2+2^2 = 6^2+3^2+3^2+3^2+1^2 = 5^2+5^2+3^2+1^2+1^2.

Formula

Asymptotic behavior for large values of n is a(n) = n^2/2-47n/2+243.
a(n) = A025429(n^2). - R. J. Mathar, Jun 26 2010
a(n) = A065459(n) - A065458(n). - Alois P. Heinz, Oct 25 2018
Previous Showing 11-14 of 14 results.

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