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.

A072474 Sum of next n squares.

Original entry on oeis.org

1, 13, 77, 294, 855, 2071, 4403, 8492, 15189, 25585, 41041, 63218, 94107, 136059, 191815, 264536, 357833, 475797, 623029, 804670, 1026431, 1294623, 1616187, 1998724, 2450525, 2980601, 3598713, 4315402, 5142019, 6090755, 7174671, 8407728, 9804817, 11381789, 13155485
Offset: 1

Views

Author

Amarnath Murthy, Jun 20 2002

Keywords

Examples

			a(1) = 1^2 = 1;
a(2) = 2^2 + 3^2 = 13;
a(3) = 4^2 + 5^2 + 6^2 = 77.

Links

Crossrefs

Cf. A000290, A000330, A072475, A050410.
Cf. A006003 (for natural numbers), A260513 (for triangular numbers), A372583 (for pentagonal numbers), A372751 (for hexagonal numbers), A075664 (for cubes).

Programs

  • Magma
    [n*(3*n^2+1)*(n^2+2)/12: n in [1..35]]; // Vincenzo Librandi, Dec 31 2024
  • Mathematica
    Table[Sum[ i^2, {i, n(n - 1)/2 + 1, n(n + 1)/2}], {n, 1, 35}]
  • PARI
    a(n) = n*(3*n^2+1)*(n^2+2)/12

Formula

a(n) = k*(k+1)*(2*k+1)/6 - r*(r+1)*(2*r+1)/6, where k = n*(n+1)/2 and r = n*(n-1)/2.
a(n) = A000330(n*(n+1)/2) - A000330(n*(n-1)/2).
a(n) = (n/12)*(3*n^2 + 1)*(n^2 + 2). - Benoit Cloitre, Jun 26 2002
G.f.: x*(1+3*x+x^2)*(1+4*x+x^2)/(1-x)^6. - Colin Barker, Mar 23 2012
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 6. - Jinyuan Wang, May 25 2020
E.g.f.: exp(x)*x*(12 + 66*x + 82*x^2 + 30*x^3 + 3*x^4)/12. - Stefano Spezia, May 14 2024

Extensions

Edited by Robert G. Wilson v, Jun 21 2002

A372583 a(n) = (3*n^5 + 5*n^3)/8.

Original entry on oeis.org

1, 17, 108, 424, 1250, 3051, 6517, 12608, 22599, 38125, 61226, 94392, 140608, 203399, 286875, 395776, 535517, 712233, 932824, 1205000, 1537326, 1939267, 2421233, 2994624, 3671875, 4466501, 5393142, 6467608, 7706924, 9129375, 10754551, 12603392, 14698233
Offset: 1

Views

Author

Kelvin Voskuijl, May 05 2024

Keywords

Comments

Sum of pentagonal numbers in increasing groups 1, 5+12, 22+35+51, 70+92+117+145 etc.

Examples

			The first ten pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, 92, 117, and 145.  Taking them in groups, respectively, of 1, 2, 3, and 4, i.e., (1), (5, 12), (22, 35, 51), and (70, 92, 117, 145), and summing each group separately gives 1, 17, 108, 424.

Links

Crossrefs

Cf. A260513 (for triangular numbers), A072474 (for squares).
Cf. A000326 (pentagonal numbers), A002411 (their partial sums).

Programs

Formula

From Stefano Spezia, May 06 2024: (Start)
G.f.: x*(1 + 11*x + 21*x^2 + 11*x^3 + x^4)/(1 - x)^6.
E.g.f.: exp(x)*x*(8 + 60*x + 80*x^2 + 30*x^3 + 3*x^4)/8. (End)

A372751 a(n) = (3*n^5 + 4*n^3 - n)/6.

Original entry on oeis.org

1, 21, 139, 554, 1645, 4031, 8631, 16724, 30009, 50665, 81411, 125566, 187109, 270739, 381935, 527016, 713201, 948669, 1242619, 1605330, 2048221, 2583911, 3226279, 3990524, 4893225, 5952401, 7187571, 8619814, 10271829, 12167995, 14334431, 16799056, 19591649
Offset: 1

Views

Author

Kelvin Voskuijl, May 12 2024

Keywords

Comments

Sums of hexagonal numbers (A000384) in successive groups of length 1,2,3,etc, so 1, 6+15, 28+45+66, 91+120+153+190, etc.

Examples

			The hexagonal numbers and their groups summed begin
  1, 6, 15, 28, 45, 66, 91, 120, 153, 190
  \/ \---/  \--------/  \---------------/
  1,   21,     139,            554

Links

Crossrefs

Cf. A000384 (hexagonal numbers), A002412 (their partial sums).
Cf. A260513 (for triangular numbers), A072474 (for squares), A372583 (for pentagonal numbers), A075664 (cubes).

Programs

Formula

From Stefano Spezia, May 12 2024: (Start)
G.f.: x*(1 + 15*x + 28*x^2 + 15*x^3 + x^4)/(1 - x)^6.
E.g.f.: exp(x)*x*(6 + 57*x + 79*x^2 + 30*x^3 + 3*x^4)/6. (End)
Showing 1-3 of 3 results.

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