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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A096000 Cupolar numbers: a(n) = (n+1)*(5*n^2 + 7*n + 3)/3.

Original entry on oeis.org

1, 10, 37, 92, 185, 326, 525, 792, 1137, 1570, 2101, 2740, 3497, 4382, 5405, 6576, 7905, 9402, 11077, 12940, 15001, 17270, 19757, 22472, 25425, 28626, 32085, 35812, 39817, 44110, 48701, 53600, 58817, 64362, 70245, 76476, 83065, 90022, 97357, 105080, 113201, 121730
Offset: 0

Views

Author

N. J. A. Sloane, in memory of Harold Scott MacDonald Coxeter [Feb 09 1907 - Mar 31 2003], May 08 2004

Keywords

Comments

Number of equal balls that will fill a triangular cupola, formed by splitting a cuboctahedron along one of its four "equilateral" hexagons.
Also as a(n) = (1/6)*(10*n^3 - 6*n^2 + 10*n), n>0: structured pentagonal anti-prism numbers (Cf. A100185 = structured anti-prisms); and structured tetragonal anti-diamond numbers (vertex structure 7) (Cf. A000447 = alternate vertex; A100188 = structured anti-diamonds). Cf. A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

References

  • H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.

Links

Crossrefs

Cf. A000447, A005902, A100145, A100185, A100188.

Programs

Formula

a(n) = (1/2)*(Q(n) + 3*n^2 + 3*n + 1), where Q(n) are the cuboctahedral numbers, A005902.
G.f.: (1+6*x+3*x^2)/(1-x)^4. - Paul Barry, Oct 28 2006
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - Wesley Ivan Hurt, May 23 2015
E.g.f.: exp(x)*(3 + 27*x + 27*x^2 + 5*x^3)/3. - Elmo R. Oliveira, Aug 11 2025

A100186 Structured heptagonal anti-diamond numbers (vertex structure 7).

Original entry on oeis.org

1, 16, 67, 176, 365, 656, 1071, 1632, 2361, 3280, 4411, 5776, 7397, 9296, 11495, 14016, 16881, 20112, 23731, 27760, 32221, 37136, 42527, 48416, 54825, 61776, 69291, 77392, 86101, 95440, 105431, 116096
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Links

Crossrefs

Cf. A063521 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers.

Programs

  • Magma
    [(1/6)*(22*n^3-24*n^2+8*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011
  • Mathematica
    Table[(22*n^3 - 24*n^2 + 8*n)/6, {n,1,40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 16, 67, 176}, 40] (* G. C. Greubel, Nov 08 2018 *)
  • PARI
    vector(40, n, (22*n^3 -24*n^2 +8*n)/6) \\ G. C. Greubel, Nov 08 2018

Formula

a(n) = (1/6)*(22*n^3 - 24*n^2 + 8*n).
G.f.: x*(1 + 12*x + 9*x^2)/(1-x)^4. - Colin Barker, Jan 19 2012
E.g.f.: (3*x +21*x^2 +11*x^3)*exp(x)/3. - G. C. Greubel, Nov 08 2018

A100187 Structured octagonal anti-diamond numbers (vertex structure 7).

Original entry on oeis.org

1, 18, 77, 204, 425, 766, 1253, 1912, 2769, 3850, 5181, 6788, 8697, 10934, 13525, 16496, 19873, 23682, 27949, 32700, 37961, 43758, 50117, 57064, 64625, 72826, 81693, 91252, 101529, 112550, 124341, 136928
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Links

Crossrefs

Cf. A063523 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers.

Programs

  • Magma
    [(1/6)*(26*n^3-30*n^2+10*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011
  • Mathematica
    Table[(26n^3-30n^2+10n)/6,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,18,77,204},40] (* Harvey P. Dale, Dec 24 2012 *)
  • PARI
    vector(40, n, (13*n^3 -15*n^2 +5*n)/3) \\ G. C. Greubel, Nov 08 2018

Formula

a(n) = (1/6)*(26*n^3 - 30*n^2 + 10*n).
G.f.: x*(1 + 14*x + 11*x^2)/(1-x)^4. - Colin Barker, Jan 19 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=1, a(2)=18, a(3)=77, a(4)=204. - Harvey P. Dale, Dec 24 2012
E.g.f.: (3*x + 24*x^2 + 13*x^3)*exp(x)/3. - G. C. Greubel, Nov 08 2018

A100189 Equatorial structured meta-anti-diamond numbers, the n-th number from an equatorial structured n-gonal anti-diamond number sequence.

Original entry on oeis.org

1, 6, 27, 92, 245, 546, 1071, 1912, 3177, 4990, 7491, 10836, 15197, 20762, 27735, 36336, 46801, 59382, 74347, 91980, 112581, 136466, 163967, 195432, 231225, 271726, 317331, 368452, 425517, 488970, 559271, 636896
Offset: 1

Views

Author

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Keywords

Examples

			There are no 1- or 2-gonal anti-diamonds, so 1 and (2n+2) are used as the first and second terms since all the sequences begin as such.

Links

Crossrefs

Cf. A000578, A096000, A051673, A005915, A100186, A100187 - "equatorial" structured anti-diamonds; A100188 - "polar" structured meta-anti-diamond numbers; A006484 for other structured meta numbers; and A100145 for more on structured numbers.

Programs

  • Magma
    [(1/6)*(4*n^4-12*n^3+20*n^2-6*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011
  • Mathematica
    Table[(4n^4-12n^3+20n^2-6n)/6,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,6,27,92,245},40] (* Harvey P. Dale, Jul 05 2011 *)

Formula

a(n) = (1/6)*(4*n^4-12*n^3+20*n^2-6*n).
a(1)=1, a(2)=6, a(3)=27, a(4)=92, a(5)=245, a(n)=5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Jul 05 2011
G.f.: x*(1+x)*(1+7*x^2)/(1-x)^5. - Colin Barker, Jan 19 2012
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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