A100189 -id:A100189 - cp's OEIS frontend

A006484 a(n) = n(n + 1)(n^2 - 3*n + 5)/6.

0, 1, 3, 10, 30, 75, 161, 308, 540, 885, 1375, 2046, 2938, 4095, 5565, 7400, 9656, 12393, 15675, 19570, 24150, 29491, 35673, 42780, 50900, 60125, 70551, 82278, 95410, 110055, 126325, 144336, 164208, 186065, 210035, 236250, 264846, 295963, 329745, 366340

Offset: 0

Dennis S. Kluk (mathemagician(AT)ameritech.net)

Structured meta-pyramidal numbers, the n-th number from an n-gonal pyramidal number sequence. - James A. Record (james.record(AT)gmail.com), Nov 07 2004

The Gi4 triangle sums of A139600 are given by the terms of this sequence. For the definitions of the Gi4 and other triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. other meta sequences: A100177: prism; A000447: "polar" diamond; A059722: "equatorial diamond"; A100185: anti-prism; A100188: "polar" anti-diamond; and A100189: "equatorial" anti-diamond. Cf. A100145 for more on structured numbers.

Cf. A000332.

[n*(n+1)*(n^2 - 3*n + 5)/6: n in [0..50]]; // Vincenzo Librandi, May 16 2011

A006484:=-(1-2*z+5*z**2)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

lst={};Do[AppendTo[lst, n*(n+1)*(n^2-3*n+5)/6], {n, 0, 4!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
Table[n(n+1) (n^2-3n+5)/6,{n,0,40}] (* Harvey P. Dale, May 29 2019 *)

a(n)=n*(n+1)*(n^2-3*n+5)/6 \\ Charles R Greathouse IV, Oct 18 2022

a(n) = (1/6)*(n^4 - 2*n^3 + 2*n^2 + 5*n). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

a(n) = binomial(n+3,4) - 2*binomial(n+2,4) + 5*binomial(n+1,4). - Johannes W. Meijer, Apr 29 2011

A100188 Polar structured meta-anti-diamond numbers, the n-th number from a polar structured n-gonal anti-diamond number sequence.

Original entry on oeis.org

1, 6, 27, 84, 205, 426, 791, 1352, 2169, 3310, 4851, 6876, 9477, 12754, 16815, 21776, 27761, 34902, 43339, 53220, 64701, 77946, 93127, 110424, 130025, 152126, 176931, 204652, 235509, 269730, 307551, 349216

Offset: 1

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

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

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A000578, A000447, A004466, A007588, A063521, A062523 - "polar" structured anti-diamonds; A100189 - "equatorial" structured meta-anti-diamond numbers; A006484 for other structured meta numbers; and A100145 for more on structured numbers.

[(1/6)*(2*n^4-2*n^2+6*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011

Table[(2n^4-2n^2+6n)/6,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1}, {1,6,27,84,205},40] (* Harvey P. Dale, May 11 2016 *)

vector(40, n, (n^4 -n^2 +3*n)/3) \\ G. C. Greubel, Nov 08 2018

a(n) = (1/6)*(2*n^4 - 2*n^2 + 6*n).
G.f.: x*(1 + x + 7*x^2 - x^3)/(1-x)^5. - Colin Barker, Apr 16 2012
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(1)=1, a(2)=6, a(3)=27, a(4)=84, a(5)=205. - Harvey P. Dale, May 11 2016
E.g.f.: (3*x + 6*x^2 + 6*x^3 + x^4)*exp(x)/3. - G. C. Greubel, Nov 08 2018

cp's OEIS Frontend

A006484 a(n) = n(n + 1)(n^2 - 3*n + 5)/6.

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A100188 Polar structured meta-anti-diamond numbers, the n-th number from a polar structured n-gonal anti-diamond number sequence.

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cp's OEIS Frontend

A006484 a(n) = n*(n + 1)*(n^2 - 3*n + 5)/6.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A100188 Polar structured meta-anti-diamond numbers, the n-th number from a polar structured n-gonal anti-diamond number sequence.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A006484 a(n) = n(n + 1)(n^2 - 3*n + 5)/6.