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

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

Author

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

Keywords

• nonn
• easy

Examples

			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.
		

Links

• 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).

Crossrefs

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.

Programs

Magma

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

Mathematica

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 *)

PARI

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

Formula

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