A008384 Crystal ball sequence for A_4 lattice.

1, 21, 131, 471, 1251, 2751, 5321, 9381, 15421, 24001, 35751, 51371, 71631, 97371, 129501, 169001, 216921, 274381, 342571, 422751, 516251, 624471, 748881, 891021, 1052501, 1235001, 1440271

Offset: 0

Author

N. J. A. Sloane and J. H. Conway

Keywords

• nonn
• nice
• easy

Comments

Partial sums of A008383.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
• H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
• Index entries for crystal ball sequences
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A008383.

Programs

Mathematica

Table[1/12 (12-50 n+85 n^2-70 n^3+35 n^4),{n,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,21,131,471,1251},30] (* Harvey P. Dale, Aug 22 2011 *)

PARI

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,-5,10,-10,5]^n*[1;21;131;471;1251])[1,1] \\ Charles R Greathouse IV, Jun 15 2015

Formula

a(n) = 1 +5*n*(n+1)*(7*n^2+7*n+10)/12. - T. D. Noe, Apr 29 2007

G.f.: (-1-x^4-16*x^3-36*x^2-16*x)/(x-1)^5. [Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009]

a(n) = 5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5), n> 4. - Harvey P. Dale, Aug 22 2011