A010025 Crystal ball sequence for squashed {D_5}^* lattice, perhaps the smallest example of a "non-superficial" lattice.

1, 13, 85, 377, 1239, 3291, 7503, 15275, 28517, 49729, 82081, 129493, 196715, 289407, 414219, 578871, 792233, 1064405, 1406797, 1832209, 2354911, 2990723, 3757095, 4673187, 5759949, 7040201, 8538713, 10282285, 12299827, 14622439, 17283491, 20318703, 23766225, 27666717

Offset

0,2

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).

Index entries for crystal ball sequences

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

From Gopinath A. R., Feb 14 2012, (Start)

G.f.: (x^5+7*x^4+42*x^3+22*x^2+7*x+1)/(x^6-6*x^5+15*x^4-20*x^3+15*x^2-6*x+1).

E.g.f.: 1/12*(8*x^5+95*x^4+320*x^3+360*x^2+144*x+12)*exp(x). (End)

a(n) = (8*n^5 + 15*n^4 + 30*n^3 + 45*n^2 + 46*n + 12)/12. - Charles R Greathouse IV, May 31 2026

Maple

seq( 2/3*n^5+5/4*n^4+5/2*n^3+15/4*n^2+23/6*n+1, n=0..33);

Mathematica

A010025[n_] := n*(n*(n*(n*(8*n + 15) + 30) + 45) + 46)/12 + 1;
Array[A010025, 35, 0] (* Paolo Xausa, Jun 13 2026 *)
(* Alternative: *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 13, 85, 377, 1239, 3291}, 35] (* Paolo Xausa, Jun 13 2026 *)

Prog

(PARI) a(n)=1+n*(46+45*n+30*n^2+15*n^3+8*n^4)/12 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Sequence in context: A225129 A297207 A222491 * A001848 A055843 A296647

Adjacent sequences: A010022 A010023 A010024 * A010026 A010027 A010028

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved