A007904 Crystal ball sequence for diamond.

1, 5, 17, 41, 83, 147, 239, 363, 525, 729, 981, 1285, 1647, 2071, 2563, 3127, 3769, 4493, 5305, 6209, 7211, 8315, 9527, 10851, 12293, 13857, 15549, 17373, 19335, 21439, 23691, 26095, 28657, 31381, 34273, 37337, 40579, 44003, 47615, 51419, 55421, 59625, 64037

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Binomial transform of [1, 4, 8, 4, 2, -4, 8, -16, 32, -64, 128, ...]. - Gary W. Adamson, Feb 07 2010

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 (3, -2, -2, 3, -1).

Crossrefs

Partial sums of A008253.

Programs

Maple

gf:= -(x^4+2*x^3+4*x^2+2*x+1)/((x-1)^2*(x^2-1)*(1-x)):
seq(coeff(series(gf,x,n+1),x,n), n=0..50);

Mathematica

b[0]=1; b[1]=4; b[2]=8; b[3]=4; b[n_] := (-1)^n*2^(n-3); a[n_] := Sum[Binomial[n, k]*b[k], {k, 0, n}]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Aug 08 2012, after Gary W. Adamson *)
LinearRecurrence[{3,-2,-2,3,-1},{1,5,17,41,83},80] (* Harvey P. Dale, Jan 22 2024 *)

Formula

G.f.: -(x^4 + 2*x^3 + 4*x^2 + 2*x + 1)/((x-1)^2*(x^2-1)*(1-x)).

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). - Wesley Ivan Hurt, Jan 20 2024