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

0,2

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).

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

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 *)

Crossrefs

Partial sums of A008253.

Sequence in context: A304079 A109722 A097121 * A139545 A106972 A086499

Adjacent sequences: A007901 A007902 A007903 * A007905 A007906 A007907

Keyword

nonn,nice

Author

N. J. A. Sloane

Status

approved