OFFSET
0,2
COMMENTS
Binomial transform of [1, 4, 8, 4, 2, -4, 8, -16, 32, -64, 128, ...]. - Gary W. Adamson, Feb 07 2010
Let P be the Promislow/Hantzsche-Wendt group defined as < p,q | q^(-1)p^2q=p^(-2), p^(-1)q^2p=q^(-2) >. Consider the Cayley Graph of P by the generating set S={p,q,p^(-1),q^(-1)}. Let B(n) denote the ball in the Cayley graph of radius n about the identity. Expectedly, for offset 0, a(n)=|B(n)| (tests run by computer-methods on n<11). - Erica Ann Hollingsworth, Sep 19 2025
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 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
a(n) = 5/6*n^3 + 5/4*n^2 + 13/6*n + O(1). - Charles R Greathouse IV, May 28 2026
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 *)
PROG
(PARI) a(n)=(10*n^3+15*n^2+26*n+12-n%2*3)/12 \\ Charles R Greathouse IV, May 28 2026
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved