OFFSET
0,3
COMMENTS
Also, coordination sequence for diamond structure D^+_8. (Edges defined by l_1 norm = 1.) - J. Serra-Sagrista (jserra(AT)ccd.uab.es). Confirmed by N. J. A. Sloane Nov 27 1998.
LINKS
Ray Chandler, 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).
Joan Serra-Sagrista, Enumeration of lattice points in l_1 norm, Inf. Proc. Lett. 76 (1-2) (2000) 39-44.
P. Solé, Counting lattice points in pyramids, Discr. Math. 139 (1995), 381-392.
Index entries for linear recurrences with constant coefficients, signature (0, 8, 0, -28, 0, 56, 0, -70, 0, 56, 0, -28, 0, 8, 0, -1).
FORMULA
a(n) = [z^(2n)] (1/2)*((1+z^2)^8+256*z^8)/(1-z^2)^8 + (1/2)*(1-z^2)^8/(1+z^2)^8.
G.f.: (x^16 + 120*x^14 + 1948*x^12 + 1024*x^11 + 11592*x^10 + 7168*x^9 + 21830*x^8 + 7168*x^7 + 11592*x^6 + 1024*x^5 + 1948*x^4 + 120*x^2 + 1)/((x - 1)^8*(x + 1)^8). - Chai Wah Wu, May 25 2026
MAPLE
1/2*((1+z^2)^8+256*z^8)/(1-z^2)^8+1/2*(1-z^2)^8/(1+z^2)^8:
seq(coeff(series(%, z, 62), z, 2*n), n=0..30);
# Alternative:
f := proc(m) local k, t1; t1 := 2^(n-1)*binomial((n+2*m)/2-1, n-1); if m mod 2 = 0 then t1 := t1+add(2^k*binomial(n, k)*binomial(m-1, k-1), k=0..n); fi; t1; end: n:=8:
seq(f(m), m=0..30);
PROG
(PARI) a(n)=if(n, my(q=n\2); q*[3072*q^6+10752*q^4+22848*q^2+3648, 1024*q^6+3584*q^5+1792*q^4-4480*q^3-3584*q^2+896*q+768][n%2+1]/315, 1) \\ Charles R Greathouse IV, May 26 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved