OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = (1/2)*(f(m,n,1)-f(m,n,2)) where f(m,n,k)=Sum((n-|kx|)*(m-|ky|)); -n<kx<n, -m<ky<m, (x,y)=1, m=8
For another more efficient formula, see Mathematica code below.
Conjectures from Colin Barker, Dec 25 2017: (Start)
G.f.: x*(1 + 66*x + 132*x^2 + 304*x^3 + 492*x^4 + 771*x^5 + 1003*x^6 + 1273*x^7 + 1390*x^8 + 1480*x^9 + 1374*x^10 + 1241*x^11 + 971*x^12 + 739*x^13 + 476*x^14 + 296*x^15 + 140*x^16 + 74*x^17 + 17*x^18 + 8*x^19 + 8*x^20) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)).
a(n) = -a(n-2) + a(n-5) + a(n-6) + 2*a(n-7) + a(n-8) + a(n-9) - a(n-11) - a(n-12) - 2*a(n-13) - a(n-14) - a(n-15) + a(n-18) + a(n-20) for n>21.
(End)
MATHEMATICA
m=8;
a[0]=0; a[1]=1;
a[2]=m^2+2;
a[3]=2*m^2+3-Mod[m, 2];
a[n_]:=a[n]=2*a[n-1]-a[n-2]+2*p1[m, n]+2*p4[m, n]
p1[m_, n_]:=Sum[p2[m, n, y], {y, 1, m-1}]
p2[m_, n_, y_]:=If[GCD[y, n-1]==1, m-y, 0]
p[i_]:=If[i>0, i, 0]
p2[m_, n_, x_, y_]:=p2[m, n, x, y]=(n-x)*(m-y)-p[n-2*x]*p[m-2*y]
p3[m_, n_, x_, y_]:=p2[m, n, x, y]-2*p2[m, n-1, x, y]+p2[m, n-2, x, y]
p4[m_, n_]:=p4[m, n]=If[Mod[n, 2]==0, 0, p42[m, n]]
p42[m_, n_]:=p42[m, n]=Sum[p43[m, n, y], {y, 1, m-1}]
p43[m_, n_, y_]:=If[GCD[(n-1)/2, y]==1, p3[m, n, (n-1)/2, y], 0]
Table[a[n], {n, 0, 40}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Seppo Mustonen, May 28 2009
STATUS
approved