1,2

A centered 11-gonal number is defined by (11*r^2-11*r+2)/2 ; a 11-gonal number by (9*p^2-7*p)/2. A number is both these numbers iff exist p and r such that (18*p-7)^2=99*(2*r-1)+22. The diophantine equation X^2=99*Y^2+22 is such that : X is given by the sequence 11, 209, 4169, 83171,... which satisfies a(n+2)=20*a(n+1)-a(n) and also a(n+1)=10*a(n)+(99*a(n)^2-2178)^0.5 ; Y is given by the sequence 1, 21, 419, 8359,... which satisfies a(n+2)=20*a(n+1)-a(n) and a(n+1)=10*a(n)+(99*a(n)^2+22)^0.5. The first equation is such that : p is given by 1, 12, 232, 4621,... which satisfies a(n+2)=20*a(n+1)-a(n)-7 and a(n+1)=10*a(n)-3.5+0.5*(396*a(n)^2-308*a(n)+33)^0.5 ; r is given by 1, 11, 210, 4180,... xhich satisfies a(n+2)=20*a(n+1)-a(n)-9 and a(n+1)=10*a(n)-4.5+0.5*(396*a(n)^2-396*a(n)+121)^0.5.

a(n+2)=398*a(n+1)-a(n)+209 a(n+1)=199*a(n)+104.5+2.5*(6336*a(n)^2+6688*a(n)+1617)^0.5 G.f.: f(z)=a(1)*z+a(2)*z^2+...=((z*(1+207*z+z^2))/((1-z)*(1-398*z+z^2))

Sequence in context: A031792 A020383 A064250 this_sequence A096525 A142554 A027514

Adjacent sequences: A131212 A131213 A131214 this_sequence A131216 A131217 A131218

nonn,uned

Richard Choulet (richardchoulet(AT)yahoo.fr), Sep 27 2007