A131216 -id:A131216 - cp's OEIS frontend

1, 606, 241396, 96075211, 38237692791, 15218505655816, 6056927013322186, 2410641732796574421, 959429352726023297581, 381850471743224475863026, 151975528324450615370186976, 60485878422659601692858553631

Offset: 1

Richard Choulet, Sep 27 2007

nonn

A centered 11-gonal number is defined by (11*r^2 - 11*r + 2)/2 = A069125(r); a 11-gonal number by (9*p^2 - 7*p)/2 = A051682(p).
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,... in A131216; Y is given by the sequence 1, 21, 419, 8359,... in A083043.
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) - 7/2 + sqrt(396*a(n)^2 - 308*a(n) + 33)/2 with g.f.  (1 -9*x +x^2)/( (1-x) * (1 -20*x + x^2) ); r is given by 1, 11, 210, 4180,... which satisfies a(n+2) = 20*a(n+1) - a(n) - 9 and a(n+1) = 10*a(n) - 9/2 + sqrt(396*a(n)^2 - 396*a(n) + 121)/2 with g.f. (1 - 10*x)/( (1-x)*(1 -20*x +x^2) ).

G. C. Greubel, Table of n, a(n) for n = 1..380
Index entries for linear recurrences with constant coefficients, signature (399,-399,1).

Cf. A128922.

a:=[1,606,241396];; for n in [4..20] do a[n]:=399*a[n-1]-399*a[n-2] +a[n-3]; od; a; # G. C. Greubel, Dec 06 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)) )); // G. C. Greubel, Dec 06 2019

seq(coeff(series(x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)), x, n+1), x, n), n = 1..20); # G. C. Greubel, Dec 06 2019

LinearRecurrence[{399,-399,1},{1,606,241396},20] (* Harvey P. Dale, Mar 04 2015 *)

my(x='x+O('x^20)); Vec(x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2))) \\ G. C. Greubel, Dec 06 2019

def A131215_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)) ).list()
a=A131215_list(20); a[1:] # G. C. Greubel, Dec 06 2019

a(n+2) = 398*a(n+1) - a(n) + 209.
a(n+1) = 199*a(n) + 209/2 + (5/2)*sqrt(6336*a(n)^2 + 6688*a(n) + 1617).
G.f.: z*(1 +207*z +z^2)/((1-z)*(1-398*z+z^2)).
a(1)=1, a(2)=606, a(3)=241396, a(n) = 399*a(n-1) - 399*a(n-2) + a(n-3). - Harvey P. Dale, Mar 04 2015

More terms from Paolo P. Lava, Sep 26 2008

cp's OEIS Frontend

A131215 Numbers which are both 11-gonal and centered 11-gonal.

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