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A131215 Numbers which are both 11-gonal and centered 11-gonal. 2
1, 606, 241396, 96075211, 38237692791, 15218505655816, 6056927013322186, 2410641732796574421, 959429352726023297581, 381850471743224475863026, 151975528324450615370186976, 60485878422659601692858553631 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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)-3.5+0.5*sqrt(396*a(n)^2-308*a(n)+33) with g.f.  ( -1+9*x-x^2 ) / ( (x-1)*(x^2-20*x+1) ); 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)-4.5+0.5*(396*a(n)^2-396*a(n)+121)^0.5 with g.f. ( -1+10*x ) / ( (x-1)*(x^2-20*x+1) ).

LINKS

Table of n, a(n) for n=1..12.

Index entries for linear recurrences with constant coefficients, signature (399,-399,1).

FORMULA

a(n+2)=398*a(n+1)-a(n)+209.

a(n+1)=199*a(n)+104.5+2.5*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(n)=-(19/36)-(11/48)*[199-60*sqrt(11)]^n*sqrt(11)+(55/72)*[199-60*sqrt(11)]^n+(11/48) *sqrt(11)*[199+60*sqrt(11)]^n+(55/72)*[199+60*sqrt(11)]^n, with n>=0 [From Paolo P. Lava, Sep 26 2008] - Paolo P. Lava, Sep 26 2008

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

MATHEMATICA

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

CROSSREFS

Cf. A128922.

Sequence in context: A064250 A186818 A283639 * A096525 A142554 A252436

Adjacent sequences:  A131212 A131213 A131214 * A131216 A131217 A131218

KEYWORD

nonn

AUTHOR

Richard Choulet, Sep 27 2007

EXTENSIONS

More terms from Paolo P. Lava, Sep 26 2008

STATUS

approved

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Last modified May 27 02:54 EDT 2019. Contains 323597 sequences. (Running on oeis4.)