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 A131215 Numbers which are both 11-gonal and centered 11-gonal. 3
 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) - 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) ). LINKS G. C. Greubel, Table of n, a(n) for n = 1..380 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) + 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(n) = -(19/36) - (11/48)*sqrt(11)*(199-60*sqrt(11))^n + (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. - 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 MAPLE 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 MATHEMATICA LinearRecurrence[{399, -399, 1}, {1, 606, 241396}, 20] (* Harvey P. Dale, Mar 04 2015 *) PROG (PARI) 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 (MAGMA) 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 (Sage) 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 (GAP) 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 CROSSREFS Cf. A128922. Sequence in context: A186818 A283639 A332160 * A096525 A142554 A345875 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 January 18 13:38 EST 2022. Contains 350455 sequences. (Running on oeis4.)