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%I #19 Jan 01 2024 23:49:19
%S 1,606,241396,96075211,38237692791,15218505655816,6056927013322186,
%T 2410641732796574421,959429352726023297581,381850471743224475863026,
%U 151975528324450615370186976,60485878422659601692858553631
%N Numbers which are both 11-gonal and centered 11-gonal.
%C 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).
%C A number is both these numbers iff exist p and r such that (18*p - 7)^2 = 99*(2*r - 1) + 22.
%C 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.
%C 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) ).
%H G. C. Greubel, <a href="/A131215/b131215.txt">Table of n, a(n) for n = 1..380</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (399,-399,1).
%F a(n+2) = 398*a(n+1) - a(n) + 209.
%F a(n+1) = 199*a(n) + 209/2 + (5/2)*sqrt(6336*a(n)^2 + 6688*a(n) + 1617).
%F G.f.: z*(1 +207*z +z^2)/((1-z)*(1-398*z+z^2)).
%F 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
%p 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
%t LinearRecurrence[{399,-399,1},{1,606,241396},20] (* _Harvey P. Dale_, Mar 04 2015 *)
%o (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
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)) )); // _G. C. Greubel_, Dec 06 2019
%o (Sage)
%o def A131215_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x*(1+207*x+x^2)/((1-x)*(1-398*x+x^2)) ).list()
%o a=A131215_list(20); a[1:] # _G. C. Greubel_, Dec 06 2019
%o (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
%Y Cf. A128922.
%K nonn
%O 1,2
%A _Richard Choulet_, Sep 27 2007
%E More terms from _Paolo P. Lava_, Sep 26 2008
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