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Numbers which are both 12-gonal and centered 12-gonal numbers.
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%I #12 Jan 02 2024 08:54:15

%S 1,793,382537,184382353,88871911921,42836077163881,20646900321079033,

%T 9951763118682930337,4796729176304851343713,2312013511215819664739641,

%U 1114385715676848773553163561,537131602942729893032960097073

%N Numbers which are both 12-gonal and centered 12-gonal numbers.

%C We write G12(r)=5*r^2-4*r and CG12(p)=6*p^2-6*p+1. A number has both properties iff there exist r and p such that 2*(5*r-2)^2=15*(2*p-1)^2+3. The Diophantine equation (2*X)^2=30*Y^2+6 gives 2 new sequences. We obtain also 2 new sequences with the indices given by r and p respectively.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (483,-483,1).

%F a(n+2)=482*a(n+1)-a(n)+312 ; a(n+1)=241*a(n)+156+44*(30*a(n)^2+39*a(n)+12)^0.5 ;

%F G.f.: (z+310*z^2+z^3)/((1-z)*(1-482*z+z^2)).

%t LinearRecurrence[{483,-483,1},{1,793,382537},20] (* _Harvey P. Dale_, Aug 27 2020 *)

%K nonn,easy

%O 1,2

%A _Richard Choulet_, Oct 16 2007