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A133274
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Numbers which are both 12-gonal and centered 12-gonal numbers.
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0
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1, 793, 382537, 184382353, 88871911921, 42836077163881, 20646900321079033, 9951763118682930337, 4796729176304851343713, 2312013511215819664739641, 1114385715676848773553163561, 537131602942729893032960097073
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=1..12.
Index entries for linear recurrences with constant coefficients, signature (483,-483,1).
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FORMULA
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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 ;
G.f.: (z+310*z^2+z^3)/((1-z)*(1-482*z+z^2)).
a(n)=-(13/20)+(33/40)*{[241+44*sqrt(30)]^n+[241-44*sqrt(30)]^n}-(3/20)*sqrt(30)*{[241-44*sqrt(30)]^n-[241+44*sqrt(30)]^n }, with n>=0 [From Paolo P. Lava, Nov 25 2008]
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MATHEMATICA
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LinearRecurrence[{483, -483, 1}, {1, 793, 382537}, 20] (* Harvey P. Dale, Aug 27 2020 *)
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CROSSREFS
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Sequence in context: A213471 A075667 A136543 * A261657 A086393 A336943
Adjacent sequences: A133271 A133272 A133273 * A133275 A133276 A133277
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KEYWORD
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nonn,easy
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AUTHOR
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Richard Choulet, Oct 16 2007
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STATUS
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approved
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