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A131751 Numbers that are both centered triangular and centered pentagonal.

%I A131751 #31 Jan 28 2025 08:39:00
%S A131751 1,31,1891,117181,7263301,450207451,27905598631,1729696907641,
%T A131751 107213302675081,6645495068947351,411913480972060651,
%U A131751 25531990325198812981,1582571486681354344141,98093900183918770523731,6080239239916282418127151,376876738974625591153359601
%N A131751 Numbers that are both centered triangular and centered pentagonal.
%C A131751 We solve 0.5*(3*p^2+3*p+2)=0.5*(5*r^2+5*r+2), i.e., 3*(2*p+1)^2=5*(2*r+1)^2-2.
%C A131751 The Diophantine equation 3*X^2=5*Y^2-2is such that : X is given by A057080 which satisfies the new formula a(n+1)=4*a(n)+(15*a(n)^2+10)^0.5, Y is given by A070997 which satisfies the new formula a(n+1)=4*a(n)+(15*a(n)^2-6)^0.5 while r is given by the sequence 0,3,27,216,1704,... which satisfies a(n+2)=8*a(n+1)-a(n)+3 and a(n+1)=4*a(n)+1.5+0.5*(60*a(n)^2+60*a(n)+9)^0.5, p is given by the sequence 0,4,35,279,2200,... which satisfies a(n+2)=8*a(n+1)-a(n)+3 and a(n+1)=4*a(n)+1.5+0.5*sqrt(60*a(n)^2+60*a(n)+25).
%H A131751 Giovanni Lucca, <a href="https://web.archive.org/web/20230807074100/http://forumgeom.fau.edu/FG2016volume16/FG2016volume16.pdf#page=423">Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences</a>, Forum Geometricorum, Volume 16 (2016) 419-427.
%H A131751 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (63,-63,1).
%F A131751 a(n+2) = 62*a(n+1) - a(n) - 30, a(n+1) = 31*a(n) - 15 + sqrt(960*a(n)^2 - 960*a(n)+225).
%F A131751 G.f.: f(z) = a(1)*z+a(2)*z^2+... = z*(1-32*z+z^2)/((1-z)*(1-62*z+z^2)).
%F A131751 A005891 INTERSECT A005448. - _R. J. Mathar_, Oct 24 2007
%p A131751 A131751 := proc(n) coeftayl(x*(1-32*x+x^2)/(1-x)/(1-62*x+x^2),x=0,n) ; end: seq(A131751(n),n=1..20) ; # _R. J. Mathar_, Oct 24 2007
%t A131751 LinearRecurrence[{63,-63,1},{1,31,1891},20] (* _Harvey P. Dale_, Oct 01 2017 *)
%Y A131751 Cf. A050807 A070997.
%K A131751 nonn,easy
%O A131751 1,2
%A A131751 _Richard Choulet_, Sep 20 2007
%E A131751 Corrected and extended by _R. J. Mathar_, Oct 24 2007