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%I #19 Jan 20 2021 07:23:36
%S 1,148,21022,2984983,423846571,60183228106,8545594544488,
%T 1213414242089197,172296276782121493,24464857888819162816,
%U 3473837523935538998386
%N Numbers simultaneously heptagonal and centered heptagonal.
%H S. C. Schlicker, <a href="http://www.jstor.org/stable/10.4169/math.mag.84.5.339">Numbers Simultaneously Polygonal and Centered Polygonal</a>, Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (143,-143,1)
%F x(n) + y(n)*sqrt(35) = (7+sqrt(35))*(6+sqrt(35))^n s(n) = (y(n)+1)/2 a(n) = (1/2)*(2+7*(s(n)^2-s(n))).
%F From _Richard Choulet_, Oct 01 2007: (Start)
%F a(n+2) = 142*a(n+1)-a(n)+7.
%F a(n+1) = 71*a(n)+3.5+1.5*(2240*a(n)^2+224*a(n)-63)^0.5.
%F G.f.: z*(1+5*z+z^2)/((1-z)*(1-142*z+z^2)). (End)
%e a(1)=148 because 148 is the seventh centered heptagonal number and the eighth heptagonal number.
%p CP := n -> 1+1/2*7*(n^2-n): N:=10: u:=6: v:=1: x:=7: y:=1: k_pcp:=[1]: for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+35*tempy*v: y:=tempx*v+tempy*u: s:=(y+1)/2: k_pcp:=[op(k_pcp),CP(s)]: end do: k_pcp;
%t Nest[Append[#,142Last[#]-#[[-2]]+7]&,{1,148},20] (* _Harvey P. Dale_, Apr 17 2011 *)
%Y Cf. A000566, A069099.
%K easy,nonn
%O 0,2
%A _Steven Schlicker_, Apr 24 2007
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