|
%I #24 Jan 06 2024 01:00:20
%S 1,451,145351,46802701,15070324501,4852597686751,1562521384809451,
%T 503127033310956601,162005342204743216201,52165217062894004660251,
%U 16797037888909664757384751
%N Numbers simultaneously 10-gonal and centered 10-gonal.
%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 (323,-323,1).
%F Let x(n) + y(n)*sqrt(80) =: (10+sqrt(80))*(9+sqrt(80))^n, s(n) = (y(n)+1)/2; then a(n) = (1/2)*(2+10*(s(n)^2-s(n))).
%F From _Richard Choulet_, Oct 01 2007: (Start)
%F a(n+2) = 322*a(n+1)-a(n)+130.
%F a(n+1) = 161*a(n)+65+9*(320*a(n)^2+260*a(n)+45)^0.5.
%F G.f.: z*(1+128*z+z^2)/((1-z)*(1-322*z+z^2)). (End)
%e a(1) = 451 because 451 is the tenth centered 10-gonal number and the eleventh 10-gonal number.
%p CP := n -> 1+1/2*10*(n^2-n): N:=10: u:=9: v:=1: x:=10: y:=1: k_pcp:=[1]: for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+80*tempy*v: y:=tempx*v+tempy*u: s:=(y+1)/2: k_pcp:=[op(k_pcp),CP(s)]: end do: k_pcp;
%Y Cf. A001107, A062786, A048909.
%K easy,nonn
%O 0,2
%A _Steven Schlicker_, Apr 24 2007
| 