A128919 Numbers simultaneously heptagonal and centered heptagonal.
1, 148, 21022, 2984983, 423846571, 60183228106, 8545594544488, 1213414242089197, 172296276782121493, 24464857888819162816, 3473837523935538998386
Offset: 0
Examples
a(1)=148 because 148 is the seventh centered heptagonal number and the eighth heptagonal number.
Links
- S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.
- Index entries for linear recurrences with constant coefficients, signature (143,-143,1)
Programs
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Maple
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;
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Mathematica
Nest[Append[#,142Last[#]-#[[-2]]+7]&,{1,148},20] (* Harvey P. Dale, Apr 17 2011 *)
Formula
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))).
From Richard Choulet, Oct 01 2007: (Start)
a(n+2) = 142*a(n+1)-a(n)+7.
a(n+1) = 71*a(n)+3.5+1.5*(2240*a(n)^2+224*a(n)-63)^0.5.
G.f.: z*(1+5*z+z^2)/((1-z)*(1-142*z+z^2)). (End)