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A128919
Numbers which are both heptagonal and centered heptagonal.
3
1, 148, 21022, 2984983, 423846571, 60183228106, 8545594544488, 1213414242089197, 172296276782121493, 24464857888819162816, 3473837523935538998386, 493260463540957718608003, 70039511985292060503338047, 9945117441447931633755394678, 1412136637173620999932762706236
OFFSET
1,2
LINKS
S. C. Schlicker, Numbers Simultaneously Polygonal and Centered Polygonal, Mathematics Magazine, Vol. 84, No. 5, December 2011, pp. 339-350.
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) = 142*a(n-1) - a(n-2) + 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)
a(n) = A000566(A133271(n)) = A069099(A133272(n)). - Andrew Howroyd, Nov 07 2025
EXAMPLE
a(1)=148 because 148 is the seventh centered heptagonal number and the eighth heptagonal number.
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;
MATHEMATICA
Nest[Append[#, 142Last[#]-#[[-2]]+7]&, {1, 148}, 20] (* Harvey P. Dale, Apr 17 2011 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Steven Schlicker, Apr 24 2007
EXTENSIONS
Offset corrected and more terms from Andrew Howroyd, Nov 07 2025
STATUS
approved