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A046196 Indices of square numbers which are also heptagonal.

%I A046196 #20 Feb 16 2025 08:32:39
%S A046196 1,9,77,1519,12987,111035,2190397,18727245,160112393,3158550955,
%T A046196 27004674303,230881959671,4554628286713,38940721617681,
%U A046196 332931625733189,6567770830889191,56152493568021699,480087173425298867,9470720983513926709,80971856784365672277
%N A046196 Indices of square numbers which are also heptagonal.
%H A046196 Colin Barker, <a href="/A046196/b046196.txt">Table of n, a(n) for n = 1..950</a>
%H A046196 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeptagonalSquareNumber.html">Heptagonal Square Number.</a>
%H A046196 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1442,0,0,-1).
%F A046196 From _Paul Weisenhorn_, May 01 2009: (Start)
%F A046196 a(n+6) = 1442*a(n+3)-a(n) with
%F A046196 a(-2)=-77; a(-1)=-9; a(0)=-1; a(1)=1; a(2)=9; a(3)=77;
%F A046196 A = (721+sqrt(10)*228)^k; B = (721-sqrt(10)*228)^k;
%F A046196 a(3*k+1) = (7*(A-B)/sqrt(10)+2*(A+B))/4;
%F A046196 a(3*k+2) = (57*(A-B)/sqrt(10)+18*(A+B))/4;
%F A046196 a(3*k) = (7*(A-B)/sqrt(10)-2*(A+B))/4;
%F A046196 (End)
%F A046196 G.f.:  x * (1 + x) * (1 + 8*x + 69*x^2 + 8*x^3 + x^4) / (1-1442*x^3 + x^6). - _Ant King_, Nov 11 2011
%p A046196 for n from 1 to 10000 do m:=sqrt((5*n*n-3*n)/2):
%p A046196 if (trunc(m)=m) then print(n,m): end if: end do: # _Paul Weisenhorn_, May 01 2009
%t A046196 LinearRecurrence[{ 0, 0, 1442, 0, 0, -1 } , {1, 9, 77, 1519, 12987, 111035 }, 17] (* _Ant King_, Nov 11 2011 *)
%o A046196 (PARI) Vec(x*(x+1)*(x^4+8*x^3+69*x^2+8*x+1)/(x^6-1442*x^3+1) + O(x^50)) \\ _Colin Barker_, Jun 23 2015
%Y A046196 Cf. A036354, A046195.
%K A046196 nonn,easy
%O A046196 1,2
%A A046196 _Eric W. Weisstein_