This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A036354 #48 Feb 16 2025 08:32:37
%S A036354 1,81,5929,2307361,168662169,12328771225,4797839017609,
%T A036354 350709705290025,25635978392186449,9976444135331412025,
%U A036354 729252434211108535809,53306479301521270428241,20744638830126197732344369,1516379800105728357531817761,110843467413344235941816109721
%N A036354 Heptagonal square numbers.
%C A036354 From _Ant King_, Nov 11 2011: (Start)
%C A036354 This sequence is also the union of the three sequences defined by:
%C A036354 a(3n-2) = ((10 - sqrt(10)) * (3 + sqrt(10))^(4*n-3) - (10 + sqrt(10)) * (-3 + sqrt(10))^(4*n-3))^2 / 1600.
%C A036354 a(3n-1) = 9/160 * ((3 + sqrt(10))^(4*n-2) - (-3 + sqrt(10))^(4*n-2))^2.
%C A036354 a(3n) = ((20 - 7*sqrt(10)) * (3 + sqrt(10))^(4*n) + (20 + 7*sqrt(10)) * (-3 + sqrt(10))^(4*n))^2 / 1600.
%C A036354 Equivalent short forms for these subsequences are:
%C A036354 a(3n-2) = floor((10 - sqrt(10))^2 * (3 + sqrt(10))^(8*n - 6) / 1600).
%C A036354 a(3n-1) = floor( 9/160 * (3 + sqrt(10))^(8*n - 4)).
%C A036354 a(3n) = floor((20 - 7*sqrt(10))^ 2 * (3 + sqrt(10))^(8*n) / 1600).
%C A036354 (End)
%C A036354 Also heptagonal numbers (A000566) which are also centered octagonal numbers (A016754). - _Colin Barker_, Jan 19 2015
%H A036354 Colin Barker, <a href="/A036354/b036354.txt">Table of n, a(n) for n = 1..475</a>
%H A036354 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeptagonalSquareNumber.html">Heptagonal Square Number.</a>
%H A036354 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,2079362,-2079362,0,-1,1).
%F A036354 O.g.f.: -x*(1 + 80*x + 5848*x^2 + 222070*x^3 + 5848*x^4 + 80*x^5 + x^6) / ( (x-1)*(x^6 - 2079362*x^3 + 1) ).
%F A036354 From _Richard Choulet_, May 08 2009: (Start)
%F A036354 With the first values, for n>=0, a(n+9) = 2079363*(a(n+6) - a(n+3)) + a(n).
%F A036354 On every bisection modulo 2: a(n+1) = 1039681*a(n) + 116964 + 164388*sqrt(40*a(n)^2 + 9*a(n)).
%F A036354 On every bisection modulo 2: a(n+2) = 2079362*a(n+1) - a(n) + 233928. (End)
%F A036354 From _Ant King_, Nov 11 2011: (Start)
%F A036354 a(n) = a(n-1) + 2079362*a(n-3) - 2079362*a(n-4) - a(n-6) + a(n-7).
%F A036354 a(n) = 2079362*a(n-3) - a(n-6) + 233928.
%F A036354 (End)
%F A036354 From _Jonathan Pappas_, Jan 16 2022: (Start)
%F A036354 Define the three sequences
%F A036354 b(n) = 1442*b(n-1) - b(n-2) for n >= 2, with b(0) = -77, b(1) = 1;
%F A036354 c(n) = 1442*c(n-1) - c(n-2) for n >= 2, with c(0) = -9, c(1) = 9; and
%F A036354 d(n) = 1442*d(n-1) - d(n-2) for n >= 2, with d(0) = -1, d(1) = 77.
%F A036354 Then, for n >= 1,
%F A036354 a(3n - 2) = b(n)^2,
%F A036354 a(3n - 1) = c(n)^2, and
%F A036354 a(3n) = d(n)^2.
%F A036354 (End)
%p A036354 A036354 := proc(n)
%p A036354 if n <= 7 then
%p A036354 op(n,[1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609]);
%p A036354 else
%p A036354 procname(n-1) +2079362 *(procname(n-3)-procname(n-4)) -procname(n-6) +procname(n-7) ;
%p A036354 end if;
%p A036354 end proc:
%p A036354 seq(A036354(n),n=1..12) ;
%t A036354 LinearRecurrence[{ 1, 0, 2079362, -2079362, 0, -1, 1 }, {1, 81, 5929, 2307361, 168662169, 12328771225, 4797839017609 }, 13] (* _Ant King_, Nov 11 2011 *)
%o A036354 (PARI) Vec(-x*(x^6+80*x^5+5848*x^4+222070*x^3+5848*x^2+80*x+1)/((x-1)*(x^6-2079362*x^3+1)) + O(x^100)) \\ _Colin Barker_, Jan 19 2015
%Y A036354 Cf. A046195, A046196, A253920.
%K A036354 nonn,easy
%O A036354 1,2
%A A036354 Jean-Francois Chariot (jeanfrancois.chariot(AT)afoc.alcatel.fr)
%E A036354 More terms from _Eric W. Weisstein_
%E A036354 One more term from _Richard Choulet_, May 08 2009