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A006061 Star numbers (A003154) that are squares.

%I A006061 M5385 #51 Feb 16 2025 08:32:29
%S A006061 1,121,11881,1164241,114083761,11179044361,1095432263641,
%T A006061 107341182792481,10518340481399521,1030690025994360601,
%U A006061 100997104206965939401,9896685522256667700721,969774184076946468731281
%N A006061 Star numbers (A003154) that are squares.
%D A006061 J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 121, p. 42, Ellipses, Paris 2008.
%D A006061 M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 22.
%D A006061 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006061 G. C. Greubel, <a href="/A006061/b006061.txt">Table of n, a(n) for n = 1..500</a>
%H A006061 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A006061 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A006061 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StarNumber.html">Star Number</a>
%H A006061 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99, -99, 1).
%F A006061 A007667 = 3*square star numbers (A006061) + 2.
%F A006061 a(n) = denominator of kappa(sqrt(6)/A054320(n)) where kappa(x) is the sum of successive remainders by computing the Euclidean algorithm for (1, x). - _Thomas Baruchel_, Nov 29 2003
%F A006061 From _Ignacio Larrosa Cañestro_, Feb 27 2000: (Start)
%F A006061 a(n) = 99*(a(n-1) - a(n-2)) + a(n-3).
%F A006061 a(n) = (5 - 2*sqrt(6))/8*(sqrt(3) + sqrt(2))^(4*n) + (5 + 2*sqrt(6))/8*(sqrt(3) - sqrt(2))^(4*n) - 1/4. (End)
%F A006061 a(n) = 98*a(n-1) - a(n-2) + 24. - _Lekraj Beedassy_, Jul 14 2008
%e A006061 a(2)=121 because this is the 2nd star number (A003154) that is a square.
%p A006061 Digits := 1000:q := seq(floor(evalf(( (5+2*sqrt(6))^n*(sqrt(6)-2)-(5-2*sqrt(6))^n*(sqrt(6)+2))^2/16)),n=1..100);
%p A006061 A006061:=-(1+22*z+z**2)/(z-1)/(z**2-98*z+1); # conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation
%t A006061 CoefficientList[Series[(1+22*x+x^2)/((1-x)*(1-98*x+x^2)), {x,0,20}], x] (* or *) LinearRecurrence[{99,-99,1}, {1,121,11881}, 20] (* _G. C. Greubel_, Jul 23 2019 *)
%o A006061 (PARI) my(x='x+O('x^20)); Vec((1+22*x+x^2)/((1-x)*(1-98*x+x^2))) \\ _G. C. Greubel_, Jul 23 2019
%o A006061 (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+22*x+x^2)/((1-x)*(1-98*x+x^2)) )); // _G. C. Greubel_, Jul 23 2019
%o A006061 (Sage) ((1+22*x+x^2)/((1-x)*(1-98*x+x^2))).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 23 2019
%o A006061 (GAP) a:=[1,121,11881];; for n in [4..20] do a[n]:=99*a[n-1]-99*a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Jul 23 2019
%Y A006061 A007667 is 3*a(n)+2, sqrt(a(n)) is A054320.
%Y A006061 Cf. A003154.
%K A006061 nonn,easy
%O A006061 1,2
%A A006061 _N. J. A. Sloane_
%E A006061 More terms from _Eric W. Weisstein_ and _Sascha Kurz_, Mar 24 2002