A006061 Star numbers (A003154) that are squares.

1, 121, 11881, 1164241, 114083761, 11179044361, 1095432263641, 107341182792481, 10518340481399521, 1030690025994360601, 100997104206965939401, 9896685522256667700721, 969774184076946468731281

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			a(2)=121 because this is the 2nd star number (A003154) that is a square.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 121, p. 42, Ellipses, Paris 2008.
• M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 22.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..500
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Eric Weisstein's World of Mathematics, Star Number
• Index entries for linear recurrences with constant coefficients, signature (99, -99, 1).

Crossrefs

A007667 is 3*a(n)+2, sqrt(a(n)) is A054320.

Cf. A003154.

Programs

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

Magma

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+22*x+x^2)/((1-x)*(1-98*x+x^2)) )); // G. C. Greubel, Jul 23 2019

Maple

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);
A006061:=-(1+22*z+z**2)/(z-1)/(z**2-98*z+1); # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

Mathematica

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 *)

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

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

Formula

A007667 = 3*square star numbers (A006061) + 2.

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

From Ignacio Larrosa Cañestro, Feb 27 2000: (Start)

a(n) = 99*(a(n-1) - a(n-2)) + a(n-3).

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)

a(n) = 98*a(n-1) - a(n-2) + 24. - Lekraj Beedassy, Jul 14 2008

Extensions

More terms from Eric W. Weisstein and Sascha Kurz, Mar 24 2002