A006061 Star numbers (A003154) that are squares. (Formerly M5385)

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

Offset

1,2

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

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

Example

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

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 *)

Prog

(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
(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
(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
(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

Crossrefs

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

Cf. A003154.

Sequence in context: A263819 A036508 A054319 * A079215 A137466 A062689

Adjacent sequences: A006058 A006059 A006060 * A006062 A006063 A006064

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved