A156066 Numbers n with property that n^2 is a square arising in A154138.

2, 3, 9, 16, 52, 93, 303, 542, 1766, 3159, 10293, 18412, 59992, 107313, 349659, 625466, 2037962, 3645483, 11878113, 21247432, 69230716, 123839109, 403506183, 721787222, 2351806382, 4206884223, 13707332109, 24519518116, 79892186272

Offset: 1

Author

Zak Seidov, Oct 21 2009

Keywords

• nonn
• easy

Comments

Except for the first term, positive values of x (or y) satisfying x^2 - 6xy + y^2 + 23 = 0. - Colin Barker, Feb 08 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Counting families of generalized balancing numbers, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Crossrefs

Cf. A154138.

Programs

GAP

a:=[2,3,9,16];; for n in [5..30] do a[n]:=6*a[n-2]-a[n-4]; od; a; # Muniru A Asiru, Sep 28 2018

Magma

I:=[2,3,9,16]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 11 2014

Maple

seq(coeff(series(-x*(x-1)*(x+2)*(2*x+1)/((x^2-2*x-1)*(x^2+2*x-1)),x,n+1), x, n), n = 1..30); # Muniru A Asiru, Sep 28 2018

Mathematica

a[1]=2;a[2]=3;a[3]=9;a[4]=16;a[n_]:=a[n]=6*a[n-2]-a[n-4];A1=Table[a[n],{n,25}]
CoefficientList[Series[-(x - 1) (x + 2) (2 x + 1)/((x^2 - 2 x - 1) (x^2 + 2 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 11 2014 *)

PARI

Vec(-x*(x-1)*(x+2)*(2*x+1)/((x^2-2*x-1)*(x^2+2*x-1)) + O(x^100)) \\ Colin Barker, Feb 08 2014

Formula

a(n) = sqrt((A154138(n)^2 + A154138(n) + 6)/2).

a(1..4) = (2,3,9,16); a(n>4) = 6*a(n-2) - a(n-4).

G.f.: -x*(x-1)*(x+2)*(2*x+1) / ((x^2-2*x-1)*(x^2+2*x-1)). - Colin Barker, Feb 08 2014

a(n) = A006452(n-1) - A006452(n) + A006452(n+1). - Carl Najafi, Sep 27 2018