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A156066

Numbers k such that k^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

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OFFSET

1,1

COMMENTS

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

From Vladimir Pletser, Sep 10 2025: (Start)

2*a(n) are the first numbers A of two numbers (A, B), such that the difference 2*A^2 - B^2 = 23. A387549(n) gives the B-values.

2*a(n) gives the y-values solving the Diophantine equation x^2 + 23 = 2*y^2 or the Pell equation x^2 - 2*y^2 = -23. A387549(n) gives the x-values. (End)

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).

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

From Vladimir Pletser, Sep 10 2025: (Start)

a(n) = a(n-1) + (A387549(n-1) + (-1)^(n-1)*A048655(n-2))/2 for n > 1.

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

a(n)*a(n+3)-a(n+1)*a(n+2) = (34+14*(-1)^n)/2. (End)

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 *)

PROG

(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

(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

(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

CROSSREFS

Cf. A154138.

Sequence in context: A324014 A289452 A086771 * A143890 A270339 A272057

Adjacent sequences: A156063 A156064 A156065 * A156067 A156068 A156069

KEYWORD

nonn,easy

AUTHOR

Zak Seidov, Oct 21 2009

STATUS

approved

URL: https://oeis.org/A156066

⇱ A156066 - OEIS