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A156066
Numbers n with property that n^2 is a square arising in A154138.
2
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
1,1
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
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.
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
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
KEYWORD
nonn,easy
AUTHOR
Zak Seidov, Oct 21 2009
STATUS
approved