

A005386


Area of nth triple of squares around a triangle.
(Formerly M3017)


8



1, 3, 16, 75, 361, 1728, 8281, 39675, 190096, 910803, 4363921, 20908800, 100180081, 479991603, 2299777936, 11018898075, 52794712441, 252954664128, 1211978608201, 5806938376875, 27822713276176, 133306628004003, 638710426743841, 3060245505715200
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OFFSET

1,2


COMMENTS

a(n)*(1)^(n+1) is the r=3 member of the rfamily of sequences S_r(n), n>=1, defined in A092184 where more information can be found.
The sequence is the case P1 = 3, P2 = 10, Q = 1 of the 3 parameter family of 4thorder linear divisibility sequences found by Williams and Guy.  Peter Bala, Apr 03 2014


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



FORMULA

G.f.: x*(1x)/((1+x)*(15*x+x^2)).
a(n) = 4*a(n1) + 4*a(n2)  a(n3), a(1)=1, a(2)=3, a(3)=16.
a(n) = (2/7)*(T(n, 5/2)  (1)^n) with twice Chebyshev's polynomials of the first kind evaluated at x=5/2: 2*T(n, 5/2) = A003501(n) = ((5+sqrt(21))^n + (5sqrt(21))^n)/2^n.  Wolfdieter Lang, Oct 18 2004
a(2n) = A003690(n). a(2n+1) = A004253(n)^2.  Alexander Evnin, Mar 11 2012
a(n) = U(n1, sqrt(3)*i/2)^2, where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 5/2; 1, 3/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4thorder linear divisibility sequences. (End)


MAPLE

a:= n> (Matrix([[0, 1, 3]]). Matrix(3, (i, j)> if (i=j1) then 1 elif j=1 then [4, 4, 1][i] else 0 fi)^(n))[1, 1]: seq(a(n), n=1..25); # Alois P. Heinz, Aug 05 2008


MATHEMATICA

a[n_]:= Module[{n1=1, n2=0}, Do[{n1, n2}={Sqrt[3]*n1+n2, n1}, {n1}]; n1^2];
Table[a[n], {n, 30}]
a[n_]:= Round[((5+Sqrt[21])/2)^n/7]; Table[a[n], {n, 30}]
Rest@(CoefficientList[Series[x/(1x*(Sqrt[3]+x)), {x, 0, 30}], x])^2
Abs[ChebyshevU[Range[1, 40]1, I*Sqrt[3]/2]]^2 (* G. C. Greubel, Nov 16 2022 *)


PROG

(Magma) I:=[1, 3, 16]; [n le 3 select I[n] else 4*Self(n1) +4*Self(n2) Self(n3): n in [1..41]]; // G. C. Greubel, Nov 16 2022
(SageMath)
def A005386(n): return abs(chebyshev_U(n1, i*sqrt(3)/2))^2


CROSSREFS



KEYWORD

nonn,easy


AUTHOR

Jean Meeus


EXTENSIONS

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 09 2004


STATUS

approved



