A005386 Area of n-th triple of squares around a triangle. (Formerly M3017)

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

Offset

1,2

Comments

a(n)*(-1)^(n+1) is the r=-3 member of the r-family 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 4th-order 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

G. C. Greubel, Table of n, a(n) for n = 1..1000

J. Meeus, Letter to N. J. A. Sloane with attachment, Mar 1975

J. C. G. Nottrot, Vierkantenkransen rond een driehoek, Pythagoras (Netherlands), 14 (1975-1976) 77-81.

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

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (4,4,-1).

Formula

G.f.: x*(1-x)/((1+x)*(1-5*x+x^2)).

a(n) = 4*a(n-1) + 4*a(n-2) - a(n-3), 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 + (5-sqrt(21))^n)/2^n. - Wolfdieter Lang, Oct 18 2004

a(2n) = A003690(n). a(2n+1) = A004253(n)^2. - Alexander Evnin, Mar 11 2012

From Peter Bala, Apr 03 2014: (Start)

a(n) = |U(n-1, 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 4th-order linear divisibility sequences. (End)

Maple

A005386:=-(-1+z)/(z+1)/(z**2-5*z+1); [Conjectured by Simon Plouffe in his 1992 dissertation.]
a:= n-> (Matrix([[0, 1, 3]]). Matrix(3, (i, j)-> if (i=j-1) 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}, {n-1}]; 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/(1-x*(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(n-1) +4*Self(n-2) -Self(n-3): n in [1..41]]; // G. C. Greubel, Nov 16 2022
(SageMath)
def A005386(n): return abs(chebyshev_U(n-1, i*sqrt(3)/2))^2
[A005386(n) for n in range(1, 40)] # G. C. Greubel, Nov 16 2022

Crossrefs

Essentially the same as A003769.

First differences of A099025.

Cf. A100047.

Sequence in context: A207836 A005947 A003769 * A053572 A329806 A371350

Adjacent sequences: A005383 A005384 A005385 * A005387 A005388 A005389

Keyword

nonn,easy

Author

Jean Meeus

Extensions

Edited by Peter J. C. Moses, Apr 23 2004

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

Status

approved