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A005386 Area of n-th triple of squares around a triangle.
(Formerly M3017)
7
1, 3, 16, 75, 361, 1728, 8281, 39675, 190096, 910803, 4363921, 20908800, 100180081, 479991603, 2299777936, 11018898075, 52794712441, 252954664128, 1211978608201, 5806938376875, 27822713276176, 133306628004003 (list; graph; refs; listen; history; text; internal format)
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

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

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

LINKS

Table of n, a(n) for n=1..22.

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for sequences related to Chebyshev polynomials.

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

FORMULA

G.f.: x*(1-x)/(x^3-4*x^2-4*x+1), a(n)=4*(a(n-1)+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..22); # 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] a[n_]:=Round[((5+Sqrt[21])/2)^n/7] (CoefficientList[Series[{(x/(1-x*(Sqrt[3]+x)))}, {x, 0, 20}], x])^2 CoefficientList[Series[{x*(1-x)/(x^3-4*x^2-4*x+1)}, {x, 0, 20}], x]

CROSSREFS

Essentially the same as A003769. First differences of A099025. A100047.

Sequence in context: A317365 A207836 A005947 * A003769 A053572 A055842

Adjacent sequences:  A005383 A005384 A005385 * A005387 A005388 A005389

KEYWORD

nonn

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

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Last modified October 21 19:27 EDT 2018. Contains 316428 sequences. (Running on oeis4.)