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A127670 Discriminants of Chebyshev S-polynomials A049310. 21
1, 4, 32, 400, 6912, 153664, 4194304, 136048896, 5120000000, 219503494144, 10567230160896, 564668382613504, 33174037869887488, 2125764000000000000, 147573952589676412928, 11034809241396899282944, 884295678882933431599104 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n-1) is the number of fixed n-cell polycubes that are proper in n - 1 dimensions (Barequet et al., 2010).

From Rigoberto Florez, Sep 02 2018: (Start)

a(n-1) is the discriminant of the Morgan-Voyce Fibonacci-type polynomial B(n).

Morgan-Voyce Fibonacci-type polynomials are defined as B(0) = 0, B(1) = 1 and B(n) = (x + 2)*B(n-1) - B(n-2) for n > 1.

The absolute value of the discriminant of Fibonacci polynomial F(n) is a(n-1).

Fibonacci polynomials are defined as F(0) = 0, F(1) = 1 and F(n) = x*F(n-1) + F(n-2) for n > 1. (End)

REFERENCES

Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf

G. Barequet, M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes, 31st International Symposium on Computational Geometry (SoCG'15).  Editors: Lars Arge and János Pach; pp. 19-22, 2015.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990; p. 219 for T and U polynomials.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Andrei Asinowski, Gill Barequet, Ronnie Barequet, Gunter Rote, Proper n-Cell Polycubes in n - 3 Dimensions, Journal of Integer Sequences, Vol. 15 (2012), #12.8.4.

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012. See Th. 1. [From N. J. A. Sloane, Oct 16 2010]

R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30 (2010), pp. 257-275.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.6.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.

R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.

Eric Weisstein's World of Mathematics, Discriminant

Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials

Eric Weisstein's World of Mathematics, Fibonacci Polynomial

FORMULA

a(n) = ((n+1)^(n-2))*2^n, n >= 1.

a(n) = (Det(Vn(xn[1],...,xn[n]))^2 with the determinant of the Vandermonde matrix Vn with elements (Vn)i,j:= xn[i]^j, i=1..n, j=0..n-1 and xn[i]:=2*cos(Pi*i/(n+1)), i=1..n, are the zeros of S(n,x):=U(n,x/2).

a(n) = ((-1)^(n*(n-1)/2))*Product_{j=1..n} ((d/dx)S(n,x)|_{x=xn[j]}), n >= 1, with the zeros xn[j], j=1..n, given above.

a(n) = A007830(n-2)*A000079(n), n >= 2. - Omar E. Pol, Aug 27 2011

E.g.f.: -LambertW(-2*x)*(2+LambertW(-2*x))/(4*x). - Vaclav Kotesovec, Jun 22 2014

EXAMPLE

n=3: The zeros are [sqrt(2),0,-sqrt(2)]. The Vn(xn[1],...,xn[n]) matrix is [[1,1,1],[sqrt(2),0,-sqrt(2)],[2,0,2]]. The squared determinant is 32 = a(3). - Wolfdieter Lang, Aug 07 2011

MATHEMATICA

Table[((n + 1)^n)/(n + 1)^2 2^n, {n, 1, 30}] (* Vincenzo Librandi, Jun 23 2014 *)

PROG

(MAGMA) [((n+1)^n/(n+1)^2)*2^n: n in [1..20]]; // Vincenzo Librandi, Jun 23 2014

CROSSREFS

Cf. A007701 (T-polynomials), A086804 (U-polynomials), A171860 and A191092 (fixed n-cell polycubes proper in n-2 and n-3 dimensions, resp.).

Cf. A243953, A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

A317403 is essentially the same sequence.

Sequence in context: A005263 A113131 A195762 * A317403 A243468 A317677

Adjacent sequences:  A127667 A127668 A127669 * A127671 A127672 A127673

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jan 23 2007

EXTENSIONS

Slightly edited by Gill Barequet, May 24 2011

STATUS

approved

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Last modified October 19 20:05 EDT 2018. Contains 316378 sequences. (Running on oeis4.)