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)
The first 6 values are the dimensions of the polynomial ring in 3n variables xi, yi, zi for 1 <= i <= n modulo the ideal generated by x1^a y1^b z1^c + ... + xn^a yn^b zn^c for 0 < a+b+c <= n (see Fact 2.8.1 in Haiman's paper). - Mike Zabrocki, Dec 31 2019
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, https://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, and 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.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, preprint, 1993.
M. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), no. 1, 17--76.
O. Khorunzhiy, Enumeration of tree-type diagrams assembled from oriented chains of edges, arXiv:2207.00766 [math.CO], 2022.
Andrew Snowden, Measures for the colored circle, arXiv:2302.08699 [math.CO], 2023.
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.
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.
Diagonal 1 of A195739.
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 23 2007
EXTENSIONS
Slightly edited by Gill Barequet, May 24 2011
STATUS
approved