A171860 -id:A171860 - cp's OEIS frontend

A127670 Discriminants of Chebyshev S-polynomials A049310.

1, 4, 32, 400, 6912, 153664, 4194304, 136048896, 5120000000, 219503494144, 10567230160896, 564668382613504, 33174037869887488, 2125764000000000000, 147573952589676412928, 11034809241396899282944, 884295678882933431599104, 75613185918270483380568064

Offset: 1

Wolfdieter Lang, Jan 23 2007

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

			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

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 and 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.

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.
Rigoberto 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

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.

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

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

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

Slightly edited by Gill Barequet, May 24 2011

A195739 Triangle read by rows: DX(n,d) = number of properly d-dimensional polyominoes with n cells, modulo translations (n>=1, 0 <= d <= n-1).

Original entry on oeis.org

1, 0, 1, 0, 1, 4, 0, 1, 17, 32, 0, 1, 61, 348, 400, 0, 1, 214, 2836, 8640, 6912, 0, 1, 758, 21225, 129288, 254800, 153664, 0, 1, 2723, 154741, 1688424, 6160640, 8749056, 4194304, 0, 1, 9908, 1123143, 20762073, 125055400, 313921008, 343901376, 136048896

Offset: 1

N. J. A. Sloane, Sep 23 2011

According to Barequet-Barequet-Rote, p. 261, the value DX(7, 6) = 134209 given by W. F. Lunnon is incorrect; it should be 153664, see A127670. - Alexander Knapp, May 13 2013

			Triangle begins with DX(1,0):
n\d 0  1   2     3      4      5      6
---------------------------------------
1...1
2...0  1
3...0  1   4
4...0  1  17    32
5...0  1  61   348    400
6...0  1 214  2836   8640   6912
7...0  1 758 21225 129288 254800 153664
...

Robert A. Russell, Table of n, a(n) for n = 1..60
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica 30 (2010), pp. 257-275.
W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (1975), no. 4, pp. 366-367.

Columns give A006762, A006763, A006764. Cf. A195738, A049430.

Diagonals (with formulas) are A127670, A171860, A191092, A259015, A290738.

A191092 Number of n-cell polycubes that are proper in n-3 dimensions.

Original entry on oeis.org

0, 1, 61, 2836, 129288, 6160640, 313921008, 17239040000, 1021644763392, 65244849242112, 4477975127425280, 329252714454974464, 25850313756000000000, 2160223055912342913024, 191558954408834121740288, 17973564914103712921681920

Offset: 3

Gill Barequet, May 25 2011

A. Asinowski, G. Barequet, R. Barequet, and G. Rote, Proper n-cell polycubes in n-3 dimensions, Proc. 17th Ann. Int. Computing and Combinatorics Conference, Dallas, TX, Lecture Notes in Computer Science, 6842, Springer-Verlag, 180-191, August 2011.
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.
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275.

Vincenzo Librandi, Table of n, a(n) for n = 3..100
A. Asinowski, G. Barequet, R. Barequet, and G. Rote, Proper n-Cell Polycubes in n-3 Dimensions, J. Int. Seq. 15 (2012) #12.8.4.
M. Shalah, Formulae and growth rates of animals on cubical and triangular lattices, PhD Thesis, Israel Inst. Techn. (2017)

Cf. A127670, A171860.

Diagonal 3 of A195739.

[2^(n-6)*n^(n-7)*(n-3)*(12*n^5-104*n^4+360*n^3-679*n^2+1122*n-1560)/3: n in [3..40]]; // Vincenzo Librandi, May 26 2011

a[n_]:=2^(n-6)*n^(n-7)*(n-3)*(12*n^5 - 104*n^4 + 360*n^3 - 679*n^2 + 1122*n - 1560)/3 ; Array[a, 40, 3] (* Stefano Spezia, Sep 09 2018 *)

a(n)=2^(n-6)*n^(n-7)*(n-3)*(12*n^5-104*n^4+360*n^3-679*n^2+1122*n-1560)/3

a(n) = 2^(n-6)*n^(n-7)*(n-3)*(12*n^5 - 104*n^4 + 360*n^3 - 679*n^2 + 1122*n - 1560)/3.

A036364 Number of free n-ominoes with cell centers determining n-2 space (proper dimension n-2).

Original entry on oeis.org

1, 4, 11, 35, 104, 319, 951, 2862, 8516, 25369, 75167, 222529, 656961, 1937393, 5704426, 16781247, 49320800, 144866243, 425263010, 1247877578, 3660478408, 10734834603, 31475111515, 92273758477, 270486112046, 792836030163, 2323835125879, 6811162237825

Offset: 3

Robert A. Russell

Lunnon's DE(n,n-2); Lunnon's DE(n,n-1) is number of free trees.

			1 tromino in 1-space;
4 nonstraight tetrominoes in 2-space;
11 nonflat pentominoes in 3-space (chiral pairs count as one).

Vaclav Kotesovec, Table of n, a(n) for n = 3..1000 (terms 3..200 from Vincenzo Librandi)
W. F. Lunnon, Counting Multidimensional Polyominoes, Computer Journal, Vol. 18 (1975), pp. 366-67.

Cf. A000081, A000055, A036365, A171860 (fixed).

sb[ n_, k_ ] := sb[ n, k ]=b[ n+1-k, 1 ]+If[ n<2k, 0, sb[ n-k, k ] ]; b[ 1, 1 ] := 1;
b[ n_, 1 ] := b[ n, 1 ]=Sum[ b[ i, 1 ]sb[ n-1, i ]i, {i, 1, n-1} ]/(n-1);
b[ n_, k_ ] := b[ n, k ]=Sum[ b[ i, 1 ]b[ n-i, k-1 ], {i, 1, n-1} ];
Table[ b[ i, 3 ]/2+5b[ i, 4 ]/8+Sum[ b[ i, j ], {j, 5, i} ]+If[ OddQ[ i ], 0, 7b[ i/2, 2 ]/8
+If[ OddQ[ i/2 ], 0, b[ i/4, 1 ]/4 ]+Sum[ b[ i/2, j ], {j, 3, i/2} ] ]
+Sum[ b[ j, 1 ](b[ i-2j, 1 ]/2+b[ i-2j, 2 ]/4)+Sum[ If[ OddQ[ k ], b[ j,
(k-1)/2 ]b[ i-2j, 1 ], 0 ], {k, 5, i} ], {j, 1, (i-1)/2} ], {i, 3, 30} ]

G.f.: B^3(x)/2 + B(x)B(x^2)/2 + 5B^4(x)/8 + B^2(x)B(x^2)/4 + 7B^2(x^2)/8 + B(x^4)/4 + B^5(x)/(1-B(x)) + (B(x)+B(x^2))B^2(x^2)/(1-B(x^2)), where B(x) is the generating function for rooted trees with n nodes (that is, B(x) is the g.f. of sequence A000081).

a(n) ~ A340310 * A051491^n / sqrt(n). - Vaclav Kotesovec, Apr 12 2021

A355998 Number of fixed orthoplex n-ominoes with cell centers determining (n-2)-space.

Original entry on oeis.org

1, 48, 1728, 62720, 2457600, 105815808, 5017600000, 261227298816, 14860167413760, 918839084134400, 61439672177393700, 4421589120000000000, 340976534987475000000, 28064307240230900000000, 2456376885785930000000000

Offset: 4

Robert A. Russell, Jul 22 2022

nonn

Orthoplex polyominoes are connected sets of cells of regular tilings with Schläfli symbols {}, {4}, {3,4}, {3,3,4}, {3,3,3,4}, etc. These are tilings of regular orthoplexes projected on their circumspheres. Orthoplex polyominoes are equivalent to multidimensional polyominoes that do not extend more than two units along any axis, i.e., fit within a 2^d cube. Two fixed polyominoes are identical only if one is a translation of the other.

			For A(4)=1, all 4 squares of the 2^2 space are used.

Cf. A171860 (multidimensional), A036367 (unoriented), A036368 (chiral), A036369 (asymmetric).

Diagonal 2 of A355997.

Table[2^(n-3) n^(n-5) (n-2) (n-3)^2, {n,4,30}]

a(n) = 2^(n-3) * n^(n-5) * (n-2) * (n-3)^2.

a(n) ~ A171860(n) / 2.

cp's OEIS Frontend

A127670 Discriminants of Chebyshev S-polynomials A049310.

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A195739 Triangle read by rows: DX(n,d) = number of properly d-dimensional polyominoes with n cells, modulo translations (n>=1, 0 <= d <= n-1).

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A191092 Number of n-cell polycubes that are proper in n-3 dimensions.

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A036364 Number of free n-ominoes with cell centers determining n-2 space (proper dimension n-2).

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A355998 Number of fixed orthoplex n-ominoes with cell centers determining (n-2)-space.

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