OFFSET
0,2
COMMENTS
Define a unit column of a binary matrix to be a column with only one 1. a(n) = number of 3 X n binary matrices with 1 unit column up to row and column permutations (if offset is 1). - Vladeta Jovovic, Sep 09 2000
Generally, number of 3 X n binary matrices with k=0,1,2,... unit columns, up to row and column permutations, is the coefficient of x^k in 1/6*(Z(S_n; 5 + 3*x,5 + 3*x^2, ...) + 3*Z(S_n; 3 + x,5 + 3*x^2,3 + x^3,5 + 3*x^4, ...) + 2*Z(S_n; 2,2,5 + 3*x^3,2,2,5 + 3*x^6, ...)), where Z(S_n; x_1,x_2,...,x_n) is the cycle index of symmetric group S_n of degree n.
First differences of a(n) give number of minimal 3-covers of an unlabeled n-set that cover 4 points of that set uniquely (if offset is 4).
Transform of tetrahedral numbers, binomial(n+3,3), under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005
Equals triangle A152205 as an infinite lower triangular matrix * [1, 2, 3, ...]. - Gary W. Adamson, Feb 14 2010
With a leading zero, number of all possible octahedra of any size, formed when intersecting a regular tetrahedron by planes parallel to its sides and dividing its edges into n equal parts. - V.J. Pohjola, Sep 13 2012
With 2 leading zeros and offset 1, the sequence becomes 0,0,1,4,11,... for n=1,2,3,... Call this b(n). Consider the partitions of n into two parts (p,q) with p <= q. Then b(n) is the total volume of the family of rectangular prisms with dimensions p, |q - p| and |q - p|. - Wesley Ivan Hurt, Apr 14 2018
Conjecture: For n > 2, a(n-3) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of an n X n square matrix M(n) defined as the n-th principal submatrix of the array A010751 whose general element is given by M[i,j] = floor((j - i + 1)/2). - Stefano Spezia, Jan 12 2022
Consider the following drawing of the complete graph on n vertices K_n: Vertices 1, 2, ..., n are on a straight line. Any pair of nonconsecutive vertices (i, j) with i < j is connected by a semicircle that goes above the line if i is odd, and below if i is even. With four leading zeros and offset 1, a(n) gives the number of edge crossings of the aforementioned drawing of K_n. - Carlo Francisco E. Adajar, Mar 17 2022
REFERENCES
T. A. Saaty, The Minimum Number of Intersections in Complete Graphs, Proc. Natl. Acad. Sci. USA., 52 (1964), 688-690.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Dragomir Z. Djokovic, Poincaré series of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 4.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 17.
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).
FORMULA
a(n) = floor(((n+3)^2 - 1)*((n+3)^2 - 3)/48).
G.f.: 1/((1+x)*(1-x)^5).
a(n) - 2*a(n-1) + a(n-2) = A002620(n+2).
a(n) + a(n-1) = A000332(n+4).
a(n) - a(n-2) = A000292(n+1).
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+4, 4). - Paul Barry, Jul 01 2003
a(n) = (3*(-1)^n + 93 + 168*n + 100*n^2 + 24*n^3 + 2*n^4)/96. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004
From Paul Barry, Apr 16 2005: (Start)
a(n) = Sum_{k=0..floor(n/2)} binomial(n-2k+3, 3).
a(n) = Sum_{k=0..n} binomial(k+3, 3)*(1-(-1)^(n+k-1))/2. (End)
a(n) = A108561(n+5,n) for n > 0. - Reinhard Zumkeller, Jun 10 2005
From Wesley Ivan Hurt, Apr 01 2015: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 5*(n-4) - 4*a(n-5) + a(n-6).
a(n) = Sum_{i=0..n+3} (n+3-i) * floor(i^2/2)/2. (End)
Boas-Buck recurrence: a(n) = (1/n)*Sum_{p=0..n-1} (5 + (-1)^(n-p))*a(p), n >= 1, a(0) = 1. See the Boas-Buck comment in A158454 (here for the unsigned column k = 2 with offset 0). - Wolfdieter Lang, Aug 10 2017
E.g.f.: ((48 + 147*x + 93*x^2 + 18*x^3 + x^4)*cosh(x) + (45 + 147*x + 93*x^2 + 18*x^3 + x^4)*sinh(x))/48. - Stefano Spezia, Jan 12 2022
EXAMPLE
There are 4 binary 3 X 2 matrices with 1 unit column up to row and column permutations:
[0 0] [0 0] [0 1] [0 1]
[0 0] [0 1] [0 1] [0 1]
[0 1] [1 1] [1 0] [1 1].
For n=5, the numbers of the octahedra, starting from the smallest size, are Te(5)=35, Te(3)=10, and Te(1)=1, the sum being 46. Te denotes the tetrahedral number A000292. - V.J. Pohjola, Sep 13 2012
MAPLE
A001752:=n->(3*(-1)^n+93+168*n+100*n^2+24*n^3+2*n^4)/96:
seq(A001752(n), n=0..50); # Wesley Ivan Hurt, Apr 01 2015
MATHEMATICA
a = {1, 4}; Do[AppendTo[a, a[[n - 2]] + (n*(n + 1)*(n + 2))/6], {n, 3, 10}]; a
(* Number of octahedra *) nnn = 100; Teo[n_] := (n - 1) n (n + 1)/6
Table[Sum[Teo[n - nn], {nn, 0, n - 1, 2}], {n, 1, nnn}]
(* V.J. Pohjola, Sep 13 2012 *)
LinearRecurrence[{4, -5, 0, 5, -4, 1}, {1, 4, 11, 24, 46, 80}, 50] (* Harvey P. Dale, Feb 07 2019 *)
PROG
(PARI) a(n)=if(n<0, 0, ((n+3)^2-1)*((n+3)^2-3)/48-if(n%2, 1/16))
(PARI) a(n)=(n^4+12*n^3+50*n^2+84*n+46)\/48 \\ Charles R Greathouse IV, Sep 13 2012
(Magma) [Floor(((n+3)^2-1)*((n+3)^2-3)/48): n in [0..40]]; // Vincenzo Librandi, Aug 15 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Formulae corrected by Bruno Berselli, Sep 13 2012
STATUS
approved