This site is supported by donations to The OEIS Foundation. Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A001752 Expansion of 1/((1+x)*(1-x)^5). 32
 1, 4, 11, 24, 46, 80, 130, 200, 295, 420, 581, 784, 1036, 1344, 1716, 2160, 2685, 3300, 4015, 4840, 5786, 6864, 8086, 9464, 11011, 12740, 14665, 16800, 19160, 21760, 24616, 27744, 31161, 34884, 38931, 43320, 48070, 53200, 58730, 64680, 71071, 77924, 85261 (list; graph; refs; listen; history; text; internal format)
 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..10000 Dragomir Z. Djokovic, Poincaré series [or Poincare series] of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 4. 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 Convolution of A000217 and A004526. - R. J. Mathar, Mar 29 2018 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 Cf. A057524, A056885, A152205, A000929, A158454 (signed column k=2), A216172, A216173, A216175, A001753 (partial sums), A002623 (first differences). Sequence in context: A006527 A167875 A057304 * A160860 A192748 A143075 Adjacent sequences:  A001749 A001750 A001751 * A001753 A001754 A001755 KEYWORD nonn,easy AUTHOR EXTENSIONS Formulae corrected by Bruno Berselli, Sep 13 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)