Number of edges in an n-dimensional hypercube.
Number of 132-avoiding permutations of [n+2] containing exactly one 123 pattern. -
Emeric Deutsch, Jul 13 2001
Number of ways to place n-1 nonattacking kings on a 2 X 2(n-1) chessboard for n >= 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001
(-1) times the determinant of matrix A_{i,j} = -|i-j|, 0 <= i,j <= n.
a(n) is the number of ones in binary numbers 1 to 111...1 (n bits). a(n) =
A000337(n) -
A000337(n-1) for n = 2,3,... . -
Emeric Deutsch, May 24 2003
The number of 2 X n 0-1 matrices containing n+1 1's and having no zero row or column. The number of spanning trees of the complete bipartite graph K(2,n). This is the case m = 2 of K(m,n). See
A072590. -
W. Edwin Clark, May 27 2003
Binomial transform of 0,1,2,3,4,5,... (
A001477). Without the initial 0, binomial transform of odd numbers.
With an additional leading zero, [0,0,1,4,...] this is the binomial transform of the integers repeated
A004526. Its formula is then (2^n*(n-1) + 0^n)/4. -
Paul Barry, May 20 2003
Number of zeros in all different (n+1)-bit integers. -
Ralf Stephan, Aug 02 2003
Final element of a summation table (as opposed to a difference table) whose first row consists of integers 0 through n (or first n+1 nonnegative integers
A001477); illustrating the case n=5:
0 1 2 3 4 5
1 3 5 7 9
4 8 12 16
12 20 28
32 48
80
and the final element is a(5)=80. (End)
This sequence and
A001871 arise in counting ordered trees of height at most k where only the rightmost branch at the root actually achieves this height and the count is by the number of edges, with k = 3 for this sequence and k = 4 for
A001871.
Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all elements x,y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. Then a(n) = |R|. -
Ross La Haye, Sep 21 2004
Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1) and (10;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in same relative order as those in the triple (x,y,z). -
Sergey Kitaev, Nov 11 2004
Number of subsequences 00 in all binary words of length n+1. Example: a(2)=4 because in 000,001,010,011,100,101,110,111 the sequence 00 occurs 4 times. -
Emeric Deutsch, Apr 04 2005
If you expand the n-factor expression (a+1)*(b+1)*(c+1)*...*(z+1), there are a(n) variables in the result. For example, the 3-factor expression (a+1)*(b+1)*(c+1) expands to abc+ab+ac+bc+a+b+c+1 with a(3) = 12 variables. -
David W. Wilson, May 08 2005
An inverse Chebyshev transform of n^2, where g(x)->(1/sqrt(1-4*x^2))*g(x*c(x^2)), c(x) the g.f. of
A000108. -
Paul Barry, May 13 2005
The number of never-decreasing positive integer sequences of length n with a maximum value of 2*n. -
Ben Paul Thurston, Nov 13 2006
Total size of all the subsets of an n-element set. For example, a 2-element set has 1 subset of size 0, 2 subsets of size 1 and 1 of size 2. -
Ross La Haye, Dec 30 2006
If M is the matrix (given by rows) [2,1;0,2] then the sequence gives the (1,2) entry in M^n. -
Antonio M. Oller-Marcén, May 21 2007
If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n > 0, a(n) is equal to the number of (n+1)-subsets of X intersecting each X_i (i=1,2,...,n). -
Milan Janjic, Jul 21 2007
Number of n-permutations of 3 objects u,v,w, with repetition allowed, containing exactly one u. Example: a(2)=4 because we have uv, vu, uw and wu. -
Zerinvary Lajos, Dec 27 2007
A member of the family of sequences defined by a(n) = n*[c(1)*...*c(r)]^(n-1); c(i) integer. This sequence has c(1)=2,
A027471 has c(1)=3. -
Ctibor O. Zizka, Feb 23 2008
a(n) is the number of ways to split {1,2,...,n-1} into two (possibly empty) complementary intervals {1,2,...,i} and {i+1,i+2,...,n-1} and then select a subset from each interval. -
Geoffrey Critzer, Jan 31 2009
Starting with offset 1 =
A059570: (1, 2, 6, 14, 34, ...) convolved with (1, 2, 2, 2, ...). -
Gary W. Adamson, May 23 2009
The number of tatami tilings of an n X n square with n monomers is n*2^(n-1). -
Frank Ruskey, Sep 25 2010
Number of Dyck (n+2)-paths with exactly one valley at height 1 and no higher valley. -
David Scambler, Nov 07 2011
Let T(n,k) be the triangle with (first column) T(n,1) = 2*n-1 for n >= 1, otherwise T(n,k) = T(n,k-1) + T(n-1,k-1), then a(n) = T(n,n). -
J. M. Bergot, Jan 17 2013
Sum of all parts of all compositions (ordered partitions) of n. The equivalent sequence for partitions is
A066186. -
Omar E. Pol, Aug 28 2013
Starting with a(1)=1: powers of 2 (
A000079) self-convolved. -
Bob Selcoe, Aug 05 2015
Coefficients of the series expansion of the normalized Schwarzian derivative -S{p(x)}/6 of the polynomial p(x) = -(x-x1)*(x-x2) with x1 + x2 = 1 (cf.
A263646). -
Tom Copeland, Nov 02 2015
a(n) is the number of North-East lattice paths from (0,0) to (n+1,n+1) that have exactly one east step below y = x-1 and no east steps above y = x+1. Details can be found in Pan and Remmel's link. -
Ran Pan, Feb 03 2016
Also the number of maximal and maximum cliques in the n-hypercube graph for n > 0. -
Eric W. Weisstein, Dec 01 2017
Let [n]={1,2,...,n}; then a(n-1) is the total number of elements missing in proper subsets of [n] that contain n to form [n]. For example, for n = 3, a(2) = 4 since the proper subsets of [3] that contain 3 are {3}, {1,3}, {2,3} and the total number of elements missing in these subsets to form [3] is 4: 2 in the first subset, 1 in the second, and 1 in the third. -
Enrique Navarrete, Aug 08 2020
Number of 3-permutations of n elements avoiding the patterns 132, 231. See Bonichon and Sun. -
Michel Marcus, Aug 19 2022
Number of ternary strings of length n with exactly one 0 and no restriction on the number of 1's and 2's. -
Enrique Navarrete, Oct 20 2025
For n = 3 and n > 5, also 1/5 of the number cubes in the n-cube-connected cycle graph. -
Eric W. Weisstein, Mar 13 2026
Number of pairs (x,X), where x is an element of X and X is a subset of {1,..,n}. (If x is an element of the n-set and x is in X, there are 2^(n-1) choices for the remaining n-1 elements of the n-set (include them or not in X). Hence there are 2^(n-1) subsets that contain x. Since we can choose x in n ways, the total count is n*2^(n-1)). E.g. n=3: 1 appears in {1}, {1,2}, {1,3} and {1,2,3}, and similarly for 2 and 3, hence a(3) = 3*2^2 = 12. -
Enrique Navarrete, Mar 16 2026
