Let M_n be the n X n matrix m_(i,j) = 4 + abs(i-j) then det(M_n) = (-1)^(n+1)*a(n+2). - Benoit Cloitre, May 28 2002
Number of ordered pairs of (possibly empty) ordered partitions, each not beginning with 1. - Christian G. Bower, Jan 23 2004
If X_1, X_2, ..., X_n are 2-blocks of a (2n+4)-set X then, for n>=1, a(n+3) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 18 2007
Except for an initial 1, this is the p-INVERT of (1,1,1,1,1,...) for p(S) = (1 - S^2)^2; see A291000. - Clark Kimberling, Aug 24 2017
Conjecture 1: For compositions of n+k-1, a(n) is the number of runs of 1 of length k. Example: Among the compositions of 4+2-1 = 5, there are a(4) = 4 runs of two 1's: 3,[1,1]; [1,1],3; 1,2,[1,1] and [1,1],2,1. - Gregory L. Simay, Feb 18 2018
Conjecture 2: More generally, let R(n,m,k) = the number of runs of k m's in all compositions of n. Then R(n,m,k) = A045623(n-m*k) - 2*A045623(n-m*(k+1)) + A045623(n-m*(k+2)). For example, R(7,1,1) = A045623(6) - 2*A045623(5) + A045623(4) = 144 - 2*64 + 28 = 44 = a(7). - Gregory L. Simay, Feb 20 2018
