OFFSET
0,3
COMMENTS
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
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Frank Ellermann, Illustration of binomial transforms.
Milan Janjic, Two Enumerative Functions.
Milan Janjic and Boris Petkovic, A Counting Function, arXiv:1301.4550 [math.CO], 2013.
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Index entries for linear recurrences with constant coefficients, signature (4,-4).
FORMULA
a(n) = Sum_{k = 0..n-3} (k+4)*binomial(n-3,k) for n >= 3. - N. J. A. Sloane, Jan 30 2008
a(n) = (n+5)*2^(n-4), n >= 3; a(0)=1, a(1)=0, a(2)=2.
G.f.: ((1-x)^2/(1-2*x))^2.
a(n) = Sum_{k=0..n} (k+1)*C(n-3,n-k). - Peter Luschny, Apr 20 2015
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=2} 1/a(n) = 512*log(2) - 74327/210.
Sum_{n>=2} (-1)^n/a(n) = 14579/70 - 512*log(3/2). (End)
E.g.f.: (1/16)*(11 - 12*x + 2*x^2 + (5+2*x)*exp(2*x)). - G. C. Greubel, Sep 27 2022
MATHEMATICA
Join[{1, 0, 2, a=4}, Table[a=(2*(n+7)*a)/(n+6), {n, 2, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 15 2011 *)
PROG
(PARI) a(n)=if(n<3, [1, 0, 2][n+1], (n+5)*2^(n-4)) \\ Charles R Greathouse IV, Jun 01 2011
(Magma) [1, 0, 2] cat [(n+5)*2^(n-4): n in [3..30]]; // G. C. Greubel, Sep 27 2022
(SageMath) [1, 0, 2]+[(n+5)*2^(n-4) for n in range(3, 30)] # G. C. Greubel, Sep 27 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
STATUS
approved