OFFSET
1,2
COMMENTS
Also the number of different strings of length n+3 obtained from "123...n" by iteratively duplicating any substring (see A137743 for comments and examples). This is the principal (although not simplest) definition of this sequence and explains why a(1)=1 and not 0.
For n >= 3, sequence appears (not yet proved by induction) to give the number of multiplications between two nonzero matrix elements in calculating the product of two n X n Hessenberg matrices (square matrices which have 0's below the subdiagonal, other elements being in general nonzero). - John M. Coffey, Jun 21 2016
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
From Bruno Berselli, Aug 23 2011: (Start)
G.f.: x*(1+4*x-5*x^2+x^4)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(-n-7) = -A000297(n). (End)
From Ilya Gutkovskiy, Jul 01 2016: (Start)
E.g.f.: 4 + x + (-24 + 24*x + 12*x^2 + x^3)*exp(x)/6.
Sum_{n>=1} 1/a(n) = 1542/1225. (End)
a(n) = binomial(n+4,3) - 2*(n+4) for n > 1. - Michael Chu, Dec 09 2021
EXAMPLE
a(5) = (5-1)*(5+4)*(5+6)/6 = 4*9*11/6 = 66. - Michael B. Porter, Jul 02 2016
MATHEMATICA
Join[{1}, Table[Binomial[n, 3]-2*n, {n, 6, 60}]] (*or*) Join[{1}, Table[(n-1)(n+4)(n+6)/6, {n, 2, 56}]] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)
PROG
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
M. F. Hasler, Feb 10 2008
STATUS
approved