OFFSET
0,2
COMMENTS
A sequence generated from the Bell difference row triangle (as a matrix).
Companion sequence A095151 has the same recursion rule but is generated from the multiplier [1 0 0] instead of [1 1 1].
a(n) is the sum of the terms in row n of a triangle with first column T(n,0) = (n+1)*(n+2)/2 and diagonal T(n,n) = n+1; T(i,j) = T(i-1,j-1) + T(i-1,j). - J. M. Bergot, Jun 26 2018
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
FORMULA
Let M = a 3 X 3 matrix formed from A095149 rows (fill in with zeros): {1, 0, 0 ; 1, 1, 0 ; 2, 1, 2}. Then M^n * {1, 1, 1} = {1, n+1, a(n)}.
From Colin Barker, Apr 23 2012: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: (1+x-x^2)/((1-x)^2*(1-2*x)). (End)
EXAMPLE
a(9) = 2547 = 3*a(8) - 2*a(7) + 1 = 3*1268 - 2*629 + 1 = 3804 - 1258 + 1.
MATHEMATICA
a[n_]:= (MatrixPower[{{1, 0, 0}, {1, 1, 0}, {2, 1, 2}}, n].{{1}, {1}, {1}})[[3, 1]]; Table[a[n], {n, 35}] (* Robert G. Wilson v, Jun 01 2004 *)
LinearRecurrence[{4, -5, 2}, {1, 5, 14}, 40] (* Harvey P. Dale, Jan 20 2021 *)
PROG
(PARI) vector(35, n, 5*2^(n-1) -(n+3)) \\ Harry J. Smith, Jun 16 2009; edited Dec 27 2021
(Magma) [5*2^n -(n+4): n in [0..35]]; // G. C. Greubel, Dec 27 2021
(Sage) [5*2^n -(n+4) for n in (0..35)] # G. C. Greubel, Dec 27 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, May 30 2004
EXTENSIONS
More terms from Robert G. Wilson v, Jun 01 2004
STATUS
approved