OFFSET
1,2
COMMENTS
Sequence generated from level 2 of the Pascal tetrahedron.
REFERENCES
Peter Hilton, Derek Holton and Jean Pederson, "Mathematical Vistas, From a Room With Many Windows"; Springer, 2000, p. 178, Fig. 14, "The Pascal Tetrahedron".
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,3,-2).
FORMULA
a(n) = 3*a(n-1) + 3*a(n-2) - 2*a(n-3) (derived from the minimal polynomial of the matrix M).
G.f.: x*(1 + 3*x - 2*x^2) / (1 - 3*x - 3*x^2 + 2*x^3). - Colin Barker, Mar 02 2017
EXAMPLE
a(4) = 73 because M^4 = [73,86,19 / 86,104,24 / 19,24,6]. Alternatively, a(4) = 3*a(3) + 3*a(2) - 2*a(1) = 57+18-2 = 73.
MAPLE
with(linalg): M[1]:=matrix(3, 3, [1, 2, 1, 2, 2, 0, 1, 0, 0]): for n from 2 to 27 do M[n]:=multiply(M[1], M[n-1]) od: seq(M[n][1, 1], n=1..27);
a[1]:=1: a[2]:=6: a[3]:=19: for n from 4 to 27 do a[n]:=3*a[n-1]+3*a[n-2]-2*a[n-3] od: seq(a[n], n=1..27);
PROG
(PARI) Vec(x*(1 + 3*x - 2*x^2) / (1 - 3*x - 3*x^2 + 2*x^3) + O(x^30)) \\ Colin Barker, Mar 02 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Nov 07 2004
EXTENSIONS
Corrected by T. D. Noe, Nov 07 2006
Edited by N. J. A. Sloane, Dec 04 2006
STATUS
approved