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A100191
The (1,1)-entry in the 3 X 3 matrix M^n, where M = [1,2,1 / 2,2,0 / 1,0,0].
2
1, 6, 19, 73, 264, 973, 3565, 13086, 48007, 176149, 646296, 2371321, 8700553, 31923030, 117128107, 429752305, 1576795176, 5785386229, 21227039605, 77883687150, 285761407807, 1048481205661, 3846960466104, 14114802199681, 51788325586033, 190015462424934
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".
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
Cf. A100190.
Sequence in context: A060579 A183326 A123950 * A191585 A359190 A220795
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