A100191 The (1,1)-entry in the 3 X 3 matrix M^n, where M = [1,2,1 / 2,2,0 / 1,0,0].

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".

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

Cf. A100190.

Sequence in context: A060579 A183326 A123950 * A191585 A359190 A220795

Adjacent sequences: A100188 A100189 A100190 * A100192 A100193 A100194

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