

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
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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(n1) + 3*a(n2)  2*a(n3) (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+182 = 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[n1]) 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[n1]+3*a[n2]2*a[n3] 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 A220795 A026545
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



