A100190 The (4,1)-entry in the 4 X 4 matrix M^n, where M = [1,0,0,0 / 3,3,0,0 / 3,6,3,0 / 1,3,3,1].

1, 20, 147, 760, 3317, 13164, 49255, 177200, 620073, 2125828, 7174523, 23914920, 78919069, 258280412, 839411151, 2711943520, 8716961105, 27894275316, 88913002339, 282429536600, 894360198981, 2824295364940, 8896530399287, 27960524111760, 87694371077497

Offset

1,2

Comments

Suggested by "Mathematical Vistas", p. 178, Fig 14: The Pascal Tetrahedron. The first few levels are (Level 0): 1; (Level 1): 1; 1, 1; (Level 2): 1; 2, 2; 1, 2, 1; (Level 3): 1; 3, 3; 3, 6, 3; 1, 3, 3, 1.

References

Peter Hilton, Derek Holton and Jean Pederson; "Mathematical Vistas, From a Room With Many Windows"; Springer, 2000; p. 178.

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (8,-22,24,-9).

Formula

G.f.: x*(1 + 12*x + 9*x^2)/((1 - 3*x)^2*(1 - x)^2).

a(n) = 8*a(n-1) - 22*a(n-2) + 24*a(n-3) - 9*a(n-4) for n>=5 (derived from the minimal polynomial of the matrix M).

a(n) = ((11 + 3^(2+n))*n - 18*(3^n - 1))/2. - Colin Barker, Feb 28 2017

Example

a(6) = 13164 because M^6 = [1,0,0,0 / 1092,729,0,0 / 10938,8748,729,0 / 13164,10938,1092,1].
Alternatively, a(6) = 8*a(5) - 22*a(4) + 24*a(3) - 9*a(2) = 26536 - 16720 + 3528 - 180 = 13164.

Maple

with(linalg): M[1]:=matrix(4, 4, [1, 0, 0, 0, 3, 3, 0, 0, 3, 6, 3, 0, 1, 3, 3, 1]): for n from 2 to 27 do M[n]:=multiply(M[1], M[n-1]) od: seq(M[n][4, 1], n=1..27);
a[1]:=1:a[2]:=20:a[3]:=147:a[4]:=760: for n from 5 to 27 do a[n]:=8*a[n-1]-22*a[n-2]+24*a[n-3]-9*a[n-4] od: seq(a[n], n=1..27);

Prog

(PARI) Vec(x*(1+12*x+9*x^2) / ((1-3*x)^2*(1-x)^2) + O(x^30)) \\ Colin Barker, Feb 28 2017

Crossrefs

Sequence in context: A238021 A183959 A094171 * A189494 A022680 A108647

Adjacent sequences: A100187 A100188 A100189 * A100191 A100192 A100193

Keyword

nonn,easy

Author

Gary W. Adamson, Nov 07 2004

Extensions

Edited by N. J. A. Sloane, Dec 04 2006

Status

approved