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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].
2
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.
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
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Nov 07 2004
EXTENSIONS
Edited by N. J. A. Sloane, Dec 04 2006
STATUS
approved