

A101822


Expansion of 1/(1x2*x^23*x^3).


2



1, 1, 3, 8, 17, 42, 100, 235, 561, 1331, 3158, 7503, 17812, 42292, 100425, 238445, 566171, 1344336, 3192013, 7579198, 17996232, 42730667, 101460725, 240910755, 572024206, 1358227891, 3225008568, 7657536968, 18182237777, 43172337417
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OFFSET

0,3


COMMENTS

The recursive multipliers "one two three" are found in the three right coefficients of the characteristic polynomial of M: x^3  x^2  2x  3, (with changed signs). a(n)/a(n1) tends to 2.3744237632...an eigenvalue of M and a root of the characteristic polynomial.


LINKS

Table of n, a(n) for n=0..29.
Index entries for linear recurrences with constant coefficients, signature (1,2,3)


FORMULA

a(n) = a(n1) + 2*a(n2) + 3*a(n3), a(0) = a(1) = 1, a(2) = 3.
a(n) = left term in M^n * [1 0 0], where M = the 3X3 matrix [1 1 1 / 2 0 0 / 0 3/2 0].


EXAMPLE

a(8) = 561 = 235 + 2*100 + 3*42 = a(7) + 2*a(6) + 3*a(5).
a(5) = 42, since M^5 * [1 0 0] = [ 42 34 24].


MATHEMATICA

a[0] = a[1] = 1; a[2] = 3; a[n_] := a[n] = a[n  1] + 2a[n  2] + 3a[n  3]; Table[ a[n], {n, 0, 29}] (* Or *)
a[n_] := (MatrixPower[{{1, 1, 1}, {2, 0, 0}, {0, 3/2, 0}}, n].{{1}, {0}, {0}})[[1, 1]]; Table[ a[n], {n, 0, 29}] (* Robert G. Wilson v, Dec 20 2004 *)
LinearRecurrence[{1, 2, 3}, {1, 1, 3}, 30] (* Harvey P. Dale, Feb 06 2019 *)


PROG

(PARI) Vec(1/(1x2*x^23*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012


CROSSREFS

The (1, 1, 1) weighted equivalent is the core tribonacci sequence A000073. The (1, 2, 3) weighted equivalent is A100550.
Sequence in context: A247374 A046994 A058811 * A088589 A319764 A063597
Adjacent sequences: A101819 A101820 A101821 * A101823 A101824 A101825


KEYWORD

nonn,easy


AUTHOR

Gary W. Adamson, Dec 17 2004


EXTENSIONS

More terms from Robert G. Wilson v, Dec 20 2004


STATUS

approved



