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A101822 Expansion of 1/(1-x-2*x^2-3*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 (list; graph; refs; listen; history; text; internal format)
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(n-1) 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(n-1) + 2*a(n-2) + 3*a(n-3), 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/(1-x-2*x^2-3*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

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Last modified May 27 21:52 EDT 2020. Contains 334671 sequences. (Running on oeis4.)