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Search: id:A101822
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| A101822 |
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a(n) = a(n-1) + 2*a(n-2) + 3*a(n-3), a(0) = a(1) = 1 & a(2) = 3. |
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2
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| 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
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0,3
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COMMENT
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A "one two three" recursive sequence.
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.
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FORMULA
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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].
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EXAMPLE
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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].
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MATHEMATICA
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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}] (from Robert G. Wilson v Dec 20 2004)
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CROSSREFS
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The (1,1,1) weighted equivalent is the core tribonacci sequence A000073. The (1,2,3) weighted equivalent is A100550.
Sequence in context: A034481 A046994 A058811 this_sequence A088589 A063597 A004210
Adjacent sequences: A101819 A101820 A101821 this_sequence A101823 A101824 A101825
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KEYWORD
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nonn
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 17 2004
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 20 2004
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