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A294627
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Expansion of x*(1 + x)/((1-2*x)*(1+x+x^2)).
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7
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0, 1, 2, 3, 7, 14, 27, 55, 110, 219, 439, 878, 1755, 3511, 7022, 14043, 28087, 56174, 112347, 224695, 449390, 898779, 1797559, 3595118, 7190235, 14380471, 28760942, 57521883, 115043767, 230087534, 460175067, 920350135, 1840700270, 3681400539, 7362801079, 14725602158, 29451204315
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OFFSET
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0,3
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COMMENTS
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A generalized tribonacci (A001590) sequence.
For n > 2, 6*a(n) is the number of quaternary sequences of length n in which all triples (q(i),q(i+1),q(i+2)) contain two (arbitrarily chosen) digits, e.g., 0 and 3.
Similarly, recurrences a(n) = a(n-1) + a(n-2) + k*a(n-3) are related to binary (k=0, the Fibonacci sequence A000045), ternary (k=1, the tribonacci sequence A001590), quinary (k=3) and so on sequences with all triples (t(i),t(i+1),t(i+2)) containing two (arbitrarily chosen) digits (usually 0 and k+1).
For n > 0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and two colors of trominoes, with the restriction that the first tile cannot be a tromino. - Greg Dresden and Bora Bursalı, Aug 31 2023
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LINKS
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FORMULA
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a(n) = a(n-1) + a(n-2) + 2*a(n-3) for n > 2.
a(n+1)/a(n) tends to 2, the unique real root of x^3 - x^2 - x - 2 = 0.
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EXAMPLE
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For n=4 there are 6*7=42 quaternary sequences of length 4 such that each triple (i.e., exactly two of them: q1,q2,q3 and q2,q3,q4) contain both 0 and 3. They are 003x, 030x, 03y0, 0330, 330x, 303x, 30y3, 3003, 0y30, 3y03, y03x, y30x, where x=0,1,2,3 and y=1,2.
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PROG
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(PARI) x='x+O('x^99); concat(0, Vec(x*(1+x)/(1-x-x^2-2*x^3))) \\ Altug Alkan, Mar 03 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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