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A318774
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Coefficients in expansion of 1/(1 - x - 3*x^4).
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3
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1, 1, 1, 1, 4, 7, 10, 13, 25, 46, 76, 115, 190, 328, 556, 901, 1471, 2455, 4123, 6826, 11239, 18604, 30973, 51451, 85168, 140980, 233899, 388252, 643756, 1066696, 1768393, 2933149, 4864417, 8064505, 13369684, 22169131, 36762382, 60955897, 101064949, 167572342, 277859488, 460727179, 763922026, 1266639052
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OFFSET
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0,5
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COMMENTS
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The coefficients in the expansion of 1/(1 - x - 3*x^4) are given by the sequence generated by the row sums in triangle A318772.
Coefficients in expansion of 1/(1 - x - 3*x^4) are given by the sum of numbers along "third Layer" skew diagonals pointing top-right in triangle A013610 ((1+3x)^n) and by the sum of numbers along "third Layer" skew diagonals pointing top-left in triangle A027465 ((3+x)^n), see links.
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REFERENCES
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Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.
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LINKS
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FORMULA
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a(n) = a(n-1) + 3*a(n-4) for n >= 0, a(n)=0 for n < 0, with a(0) = a(1) = a(2) = a(3) = 1.
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MATHEMATICA
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CoefficientList[Series[1/(1-x-3x^4), {x, 0, 50}], x]
a[n_]:= a[n]= If[n<4, 1, a[n-1] + 3*a[n-4]]; Table[a[n], {n, 0, 50}]
LinearRecurrence[{1, 0, 0, 3}, {1, 1, 1, 1}, 51]
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PROG
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(Magma) [n le 4 select 1 else Self(n-1) +3*Self(n-4): n in [1..51]]; // G. C. Greubel, May 08 2021
(Sage)
def a(n): return 1 if (n<4) else a(n-1) + 3*a(n-4)
(PARI) my(p=Mod('x, x^4-'x^3-3)); a(n) = vecsum(Vec(lift(p^n))); \\ Kevin Ryde, May 11 2021
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CROSSREFS
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Essentially a duplicate of A143454.
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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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