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A175782
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Expansion of 1/(1 - x - x^20 - x^39 + x^40).
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23
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 27, 31, 36, 42, 49, 57, 66, 76, 87, 99, 112, 126, 141, 157, 174, 192, 211, 231, 254, 279, 307, 339, 376, 419, 469, 527, 594
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OFFSET
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0,21
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COMMENTS
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Limiting ratio of a(n)/a(n-1) = 1.119189829034646... .
A quasi - Salem polynomial based on the symmetrical polynomial defined by p(x,0) = 1, p(x,n) = x^(2*n) - x^(2*n - 1) - x^n - x + 1 for n>=1.
The polynomial has one real and two complex roots outside the unit circle.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).
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FORMULA
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MAPLE
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gf:= 1/(1-x-x^20-x^39+x^40):
a:= n-> coeff(series(gf, x, n+1), x, n):
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MATHEMATICA
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CoefficientList[Series[1/(1 - x - x^20 - x^39 + x^40), {x, 0, 50}], x]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22}, 70] (* Harvey P. Dale, Jun 30 2023 *)
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PROG
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(PARI) Vec(O(x^99)+1/(1 - x - x^20 - x^39 + x^40)) \\ N.B.: This yields a vector whose first component v[1] equals a(0), i.e., the offset is shifted by one. - M. F. Hasler, Dec 11 2010
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x-x^20-x^39+x^40))); // G. C. Greubel, Nov 03 2018
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CROSSREFS
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Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A181600, A204631, A225391, A225393, A225394, A225482, A225499.
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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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