A175782 Expansion of 1/(1 - x - x^20 - x^39 + x^40).

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

Offset

0,21

Comments

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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).

Formula

a(n) = a(n-1) + a(n-20) + a(n-39) - a(n-40). - Franck Maminirina Ramaharo, Oct 31 2018

Maple

gf:= 1/(1-x-x^20-x^39+x^40):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 27 2012

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A175739.

Cf. A029826, A117791, A143419, A143438, A143472, A143619, A143644, A147663, A173908, A173911, A173924, A173925, A174522, A175740, A175772, A175773, A181600, A204631, A225391, A225393, A225394, A225482, A225499.

Sequence in context: A178157 A246083 A246090 * A272325 A358140 A249817

Adjacent sequences: A175779 A175780 A175781 * A175783 A175784 A175785

Keyword

nonn,easy

Author

Roger L. Bagula, Dec 04 2010

Status

approved