A175782 - OEIS

%I #32 Jun 30 2023 18:02:42

%S 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,

%T 14,15,16,17,18,19,20,22,24,27,31,36,42,49,57,66,76,87,99,112,126,141,

%U 157,174,192,211,231,254,279,307,339,376,419,469,527,594

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

%C Limiting ratio of a(n)/a(n-1) = 1.119189829034646... .

%C 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.

%C The polynomial has one real and two complex roots outside the unit circle.

%H Alois P. Heinz, <a href="/A175782/b175782.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_40">Index entries for linear recurrences with constant coefficients</a>, 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).

%F a(n) = a(n-1) + a(n-20) + a(n-39) - a(n-40). - _Franck Maminirina Ramaharo_, Oct 31 2018

%p gf:= 1/(1-x-x^20-x^39+x^40):

%p a:= n-> coeff(series(gf, x, n+1), x, n):

%p seq(a(n), n=0..100);  # _Alois P. Heinz_, Jul 27 2012

%t CoefficientList[Series[1/(1 - x - x^20 - x^39 + x^40), {x, 0, 50}], x]

%t 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 *)

%o (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

%o (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

%Y Cf. A175739.

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

%K nonn,easy

%O 0,21

%A _Roger L. Bagula_, Dec 04 2010