|
%I #17 Sep 08 2022 08:44:36
%S 1,2,3,5,8,11,15,20,26,33,41,50,61,73,86,101,118,136,156,178,202,228,
%T 256,286,319,354,391,431,474,519,567,618,672,729,789,852,919,989,1062,
%U 1139,1220,1304,1392,1484,1580,1680,1784,1892,2005,2122,2243,2369,2500,2635,2775,2920,3070
%N Expansion of (1+x)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
%H Harvey P. Dale, <a href="/A008762/b008762.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-1,-1,1,-1,2,-1).
%F a(0)=1, a(1)=2, a(2)=3, a(3)=5, a(4)=8, a(5)=11, a(6)=15, a(7)=20, a(8)=26, a(n) = 2*a(n-1) -a(n-2) +a(n-3) -a(n-4) -a(n-5) +a(n-6) -a(n-7) +2*a(n-8) -a(n-9). - _Harvey P. Dale_, Mar 19 2013
%F G.f.: 1/( (1+x)*(1+x^2)*(1+x+x^2)*(1-x)^4 ). - _R. J. Mathar_, Aug 06 2013
%p seq(coeff(series( (1+x)/mul(1-x^j, j=1..4) , x, n+1), x, n), n = 0..60); # _G. C. Greubel_, Sep 09 2019
%t CoefficientList[Series[(x+1)/Times@@(1-x^Range[4]),{x,0,60}],x] (* or *) LinearRecurrence[{2,-1,1,-1,-1,1,-1,2,-1},{1,2,3,5,8,11,15,20,26},60] (* _Harvey P. Dale_, Mar 19 2013 *)
%o (PARI) my(x='x+O('x^60)); Vec( (1+x)/prod(j=1,4,1-x^j) ) \\ _G. C. Greubel_, Sep 09 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x)/(&*[1-x^j: j in [1..4]]) )); // _G. C. Greubel_, Sep 09 2019
%o (Sage)
%o def AA008762_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+x)/prod(1-x^j for j in (1..4)) ).list()
%o AA008762_list(60) # _G. C. Greubel_, Sep 09 2019
%o (GAP) a:=[1,2,3,5,8,11,15,20,26];; for n in [10..60] do a[n]:=2*a[n-1] -a[n-2]+a[n-3]-a[n-4]-a[n-5]+a[n-6]-a[n-7]+2*a[n-8]-a[n-9]; od; a; # _G. C. Greubel_, Sep 09 2019
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Terms a(43) onward added by _G. C. Greubel_, Sep 09 2019
|
