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A008811 Expansion of x*(1+x^4)/((1-x)^2*(1-x^4)).

%I A008811 #42 Oct 05 2023 14:34:18
%S A008811 0,1,2,3,4,7,10,13,16,21,26,31,36,43,50,57,64,73,82,91,100,111,122,
%T A008811 133,144,157,170,183,196,211,226,241,256,273,290,307,324,343,362,381,
%U A008811 400,421,442,463,484,507,530,553,576,601,626,651,676,703,730,757,784,813
%N A008811 Expansion of x*(1+x^4)/((1-x)^2*(1-x^4)).
%C A008811 Number of 0..n-1 arrays of 5 elements with zero 2nd differences. - _R. H. Hardin_, Nov 15 2011
%H A008811 G. C. Greubel, <a href="/A008811/b008811.txt">Table of n, a(n) for n = 0..1000</a>
%H A008811 Daniel Gabric and Joe Sawada, <a href="http://www.cis.uoguelph.ca/~sawada/papers/discrep.pdf">Investigating the discrepancy property of de Bruijn sequences</a>, University of Guelph (Canada, 2020).
%H A008811 János Pach and Pankaj K. Agarwal, <a href="https://archive.org/details/combinatorialgeo0000pach/page/220/mode/2up">Combinatorial Geometry</a>, p. 220, 1995, Problem 13.10.
%H A008811 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,1,-2,1).
%F A008811 G.f.: x*(1+x^4)/((1-x)^2*(1-x^4)).
%F A008811 a(n) = 2*a(n-1) -a(n-2) +a(n-4) -2*a(n-5) +a(n-6). - _R. H. Hardin_, Nov 15 2011
%F A008811 a(n) = (-2*(1+(-1)^n)*(-1)^floor(n/2) + 2*n^2 + 5 - (-1)^n)/8. - _Tani Akinari_, Jul 24 2013
%F A008811 E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x) - 2*cos(x))/4. - _Stefano Spezia_, May 26 2021
%F A008811 Sum_{n>=1} 1/a(n) = Pi^2/24 + tanh(Pi/2)*Pi/4 + tanh(sqrt(3)*Pi/2)*Pi/sqrt(3). - _Amiram Eldar_, Aug 25 2022
%F A008811 a(n) = 2*floor((n^2 + 4)/8) + (n mod 2). - _Ridouane Oudra_, Sep 08 2023
%p A008811 f := n->n^2/4+3*n/2+g(n);
%p A008811 g := n->if n mod 2 = 0 then 3 elif n mod 4 = 1 then 9/4 else 13/4; fi;
%p A008811 seq(f(n), n=-3..50);
%t A008811 CoefficientList[Series[x*(1+x^4)/((1-x)^2*(1-x^4)), {x,0,60}], x] (* _G. C. Greubel_, Sep 12 2019 *)
%o A008811 (PARI) concat([0], Vec(x*(1+x^4)/((1-x)^2*(1-x^4))+O(x^60))) \\ _Charles R Greathouse IV_, Sep 26 2012, modified by _G. C. Greubel_, Sep 12 2019
%o A008811 (Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1+x^4)/((1-x)^2*(1-x^4)) )); // _G. C. Greubel_, Sep 12 2019
%o A008811 (Sage)
%o A008811 def A008811_list(prec):
%o A008811     P.<x> = PowerSeriesRing(ZZ, prec)
%o A008811     return P(x*(1+x^4)/((1-x)^2*(1-x^4))).list()
%o A008811 A008811_list(60) # _G. C. Greubel_, Sep 12 2019
%o A008811 (GAP) a:=[0,1,2,3,4,7];; for n in [7..60] do a[n]:=2*a[n-1]-a[n-2] +a[n-4]-2*a[n-5]+a[n-6]; od; a; # _G. C. Greubel_, Sep 12 2019
%Y A008811 Cf. A129756 (first differences).
%Y A008811 Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), this sequence (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).
%K A008811 nonn,easy
%O A008811 0,3
%A A008811 _N. J. A. Sloane_