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%I #45 Sep 08 2022 08:44:36
%S 1,2,3,4,5,8,11,14,17,20,25,30,35,40,45,52,59,66,73,80,89,98,107,116,
%T 125,136,147,158,169,180,193,206,219,232,245,260,275,290,305,320,337,
%U 354,371,388,405,424,443,462,481,500,521,542,563,584,605,628,651,674
%N Expansion of (1+x^5)/((1-x)^2*(1-x^5)).
%C Number of 0..n arrays of six elements with zero second differences. - _R. H. Hardin_, Nov 16 2011
%C Also number of ordered triples (w,x,y) with all terms in {1,...,n+1} and w + 4*x = 5*y. Also the number of 3-tuples (w,x,y) with all terms in {1,...,n+1} and 5*w = 2*x +3*y. - _Clark Kimberling_, Apr 15 2012 [Corrected by _Pontus von Brömssen_, Jan 26 2020]
%C a(n) is also the number of 5 boxes polyomino (zig-zag patterns) packing into (n+3) X (n+3) square. See illustration in links. - _Kival Ngaokrajang_, Nov 10 2013
%C Also, number of ordered pairs (x,y) with both terms in {1,...,n+1} and x+4*y divisible by 5; or number of ordered pairs (x,y) with both terms in {1,...,n+1} and 2*x+3*y divisible by 5. - _Pontus von Brömssen_, Jan 26 2020
%H G. C. Greubel, <a href="/A008812/b008812.txt">Table of n, a(n) for n = 0..1000</a>
%H Kival Ngaokrajang, <a href="/A008812/a008812.pdf">Illustration of initial terms</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,1,-2,1).
%F G.f.: (1+x^5)/((1-x)^2*(1-x^5)).
%F a(n) = 2*a(n-1) -a(n-2) +a(n-5) -2*a(n-6) +a(n-7). - _R. H. Hardin_, Nov 16 2011
%e For n = 5 there are 8 0..5 arrays of six elements with zero second differences: [0,0,0,0,0,0], [0,1,2,3,4,5], [1,1,1,1,1,1], [2,2,2,2,2,2], [3,3,3,3,3,3], [4,4,4,4,4,4], [5,4,3,2,1,0], [5,5,5,5,5,5].
%p seq(coeff(series((1+x^5)/((1-x)^2*(1-x^5)), x, n+1), x, n), n = 0..65); # _G. C. Greubel_, Sep 12 2019
%t CoefficientList[Series[(1+x^5)/(1-x)^2/(1-x^5),{x,0,65}],x] (* or *) LinearRecurrence[{2,-1,0,0,1,-2,1}, {1,2,3,4,5,8,11}, 65] (* _Harvey P. Dale_, Apr 17 2015 *)
%o (PARI) Vec((1+x^5)/(1-x)^2/(1-x^5)+O(x^65)) \\ _Charles R Greathouse IV_, Sep 25 2012
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1+x^5)/((1-x)^2*(1-x^5)) )); // _G. C. Greubel_, Sep 12 2019
%o (Sage)
%o def A008812_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1+x^5)/((1-x)^2*(1-x^5))).list()
%o A008812_list(65) # _G. C. Greubel_, Sep 12 2019
%o (GAP) a:=[1,2,3,4,5,8,11];; for n in [8..65] do a[n]:=2*a[n-1]-a[n-2] +a[n-5]-2*a[n-6]+a[n-7]; od; a; # _G. C. Greubel_, Sep 12 2019
%Y Cf. A130497 (first differences).
%Y Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), this sequence (m=5), A008813 (m=6), A008814 (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms added by _G. C. Greubel_, Sep 12 2019
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