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A299273 Partial sums of A299272. 51


%S 1,7,25,62,125,224,366,555,804,1121,1505,1973,2535,3183,3939,4816,

%T 5797,6910,8172,9555,11094,12811,14665,16699,18941,21335,23933,26770,

%U 29773,33004,36506,40187,44120,48357,52785,57489,62531,67775,73319,79236,85365,91818,98680,105763,113194,121071,129177

%N Partial sums of A299272.

%H G. C. Greubel, <a href="/A299273/b299273.txt">Table of n, a(n) for n = 0..5000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,3,-3,0,-3,3,0,1,-1).

%F Conjectures from _Colin Barker_, Feb 11 2018: (Start)

%F G.f.: (1 + x)^3*(1 + x^2)*(1 + 3*x + 5*x^2 + 3*x^3 + x^4) / ((1 - x)^4*(1 + x + x^2)^3).

%F a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10) for n>9.

%F (End)

%F These conjectures are correct. - _N. J. A. Sloane_, Feb 12 2018

%t CoefficientList[Series[(1+x)^3*(1+x^2)*(1+3*x+5*x^2+3*x^3+x^4)/((1-x)^4*(1+x+x^2)^3), {x, 0, 50}], x] (* _G. C. Greubel_, Feb 20 2018 *)

%o (PARI) x='x+O('x^30); Vec((1+x)^3*(1+x^2)*(1+3*x+5*x^2+3*x^3+x^4)/((1-x)^4*(1+x+x^2)^3)) \\ _G. C. Greubel_, Feb 20 2018

%o (MAGMA) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1+x)^3*(1+x^2)*(1+3*x+5*x^2+3*x^3+x^4)/((1-x)^4*(1+x+x^2)^3))) // _G. C. Greubel_, Feb 20 2018

%Y Cf. A299272.

%Y The 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e: A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Feb 10 2018

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Last modified February 24 18:46 EST 2021. Contains 341584 sequences. (Running on oeis4.)