A299263 - OEIS

%I #12 Feb 12 2021 11:07:33

%S 1,6,18,40,76,132,214,325,469,652,878,1150,1474,1856,2298,2803,3379,

%T 4032,4762,5572,6472,7468,8558,9745,11041,12452,13974,15610,17374,

%U 19272,21298,23455,25759,28216,30818,33568,36484,39572,42822,46237,49837,53628,57598

%N Partial sums of A299257.

%H Colin Barker, <a href="/A299263/b299263.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-8,12,-14,12,-8,4,-1).

%F From _Colin Barker_, Feb 09 2018: (Start)

%F G.f.: (1 + x)*(1 + x + x^2 + 3*x^3 - x^4 + 5*x^5 - 3*x^6 + 4*x^7 - 2*x^8) / ((1 - x)^4*(1 + x^2)^2).

%F a(n) = 4*a(n-1) - 8*a(n-2) + 12*a(n-3) - 14*a(n-4) + 12*a(n-5) - 8*a(n-6) + 4*a(n-7) - a(n-8) for n>8.

%F (End)

%F 5*a(n) = 2*(2*n+1)*(2*n^2+2*n+9)/3 - A138019(n). - _R. J. Mathar_, Feb 12 2021

%o (PARI) Vec((1 + x)*(1 + x + x^2 + 3*x^3 - x^4 + 5*x^5 - 3*x^6 + 4*x^7 - 2*x^8) / ((1 - x)^4*(1 + x^2)^2) + O(x^60)) \\ _Colin Barker_, Feb 09 2018

%Y Cf. A299257.

%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,easy

%O 0,2

%A _N. J. A. Sloane_, Feb 07 2018