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%I #18 Sep 08 2022 08:46:20
%S 1,6,19,45,90,159,257,390,563,781,1050,1375,1761,2214,2739,3341,4026,
%T 4799,5665,6630,7699,8877,10170,11583,13121,14790,16595,18541,20634,
%U 22879,25281,27846,30579,33485,36570,39839,43297,46950,50803,54861,59130
%N Partial sums of A299259.
%C Euler transform of length 4 sequence [6, -2, 1, -1]. - _Michael Somos_, Oct 03 2018
%H Colin Barker, <a href="/A299265/b299265.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F From _Colin Barker_, Feb 09 2018: (Start)
%F G.f.: (1 + x)^3*(1 + x^2) / ((1 - x)^4*(1 + x + x^2)).
%F a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6) for n>5.
%F (End)
%F a(n) = -a(-1-n) for all n in Z. - _Michael Somos_, Oct 03 2018
%t CoefficientList[Series[(1+x)^3*(1+x^2)/((1-x)^4*(1+x+x^2)), {x, 0, 50}], x] (* _G. C. Greubel_, Feb 20 2018 *)
%t a[ n_] := (8 n^3 + 12 n^2 + 24 n + 9 + Mod[n, 3]) / 9; (* _Michael Somos_, Oct 03 2018 *)
%t LinearRecurrence[{3,-3,2,-3,3,-1},{1,6,19,45,90,159},50] (* _Harvey P. Dale_, Dec 11 2018 *)
%o (PARI) Vec((1 + x)^3*(1 + x^2) / ((1 - x)^4*(1 + x + x^2)) + O(x^60)) \\ _Colin Barker_, Feb 09 2018
%o (PARI) {a(n) = (8*n^3 + 12*n^2 + 24*n + 9 + (n%3)) / 9}; /* _Michael Somos_, Oct 03 2018 */
%o (Magma) I:=[19, 45, 90, 159, 257, 390]; [1,6] cat [n le 6 select I[n] else 3*Self(n-1) - 3*Self(n-2) +2*Self(n-3) - 3*Self(n-4) + 3*Self(n-5) - Self(n-6): n in [1..30]];
%Y Cf. A299259.
%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
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