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Partial sums of A299268.
51

%I #22 Mar 14 2024 23:25:40

%S 1,7,25,73,151,277,459,699,1029,1419,1941,2517,3275,4073,5111,6167,

%T 7529,8879,10609,12289,14431,16477,19075,21523,24621,27507,31149,

%U 34509,38739,42609,47471,51887,57425,62423,68681,74297,81319,87589,95419,102379,111061

%N Partial sums of A299268.

%H Colin Barker, <a href="/A299269/b299269.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).

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

%F G.f.: (1 + 6*x + 15*x^2 + 30*x^3 + 27*x^4 + x^6) / ((1 - x)^4*(1 + x)^3).

%F a(n) = (20*n^3 + 33*n^2 - 2*n + 12) / 12 for n even.

%F a(n) = (20*n^3 + 27*n^2 + 28*n + 9) / 12 for n odd.

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

%F E.g.f.: ((12 + 75*x + 93*x^2 + 20*x^3)*cosh(x) + (9 + 51*x + 87*x^2 + 20*x^3)*sinh(x))/12. - _Stefano Spezia_, Mar 14 2024

%t CoefficientList[Series[(1+6*x+15*x^2+30*x^3+27*x^4+x^6)/((1-x)^4*(1+ x)^3), {x, 0, 50}], x] (* _G. C. Greubel_, Feb 20 2018 *)

%o (PARI) Vec((1 + 6*x + 15*x^2 + 30*x^3 + 27*x^4 + x^6) / ((1 - x)^4*(1 + x)^3) + O(x^60)) \\ _Colin Barker_, Feb 09 2018

%o (Magma) I:=[25,73,151,277,459,699,1029]; [1,7] cat [n le 7 select I[n] else Self(n-1) + 3*Self(n-2) - 3*Self(n-3) - 3*Self(n-4) + 3*Self(n-5) + Self(n-6) - Self(n-7): n in [1..30]]; // _G. C. Greubel_, Feb 20 2018

%Y Cf. A299268.

%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