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%I #21 Feb 08 2022 00:24:10
%S 1,2,6,14,27,46,72,106,149,202,266,342,431,534,652,786,937,1106,1294,
%T 1502,1731,1982,2256,2554,2877,3226,3602,4006,4439,4902,5396,5922,
%U 6481,7074,7702,8366,9067,9806,10584,11402,12261,13162,14106,15094,16127
%N a(n) = 2*a(n-1) - a(n-2) + n + 1.
%C Equals binomial transform of [1, 1, 3, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Aug 25 2010
%H G. C. Greubel, <a href="/A121968/b121968.txt">Table of n, a(n) for n = 1..5000</a>
%F a(n) = (6 + n(-1 + n(6 + n)))/6 = C(n+1, 3) + n^2 + 1 = C(n+2, 3) + C(n, 2) + 1.
%F G.f.: (1 - 2*x + 4*x^2 - 2*x^3)/(1 - x)^4. - _Robert G. Wilson v_, Sep 11 2006
%t a[n_] := (6 + n(-1 + n(6 + n)))/6; Table[a[n], {n, 0, 45}] (* or *)
%t a[0] = 1; a[1] = 2; a[n_] := a[n] = 2a[n - 1] - a[n - 2] + n + 1; Table[ a[n], {n, 0, 45}] (* or *)
%t CoefficientList[ Series[(1 - 2x + 4x^2 - 2x^3)/(x - 1)^4, {x, 0, 45}], x] (* _Robert G. Wilson v_, Sep 11 2006 *)
%o (PARI) x='x+O('x^50); Vec((1-2*x+4*x^2-2*x^3)/(1-x)^4) \\ _G. C. Greubel_, Sep 13 2017
%Y Cf. cake numbers A000125.
%K nonn,easy
%O 1,2
%A _Roger L. Bagula_, Sep 04 2006
%E Edited and extended by _Robert G. Wilson v_, Sep 11 2006
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