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a(n) = (n^3 - 3n^2 + 14n - 6)/6.
2

%I #25 Nov 14 2023 21:26:21

%S 1,3,6,11,19,31,48,71,101,139,186,243,311,391,484,591,713,851,1006,

%T 1179,1371,1583,1816,2071,2349,2651,2978,3331,3711,4119,4556,5023,

%U 5521,6051,6614,7211,7843,8511,9216,9959,10741,11563,12426,13331,14279,15271,16308

%N a(n) = (n^3 - 3n^2 + 14n - 6)/6.

%C Binomial transform of 0,1,1,0,bar(1,-1), where bar(..) denotes a periodically repeated sequence.

%C If the offset is set to 0, this is the binomial transform of the quasi-finite sequence 1,2,1,1,bar(0).

%H Paolo P. Lava, <a href="/A180415/b180415.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%e Representation as binomial transform: a(4) = 11 = (1, 2, 1, 1) dot (1, 3, 3, 1) = (1 + 6 + 3 + 1).

%t Table[(n^3-3n^2+14n-6)/6,{n,60}] (* _Harvey P. Dale_, Mar 04 2011 *)

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Sep 02 2010

%E Sequence extended, nomenclature of variables normalized, g.f. multiplied by x, binomial transforms clarified - _R. J. Mathar_, Sep 04 2010