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A006004 a(n) = C(n+2,3) + C(n,3) + C(n-1,3).

%I A006004 M3412 #34 Jan 22 2018 21:12:03
%S A006004 1,4,11,25,49,86,139,211,305,424,571,749,961,1210,1499,1831,2209,2636,
%T A006004 3115,3649,4241,4894,5611,6395,7249,8176,9179,10261,11425,12674,14011,
%U A006004 15439,16961,18580,20299,22121,24049,26086,28235,30499,32881,35384,38011,40765
%N A006004 a(n) = C(n+2,3) + C(n,3) + C(n-1,3).
%C A006004 Equals binomial transform of [1, 3, 4, 3, 0, 0, 0, ...]. Example: a(4) = 25 = (1, 3, 3, 1) dot (1, 3, 4, 3) = (1 + 9 + 12 + 3). - _Gary W. Adamson_, Jul 25 2008
%D A006004 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
%D A006004 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006004 T. D. Noe, <a href="/A006004/b006004.txt">Table of n, a(n) for n = 1..1000</a>
%H A006004 S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
%H A006004 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, -1).
%F A006004 a(n) = (n^3 - 2n^2 + 5n - 2)/2.
%F A006004 G.f.: (x^3+x^2+1)/(x-1)^4. - _Harvey P. Dale_, Jun 15 2011
%F A006004 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(0)=1, a(1)=4, a(2)=11, a(3)=25. - _Harvey P. Dale_, Jun 15 2011
%p A006004 A006004:=n->(n^3 - 2*n^2 + 5*n - 2)/2; seq(A006004(n), n=1..50); # _Wesley Ivan Hurt_, Feb 09 2014
%t A006004 Table[Binomial[n+2,3]+Binomial[n,3]+Binomial[n-1,3],{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,4,11,25},50] (* _Harvey P. Dale_, Jun 15 2011 *)
%o A006004 (PARI) a(n) = (n^3 - 2*n^2 + 5*n - 2)/2 \\ _Charles R Greathouse IV_, Feb 10 2017
%K A006004 nonn,easy,nice
%O A006004 1,2
%A A006004 _N. J. A. Sloane_
%E A006004 Terms added by _Wesley Ivan Hurt_, Feb 09 2014