%I A006000 M3436
%S A006000 1,4,12,28,55,96,154,232,333,460,616,804,1027,1288,1590,1936,2329,2772,
%T A006000 3268,3820,4431,5104,5842,6648,7525,8476,9504,10612,11803,13080,14446,
%U A006000 15904,17457,19108,20860,22716,24679,26752,28938,31240,33661,36204,38872,
               41668,44595,47656,50854,54192
%N A006000 a(n) = (n+1)*(n^2+n+2)/2; g.f.: (1 + 2 x^2 ) / ( 1 - x )^4.
%C A006000 Enumerates certain paraffins.
%C A006000 a(n) is the (n+1)st (n+3)-gonal number. - Floor van Lamoen (fvlamoen(AT)hotmail.com), 
               Oct 20 2001
%C A006000 Sum of n terms of an arithmetic progression with the first term 1 and 
               the common difference n: a(1)=1 a(2) = 1+3 a(3) = 1+4+7 a(4) = 1+5+9+13 
               etc. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 25 2004
%C A006000 This is identical to: 1st triangular number A000217, 2nd square number 
               A000290, 3rd pentagonal number A000326, 4th hexagonal number A000384, 
               5th heptagonal number A000566, 6th octagonal number A000567, ..., 
               (n+1)-th (n+3)-gonal number = main diagonal of rectangular array 
               T(n,k) of polygonal numbers, by diagonals, referred to in A086271. 
               - Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 19 2007
%D A006000 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006000 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, 
               Chem. Ber. 30 (1897), 1917-1926.
%D A006000 P. A. MacMahon, Properties of prime numbers deduced from the calculus 
               of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. 
               [Coll. Papers, Vol. II, pp. 354-382] [See p. 301]
%H A006000 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A006000 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A006000 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A006000 a:=n->sum((binomial(0,0*j)+binomial(n+1,2)),j=1..n+1): seq(a(n), n>=1). 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 25 2006
%p A006000 a:=n->sum((binomial(0,0*j)+binomial(n+1,2)),j=1..n+1): seq(a(n), n=1..49); 
               - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 25 2006
%p A006000 seq(add(k+add(l, k=0..n), l=0..n), n=0..47); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Oct 04 2007
%p A006000 A006000:=(1+2*z**2)/(z-1)**4; [S. Plouffe in his 1992 dissertation.]
%p A006000 with (combinat):seq((fibonacci(4, n)-n^2)/2, n=0..48); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 07 2008
%Y A006000 Cf. A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, 
               A086271.
%Y A006000 Sequence in context: A109629 A112087 A166019 this_sequence A161216 A085622 
               A011940
%Y A006000 Adjacent sequences: A005997 A005998 A005999 this_sequence A006001 A006002 
               A006003
%K A006000 nonn,easy
%O A006000 0,2
%A A006000 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A006000 More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2001