%I A019298
%S A019298 0,1,4,11,23,42,69,106,154,215,290,381,489,616,763,932,
%T A019298 1124,1341,1584,1855,2155,2486,2849,3246,3678,4147,4654,
%U A019298 5201,5789,6420,7095,7816,8584,9401,10268,11187,12159,13186
%N A019298 Balls in pyramid with base either a regular hexagon or a hexagon with 
               alternate sides differing by 1 (balls in hexagonal pyramid of height 
               n taken from hexagonal close-packing).
%C A019298 Alternately add and subtract successively longer sets of integers: 0; 
               1=0+1; -4=1-2-3; 11=-4+4+5+6; -23=11-7-8-9-10; 42=-23+11+12+13+14+15; 
               -69=42-16-17-18-19-20-21; ... them take absolute values. - Walter 
               G. Carlini (541carlini(AT)charter.net), Aug 28 2003
%C A019298 Number of 3 X 3 symmetric matrices with nonnegative integer entries, 
               such that every row (and column) sum equals n-1.
%C A019298 Equals sum_{0..n} of "three-quarter squares" sequence (A077043) - Philipp 
               M. Buluschek (kitschen(AT)romandie.com), Aug 12 2007
%D A019298 R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see 
               Prop. 4.6.21, p. 235, G_3(lambda).
%D A019298 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Problem 7.14(a), p. 452.
%H A019298 G. E. Andrews, P. Paule and A. Riese, <a href="http://www.risc.uni-linz.ac.at/
               research/combinat/risc/publications/#ppaule">MacMahon's partition 
               analysis III. The Omega package</a>, p. 13.
%F A019298 Floor ( (n^2+1)(2n+3)/8 ). G.f.: x(x^2+x+1)/((x+1)(x-1)^4).
%F A019298 Floor [ ( 2n^3 + 3n^2 + 2n )/8 ]; also nearest integer to ( (n+1)^4 - 
               n^4 )/16.
%F A019298 a(n)=(4n^3+6n^2+4n+1-(-1)^n)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), 
               Mar 06 2004
%p A019298 series(x*(x^2+x+1)/(x+1)/(x-1)^4,x,80);
%Y A019298 Cf. A053493.
%Y A019298 Cf. A077043.
%Y A019298 Sequence in context: A027378 A092498 A131177 this_sequence A014242 A008181 
               A008214
%Y A019298 Adjacent sequences: A019295 A019296 A019297 this_sequence A019299 A019300 
               A019301
%K A019298 nonn,easy,nice
%O A019298 0,3
%A A019298 Eric E Blom (eblom(AT)REM.re.uokhsc.edu)
%E A019298 Error in n=8 term corrected May 15 1997