%I A002717 M3827 N1569
%S A002717 0,1,5,13,27,48,78,118,170,235,315,411,525,658,812,988,1188,1413,1665,
%T A002717 1945,2255,2596,2970,3378,3822,4303,4823,5383,5985,6630,7320,8056,8840,
%U A002717 9673,10557,11493,12483,13528,14630,15790,17010,18291,19635,21043,22517
%N A002717 Floor(n(n+2)(2n+1)/8).
%C A002717 Number of triangles in triangular matchstick arrangement of side n.
%D A002717 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002717 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002717 J. H. Conway and R. K. Guy, The Book of Numbers, p. 83.
%D A002717 F. Gerrish, How many triangles, Math. Gaz., 54 (1970), 241-246.
%D A002717 J. Halsall, An interesting series, Math. Gaz., 46 (1962), 55-56.
%D A002717 M. E. Larsen, The eternal triangle - a history of a counting problem, 
               College Math. J., 20 (1989), 370-392.
%D A002717 B. D. Mastrantone, Comment, Math. Gaz., 55 (1971), 438-440.
%D A002717 Problem 889, Math. Mag., 47 (1974), 289-292.
%D A002717 L. Smiley, "A Quick Solution of Triangle Counting", Mathematics Magazine, 
               66, #1, Feb '93, p. 40.
%H A002717 Hugo Pfoertner, <a href="a002717.pdf">Illustration of A002717(5) and 
               A002717(6)</a>
%H A002717 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 A002717 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.
%H A002717 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TriangleTiling.html">Link to a section of The World of Mathematics.</
               a>
%F A002717 a(n)=(1/16)*[2n(2n+1)(n+2)+cos(pi*n)-1] - Justin C. Bozonier (justinb67(AT)excite.com), 
               Dec 05 2000
%F A002717 a(m+1)-2a(m)+2a(m-2)-a(m-3)=3. - Len Smiley (smiley(AT)math.uaa.alaska.edu), 
               Oct 08 2001
%F A002717 G.f.: x(1+2x)/((1+x)(1-x)^4).
%F A002717 a(n) = (2n(2n+1)(n+2)+(-1)^n-1)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), 
               Oct 25 2003
%F A002717 a(n)=A000292(n-1)+A002623(n-2). - Hugo Pfoertner (hugo(AT)pfoertner.org), 
               Mar 06 2004
%F A002717 a(n) = Sum_{k=0..n} (-1)^(n-k)*k*binomial(k+1,2).
%F A002717 G.f.: x(1+2x)/((1+x)(1-x)^4). - S. Plouffe in his 1992 dissertation (with 
               a different offset).
%e A002717 f(3)=13 because the following figure contains 13 triangles:
%e A002717 ....... /\
%e A002717 ...... /\/\
%e A002717 ..... /\/\/\
%p A002717 A002717:=n->floor(n*(n+2)*(2*n+1)/8);
%o A002717 (PARI) a(n)=if(n<0,0,n*(n+2)*(2*n+1)\8)
%Y A002717 Cf. A000292 number of triangles with same orientation as largest triangle, 
               A002623 number of triangles pointing in opposite direction to largest 
               triangle, A085691 number of triangles of side k in arrangement of 
               side n.
%Y A002717 Bisections: A135712, A135713.
%Y A002717 Sequence in context: A008580 A123326 A025193 this_sequence A023541 A079989 
               A062480
%Y A002717 Adjacent sequences: A002714 A002715 A002716 this_sequence A002718 A002719 
               A002720
%K A002717 nonn,easy,nice
%O A002717 0,3
%A A002717 N. J. A. Sloane (njas(AT)research.att.com).
%E A002717 Plouffe g.f. edited by R. J. Mathar, May 12 2008