0,3

Number of triangles in triangular matchstick arrangement of side n.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

J. H. Conway and R. K. Guy, The Book of Numbers, p. 83.

F. Gerrish, How many triangles, Math. Gaz., 54 (1970), 241-246.

J. Halsall, An interesting series, Math. Gaz., 46 (1962), 55-56.

M. E. Larsen, The eternal triangle - a history of a counting problem, College Math. J., 20 (1989), 370-392.

B. D. Mastrantone, Comment, Math. Gaz., 55 (1971), 438-440.

Problem 889, Math. Mag., 47 (1974), 289-292.

L. Smiley, "A Quick Solution of Triangle Counting", Mathematics Magazine, 66, #1, Feb '93, p. 40.

Hugo Pfoertner, Illustration of A002717(5) and A002717(6)

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

a(n)=(1/16)*[2n(2n+1)(n+2)+cos(pi*n)-1] - Justin C. Bozonier (justinb67(AT)excite.com), Dec 05 2000

a(m+1)-2a(m)+2a(m-2)-a(m-3)=3. - Len Smiley (smiley(AT)math.uaa.alaska.edu), Oct 08 2001

G.f.: x(1+2x)/((1+x)(1-x)^4).

a(n) = (2n(2n+1)(n+2)+(-1)^n-1)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), Oct 25 2003

a(n)=A000292(n-1)+A002623(n-2). - Hugo Pfoertner (hugo(AT)pfoertner.org), Mar 06 2004

a(n) = Sum_{k=0..n} (-1)^(n-k)*k*binomial(k+1,2).

G.f.: x(1+2x)/((1+x)(1-x)^4). - S. Plouffe in his 1992 dissertation (with a different offset).

f(3)=13 because the following figure contains 13 triangles:

....... /\

...... /\/\

..... /\/\/\

A002717:=n->floor(n*(n+2)*(2*n+1)/8);

(PARI) a(n)=if(n<0, 0, n*(n+2)*(2*n+1)\8)

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.

Bisections: A135712, A135713.

Sequence in context: A008580 A123326 A025193 this_sequence A023541 A079989 A062480

Adjacent sequences: A002714 A002715 A002716 this_sequence A002718 A002719 A002720

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

Plouffe g.f. edited by R. J. Mathar, May 12 2008