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%I M3827 N1569 #118 Jun 09 2023 03:17:20
%S 0,1,5,13,27,48,78,118,170,235,315,411,525,658,812,988,1188,1413,1665,
%T 1945,2255,2596,2970,3378,3822,4303,4823,5383,5985,6630,7320,8056,
%U 8840,9673,10557,11493,12483,13528,14630,15790,17010,18291,19635,21043,22517,24058
%N a(n) = floor(n(n+2)(2n+1)/8).
%C Number of triangles in triangular matchstick arrangement of side n, for n >= 1. Row sums of A085691.
%C We observe that the sequence is the transform of A006578 by the following transform T: T(u_0,u_1,u_2,u_3,...)=(u_0,u_0+u_1, u_0+u_1+u_2, u_0+u_1+u_2+u_3+u_4,...). In another terms v_p=sum(u_k,k=0..p) and the G.f phi_v of v is given by: phi_v=phi_u/(1-z). - _Richard Choulet_, Jan 28 2010
%C Row sums of A220053, for n > 0. - _Reinhard Zumkeller_, Dec 03 2012
%C a(n) has the expansion (1*0)+(1*1)+(4*1)+(4*2)+(7*2)+(7*3)+... ,where the expansion stops when a(n) has n+1 number of terms. The expansion starts at (1*0), and progresses by alternating addition of 1 to the second number and 3 to the first number. - _Arlu Genesis A. Padilla_, Jun 04 2014
%C Taking the absolute values of each n-th difference and excluding the first n terms of each mentioned sequence, A002717 has the first difference A006578 (see formula of Michael Somos dated Jun 09 2014), the second difference A032766 (see 'partial sum' crossref), the third difference A000034, the fourth difference A000012, and the fifth to n-th difference A000004. - _Arlu Genesis A. Padilla_, Jun 12 2014
%D J. H. Conway and R. K. Guy, The Book of Numbers, p. 83.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002717/b002717.txt">Table of n, a(n) for n = 0..1000</a>
%H Ralph E. Edwards et al., <a href="http://www.jstor.org/stable/2688056">Problem 889: A well-known problem</a>, Math. Mag., 47 (1974), 289-292.
%H F. Gerrish, <a href="http://www.jstor.org/stable/3613774">How many triangles</a>, Math. Gaz., 54 (1970), 241-246.
%H J. Halsall, <a href="http://www.jstor.org/stable/3613117">An interesting series</a>, Math. Gaz., 46 (1962), 55-56.
%H J. Halsall, <a href="/A002717/a002717_1.pdf">An interesting series</a>, Math. Gaz., 46 (1962), 55-56. [Annotated scanned copy]
%H M. E. Larsen, <a href="http://www.math.ku.dk/~mel/mel.pdf">The eternal triangle - a history of a counting problem</a>, College Math. J., 20 (1989), 370-392.
%H B. D. Mastrantone, <a href="http://www.jstor.org/stable/3612392">How Many Triangles?</a>, Math. Gaz., 55 (1971), 438-440.
%H Hugo Pfoertner, <a href="/A002717/a002717.pdf">Illustration of A002717(5) and A002717(6)</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H L. Smiley, <a href="http://www.jstor.org/stable/2690473">A Quick Solution of Triangle Counting</a>, Mathematics Magazine, 66, #1, Feb '93, p. 40.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangleTiling.html">Triangle Tiling.</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F a(n) = (1/16)*[2n(2n+1)(n+2)+cos(Pi*n)-1]. - Justin C. Bozonier (justinb67(AT)excite.com), Dec 05 2000
%F a(m+1)-2a(m)+2a(m-2)-a(m-3) = 3. - _Len Smiley_, Oct 08 2001
%F a(n) = (2n(2n+1)(n+2)+(-1)^n-1)/16. - Wesley Petty (Wesley.Petty(AT)mail.tamucc.edu), Oct 25 2003
%F a(n) = A000292(n-1) + A002623(n-2). - _Hugo Pfoertner_, Mar 06 2004
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*k*binomial(k+1,2).
%F G.f.: x(1+2x)/((1+x)(1-x)^4). - _Simon Plouffe_ in his 1992 dissertation (with a different offset).
%F a(0)=0, a(1)=1, a(2)=5, a(3)=13, a(4)=27, a(n)=3*a(n-1)-2*a(n-2)-2*a(n-3)+ 3*a(n-4)- a(n-5). - _Harvey P. Dale_, Jan 20 2013
%F a(n) = a(n-1) + A016777(floor(0.5n)*floor(0.5+0.5n). - _Arlu Genesis A. Padilla_, Jun 04 2014
%F a(-n) = - A045947(n). a(n) = a(n-1) + A006578(n). - _Michael Somos_, Jun 09 2014
%F a(n) = Sum_{i=1..n} T(n-i+1)+T(n-2*i+1), where T(n)=n*(n+1)/2=A000217(n) if n>0 and 0 if n<=0. So we have a(n+2)-a(n)=(n+2)^2+(n+1)*(n+2)/2. - _Maurice Mischler_, Sep 08 2014
%F E.g.f.: (x*(2*x^2 + 11*x + 9)*cosh(x) + (2*x^3 + 11*x^2 + 9*x - 1)*sinh(x))/8. - _Stefano Spezia_, Jul 19 2022
%e f(3)=13 because the following figure contains 13 triangles if horizontal bars are added:
%e ....... /\
%e ...... /\/\
%e ..... /\/\/\
%e G.f. = x + 5*x^2 + 13*x^3 + 27*x^4 + 48*x^5 + 78*x^6 + 118*x^7 + 170*x^8 + ...
%p A002717:=n->floor(n*(n+2)*(2*n+1)/8); seq(A002717(n), n=0..100);
%t Table[Floor[n(n+2)(2n+1)/8],{n,0,50}] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,1,5,13,27},50] (* _Harvey P. Dale_, Jan 20 2013 *)
%o (PARI) {a(n) = n * (n+2) * (2*n+1) \ 8};
%o (Magma) [Floor(n*(n+2)*(2*n+1)/8): n in [0..50]]; // _Wesley Ivan Hurt_, Jun 04 2014
%Y 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 Bisections: A135712 (odd part), A135713 (even part).
%Y Cf. A006578, A032766, A000034, A070893. - _Richard Choulet_, Jan 28 2010
%Y Cf. A045947, A016777.
%Y Cf. A000004, A000012, A000217, A220053.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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