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A002717
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Floor(n(n+2)(2n+1)/8).
(Formerly M3827 N1569)
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15
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0, 1, 5, 13, 27, 48, 78, 118, 170, 235, 315, 411, 525, 658, 812, 988, 1188, 1413, 1665, 1945, 2255, 2596, 2970, 3378, 3822, 4303, 4823, 5383, 5985, 6630, 7320, 8056, 8840, 9673, 10557, 11493, 12483, 13528, 14630, 15790, 17010, 18291, 19635, 21043, 22517
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of triangles in triangular matchstick arrangement of side n.
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). [From Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 28 2010]
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REFERENCES
| 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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| 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.
Index to sequences with linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).
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FORMULA
| 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
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).
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EXAMPLE
| f(3)=13 because the following figure contains 13 triangles:
....... /\
...... /\/\
..... /\/\/\
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MAPLE
| A002717:=n->floor(n*(n+2)*(2*n+1)/8);
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PROG
| (PARI) a(n)=if(n<0, 0, n*(n+2)*(2*n+1)\8)
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CROSSREFS
| 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.
Cf. A006578, A032766, A000034, A070893. [From Richard Choulet (richardchoulet(AT)yahoo.fr), Jan 28 2010]
Sequence in context: A008580 A123326 A025193 * A023541 A079989 A062480
Adjacent sequences: A002714 A002715 A002716 * A002718 A002719 A002720
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Plouffe g.f. edited by R. J. Mathar, May 12 2008
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