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A002717 Floor(n(n+2)(2n+1)/8).
(Formerly M3827 N1569)
18
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; text; internal format)
OFFSET

0,3

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, Jan 28 2010]

Row sums of A220053, for n > 0. - Reinhard Zumkeller, Dec 03 2012

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein's World of Mathematics, Triangle Tiling.

Index to sequences with linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

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, 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). - Simon Plouffe in his 1992 dissertation (with a different offset).

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

EXAMPLE

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

....... /\

...... /\/\

..... /\/\/\

MAPLE

A002717:=n->floor(n*(n+2)*(2*n+1)/8); seq(A002717(n), n=0..100);

MATHEMATICA

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 *)

PROG

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

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, Jan 28 2010]

Sequence in context: A212151 A123326 A025193 * A023541 A079989 A062480

Adjacent sequences:  A002714 A002715 A002716 * A002718 A002719 A002720

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 18 19:18 EDT 2014. Contains 240733 sequences.