A005581 a(n) = (n-1)*n*(n+4)/6.

0, 0, 2, 7, 16, 30, 50, 77, 112, 156, 210, 275, 352, 442, 546, 665, 800, 952, 1122, 1311, 1520, 1750, 2002, 2277, 2576, 2900, 3250, 3627, 4032, 4466, 4930, 5425, 5952, 6512, 7106, 7735, 8400, 9102, 9842, 10621, 11440, 12300, 13202, 14147, 15136, 16170

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

A class of Boolean functions of n variables and rank 2.

Also, number of inscribable triangles within a (n+4)-gon sharing with them its vertices but not its sides. - Lekraj Beedassy, Nov 14 2003

a(n) = A111808(n,3) for n > 2. - Reinhard Zumkeller, Aug 17 2005

If X is an n-set and Y a fixed 2-subset of X then a(n-2) is equal to the number of (n-3)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

The sequence starting with offset 2 = binomial transform of [2, 5, 4, 1, 0, 0, 0, ...]. - Gary W. Adamson, Mar 20 2009

Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 4, a(n-4) is the number of (0,1) n X n matrices A <= P^(-1) + I + P having exactly two 1's in every row and column with perA=8. - Vladimir Shevelev, Apr 12 2010

Also arises as the number of triples of edges which can be chosen as the cut-points in the "three-opt" heuristic for a traveling salesman problem on (n+4) nodes. - James McDermott, Jul 10 2015

a(n) = risefac(n, 3)/3! - n is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 3 and dimension n. Here risefac is the rising factorial. - Wolfdieter Lang, Dec 10 2015

For n >= 2, a(n) is the number of characters in a word Q formed by concatenating all 'directed' ( left to right or vice versa), unrearranged subwords, from length 1 to (n-1), of a length (n-1) word q- allowing for the appearance of repeated subwords- and simply inserting an extra character for all subwords thus concatenated. - Christopher Hohl, May 30 2019

Examples

			In hexagon ABCDEF, the "interior" triangles are ACE and BDF, and a(6-4)=a(2)=2. - _Toby Gottfried_, Nov 12 2011
G.f. = 2*x^2 + 7*x^3 + 16*x^4 + 30*x^5 + 50*x^6 + 77*x^7 + 112*x^8 + ...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
• Joseph D. Konhauser, Dan Velleman and Stan Wagon,, Which Way Did the Bicycle Go?, MAA, 1996, p. 177.
• V. S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, Vol. 3 (1992), pp. 15-19. - Vladimir Shevelev, Apr 12 2010
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• A. M. Yaglom and I. M. Yaglom, Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the case k=3) (First published: San Francisco: Holden-Day, Inc., 1964).

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972. [Alternative scanned copy]
• Armen G. Bagdasaryan and Ovidiu Bagdasar, On some results concerning generalized arithmetic triangles, Electronic Notes in Discrete Mathematics, Vol. 67 (2018), pp. 71-77.
• Beáta Bényi, Miguel Méndez, José L. Ramírez and Tanay Wakhare, Restricted r-Stirling Numbers and their Combinatorial Applications, arXiv:1811.12897 [math.CO], 2018.
• John Elias, Illustration: Triangular Staircasing
• Richard K. Guy, Letter to N. J. A. Sloane, 1987.
• Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988.
• F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart., Vol. 49, No. 3 (2011), pp. 231-243.
• Milan Janjic, Two Enumerative Functions.
• V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, Vol. 11, No. 4 (1999), pp. 127-138.
• V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, Vol. 9, No. 6 (1999), pp. 593-605.
• Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
• Alice McLeod and William Moser, Counting cyclic binary strings, Math. Mag., Vol. 80, No. 1 (2007), pp. 29-37.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Dead link]
• C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Cached copy, May 15 2013]
• Eric Weisstein's World of Mathematics, Trinomial Coefficient.
• Index entries for sequences related to Boolean functions.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Cf. A000217, A000292, A005582, A005586, A111808.

Cf. A176222, A000211, A052928, A128209.

Cf. A127672 (Chebyshev C).

Programs

Magma

[(n-1)*n*(n+4)/6 : n in [0..50]]; // Wesley Ivan Hurt, Jul 10 2015

Maple

A005581 := n->(n-1)*n*(n+4)/6: seq(A005581(n), n=0..50);
a:=n->sum ((j+3)*j/2,j=0..n): seq(a(n),n=-1..49); # Zerinvary Lajos, Dec 17 2006
seq((n+3)*binomial(n,3)/n, n=1..46); # Zerinvary Lajos, Feb 28 2007
A005581:=-(-2+z)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
seq(sum(binomial(n,m), m=1..3)+n^2,n=-1..44); # Zerinvary Lajos, Jun 19 2008
A005581 := n -> GegenbauerC(`if`(3A005581(n)), n=0..50); # Peter Luschny, May 10 2016

Mathematica

Table[(n-1)*n*(n+4)/6, {n,0,50}] (* Stefan Steinerberger, Apr 10 2006 *)
LinearRecurrence[{4,-6,4,-1},{0,0,2,7},50] (* Harvey P. Dale, Sep 22 2012 *)

Maxima

A005581(n):=(n-1)*n*(n+4)/6$ makelist(A005581(n),n,0,50); /* Martin Ettl, Dec 18 2012 */

PARI

{a(n) = n * (n+4) * (n-1) / 6}; /* Michael Somos, Apr 13 2007 */

PARI

concat([0, 0], Vec((x^2)*(2-x)/(1-x)^4 + O(x^50))) \\ Altug Alkan, Dec 10 2015

Sage

[(n-1)*n*(n+4)/6 for n in range(50)] # Danny Rorabaugh, Apr 20 2015

Formula

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

a(n) = binomial(n+1, n-2) + binomial(n, n-2).

a(n) = A027907(n, 3), n >= 0 (fourth column of trinomial coefficients). - N. J. A. Sloane, May 16 2003

Convolution of {1, 2, 3, ...} with {2, 3, 4, ...}. - Jon Perry, Jun 25 2003

a(n+2) = 2*te(n) - te(n-1), e.g., a(5) = 2*te(3) - te(2) = 2*20 - 10 = 30, where te(n) are the tetrahedral numbers A000292. - Jon Perry, Jul 23 2003

a(n) is the coefficient of x^3 in the expansion of (1+x+x^2)^n. For example, a(1)=0 since (1+x+x^2)^1=1+x+x^2. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 09 2007

E.g.f.: (x^2 + x^3/6) * exp(x). - Michael Somos, Apr 13 2007

a(n) = - A005586(-4-n) for all n in Z. - Michael Somos, Apr 13 2007

a(n) = C(4+n,3)-(n+4)*(n+1), since C(4+n,3) = number of all triangles in (n+4)-gon, and (n+4)*(n+1)=number of triangles with at least one of the edges included. Example: n=0,in a square, all 4 possible triangles include some of the square's edges and C(4+n,3)-(n+4)*(n+1)=4-4*1=0 = number of other triangles = a(0). - Toby Gottfried, Nov 12 2011

a(n) = 2*binomial(n,2) + binomial(n,3). - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012

a(0)=0, a(1)=0, a(2)=2, a(3)=7, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Sep 22 2012

a(n) = A000292(n-1) + A000217(n-1) for all n in Z. - Michael Somos, Jul 29 2015

a(n+2) = -A127672(6+n, n), n >= 0, with A127672 giving the coefficients of Chebyshev's C polynomials. See the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 10 2015

a(n) = GegenbauerC(N, -n, -1/2) where N = 3 if 3Peter Luschny, May 10 2016

From Amiram Eldar, Jan 09 2022: (Start)

Sum_{n>=2} 1/a(n) = 163/200.

Sum_{n>=2} (-1)^n/a(n) = 12*log(2)/5 - 253/200. (End)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000