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A005581 a(n) = (n-1)*n*(n+4)/6.
(Formerly M1744)
48
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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
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
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.
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.
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.
Eric Weisstein's World of Mathematics, Trinomial Coefficient.
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 3<n else 2*n-3. - Peter 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)
EXAMPLE
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 + ...
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`(3<n, 3, 2*n-3), -n, -1/2):
seq(simplify(A005581(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 *)
PROG
(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
(Maxima) A005581(n):=(n-1)*n*(n+4)/6$ makelist(A005581(n), n, 0, 50); /* Martin Ettl, Dec 18 2012 */
(Sage) [(n-1)*n*(n+4)/6 for n in range(50)] # Danny Rorabaugh, Apr 20 2015
(Magma) [(n-1)*n*(n+4)/6 : n in [0..50]]; // Wesley Ivan Hurt, Jul 10 2015
CROSSREFS
Cf. A127672 (Chebyshev C).
Sequence in context: A162420 A130883 A360284 * A064468 A225311 A241526
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)