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%I M1744 #163 Apr 28 2023 17:44:44
%S 0,0,2,7,16,30,50,77,112,156,210,275,352,442,546,665,800,952,1122,
%T 1311,1520,1750,2002,2277,2576,2900,3250,3627,4032,4466,4930,5425,
%U 5952,6512,7106,7735,8400,9102,9842,10621,11440,12300,13202,14147,15136,16170
%N a(n) = (n-1)*n*(n+4)/6.
%C A class of Boolean functions of n variables and rank 2.
%C Also, number of inscribable triangles within a (n+4)-gon sharing with them its vertices but not its sides. - _Lekraj Beedassy_, Nov 14 2003
%C a(n) = A111808(n,3) for n > 2. - _Reinhard Zumkeller_, Aug 17 2005
%C 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
%C The sequence starting with offset 2 = binomial transform of [2, 5, 4, 1, 0, 0, 0, ...]. - _Gary W. Adamson_, Mar 20 2009
%C 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
%C 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
%C 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
%C 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
%D 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.
%D Joseph D. Konhauser, Dan Velleman and Stan Wagon,, Which Way Did the Bicycle Go?, MAA, 1996, p. 177.
%D 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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).
%H T. D. Noe, <a href="/A005581/b005581.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972. [Alternative scanned copy]
%H Armen G. Bagdasaryan and Ovidiu Bagdasar, <a href="https://doi.org/10.1016/j.endm.2018.05.012">On some results concerning generalized arithmetic triangles</a>, Electronic Notes in Discrete Mathematics, Vol. 67 (2018), pp. 71-77.
%H Beáta Bényi, Miguel Méndez, José L. Ramírez and Tanay Wakhare, <a href="https://arxiv.org/abs/1811.12897">Restricted r-Stirling Numbers and their Combinatorial Applications</a>, arXiv:1811.12897 [math.CO], 2018.
%H John Elias, <a href="/A005581/a005581.png">Illustration: Triangular Staircasing</a>
%H Richard K. Guy, <a href="/A005712/a005712.pdf">Letter to N. J. A. Sloane/a>, 1987.
%H Richard K. Guy, <a href="/A005581/a005581_1.pdf">Letter to N. J. A. Sloane</a>, Feb 1988.
%H F. T. Howard and Curtis Cooper, <a href="http://www.fq.math.ca/Papers1/49-3/HowardCooper.pdf">Some identities for r-Fibonacci numbers</a>, Fibonacci Quart., Vol. 49, No. 3 (2011), pp. 231-243.
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H V. Jovovic and G. Kilibarda, <a href="http://dx.doi.org/10.4213/dm398">On the number of Boolean functions in the Post classes F^{mu}_8</a>, Diskretnaya Matematika, Vol. 11, No. 4 (1999), pp. 127-138.
%H V. Jovovic and G. Kilibarda, <a href="http://dx.doi.org/10.1515/dma.1999.9.6.593">On the number of Boolean functions in the Post classes F^{mu}_8</a>, (English translation), Discrete Mathematics and Applications, Vol. 9, No. 6 (1999), pp. 593-605.
%H Kyu-Hwan Lee and Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%H Alice McLeod and William Moser, <a href="http://www.jstor.org/stable/27642988">Counting cyclic binary strings</a>, Math. Mag., Vol. 80, No. 1 (2007), pp. 29-37.
%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 C. Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [Dead link]
%H C. Rossiter, <a href="/A005581/a005581.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [Cached copy, May 15 2013]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>.
%H <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F G.f.: (x^2)*(2-x)/(1-x)^4.
%F a(n) = binomial(n+1, n-2) + binomial(n, n-2).
%F a(n) = A027907(n, 3), n >= 0 (fourth column of trinomial coefficients). - _N. J. A. Sloane_, May 16 2003
%F Convolution of {1, 2, 3, ...} with {2, 3, 4, ...}. - _Jon Perry_, Jun 25 2003
%F 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
%F 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
%F E.g.f.: (x^2 + x^3/6) * exp(x). - _Michael Somos_, Apr 13 2007
%F a(n) = - A005586(-4-n) for all n in Z. - _Michael Somos_, Apr 13 2007
%F 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
%F a(n) = 2*binomial(n,2) + binomial(n,3). - _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 22 2012
%F 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
%F a(n) = A000292(n-1) + A000217(n-1) for all n in Z. - _Michael Somos_, Jul 29 2015
%F 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
%F a(n) = GegenbauerC(N, -n, -1/2) where N = 3 if 3<n else 2*n-3. - _Peter Luschny_, May 10 2016
%F From _Amiram Eldar_, Jan 09 2022: (Start)
%F Sum_{n>=2} 1/a(n) = 163/200.
%F Sum_{n>=2} (-1)^n/a(n) = 12*log(2)/5 - 253/200. (End)
%e In hexagon ABCDEF, the "interior" triangles are ACE and BDF, and a(6-4)=a(2)=2. - _Toby Gottfried_, Nov 12 2011
%e G.f. = 2*x^2 + 7*x^3 + 16*x^4 + 30*x^5 + 50*x^6 + 77*x^7 + 112*x^8 + ...
%p A005581 := n->(n-1)*n*(n+4)/6: seq(A005581(n), n=0..50);
%p a:=n->sum ((j+3)*j/2,j=0..n): seq(a(n),n=-1..49); # _Zerinvary Lajos_, Dec 17 2006
%p seq((n+3)*binomial(n,3)/n, n=1..46); # _Zerinvary Lajos_, Feb 28 2007
%p A005581:=-(-2+z)/(z-1)**4; # _Simon Plouffe_ in his 1992 dissertation
%p seq(sum(binomial(n,m), m=1..3)+n^2,n=-1..44); # _Zerinvary Lajos_, Jun 19 2008
%p A005581 := n -> GegenbauerC(`if`(3<n,3,2*n-3), -n, -1/2):
%p seq(simplify(A005581(n)), n=0..50); # _Peter Luschny_, May 10 2016
%t Table[(n-1)*n*(n+4)/6, {n,0,50}] (* _Stefan Steinerberger_, Apr 10 2006 *)
%t LinearRecurrence[{4,-6,4,-1},{0,0,2,7},50] (* _Harvey P. Dale_, Sep 22 2012 *)
%o (PARI) {a(n) = n * (n+4) * (n-1) / 6}; /* _Michael Somos_, Apr 13 2007 */
%o (PARI) concat([0, 0], Vec((x^2)*(2-x)/(1-x)^4 + O(x^50))) \\ _Altug Alkan_, Dec 10 2015
%o (Maxima) A005581(n):=(n-1)*n*(n+4)/6$ makelist(A005581(n),n,0,50); /* _Martin Ettl_, Dec 18 2012 */
%o (Sage) [(n-1)*n*(n+4)/6 for n in range(50)] # _Danny Rorabaugh_, Apr 20 2015
%o (Magma) [(n-1)*n*(n+4)/6 : n in [0..50]]; // _Wesley Ivan Hurt_, Jul 10 2015
%Y Cf. A000217, A000292, A005582, A005586, A111808.
%Y Cf. A176222, A000211, A052928, A128209. - _Vladimir Shevelev_, Apr 12 2010
%Y Cf. A127672 (Chebyshev C).
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000
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