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A005581 (n-1)*n*(n+4)/6.
(Formerly M1744)
31
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

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

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

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 797.

V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

J. D. E. Konhauser et al., 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, 3(1992),15-19. [From 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].

F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers.

Milan Janjic, Two Enumerative Functions

A. McLeod and W. O. J. Moser, Counting cyclic binary strings, Math. Mag., 80 (No. 1, 2007), 29-37.

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.

C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube.

Eric Weisstein's World of Mathematics, Trinomial Coefficient

Index entries for sequences related to Boolean functions

FORMULA

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

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

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) = 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

EXAMPLE

In hexagon ABCDEF, the "interior" triangles are ACE and BDF, and a(6-4)=a(2)=2. - Toby Gottfried, Nov 12 2011

MAPLE

A005581 := n->(n-1)*n*(n+4)/6;

a:=n->sum ((j+3)*j/2, j=0..n): seq(a(n), n=-1..44); # 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

MATHEMATICA

Table[(n - 1)*n*(n + 4)/6, {n, 0, 40}] (* 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 */

(Maxima) A005581(n):=(n-1)*n*(n+4)/6$ makelist(A005581(n), n, 0, 20); /* Martin Ettl, Dec 18 2012 */

CROSSREFS

Cf. A005582. a(n)= A027907(n, 3), n >= 0 (fourth column of trinomial coefficients).

Cf. A000292.

A005586(n)= -a(-4-n).

Cf. A176222, A000211, A052928, A128209. - Vladimir Shevelev, Apr 12 2010

Sequence in context: A070169 A162420 A130883 * A064468 A225311 A241526

Adjacent sequences:  A005578 A005579 A005580 * A005582 A005583 A005584

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified September 2 19:51 EDT 2014. Contains 246367 sequences.