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A028723 a(n) = (1/4)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2). 7
0, 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150, 225, 315, 441, 588, 784, 1008, 1296, 1620, 2025, 2475, 3025, 3630, 4356, 5148, 6084, 7098, 8281, 9555, 11025, 12600, 14400, 16320, 18496, 20808, 23409, 26163, 29241, 32490, 36100, 39900, 44100, 48510, 53361, 58443 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

It is not known if A000241 and A028723 agree.

Conjectured to be crossing number of complete graph K_n, see A000241.

Also the maximum number of rectangles that can be formed from n lines. - Erich Friedman

Number of symmetric Dyck paths of semilength n and having five peaks. E.g., a(6)=3 because we have U*DU*DUU*DDU*DU*D, U*DUU*DU*DU*DDU*D and UU*DU*DU*DU*DU*DD, where U=(1,1), D=(1,-1) and * indicates a peak. - Emeric Deutsch, Jan 12 2004

a(n-5) is the number of length n words, w(1),w(2),...,w(n) on alphabet {0,1,2} such that w(i)>=w(i+2) for all i. - Geoffrey Critzer, Mar 15 2014

a(n-1) is the number of length n binary strings beginning with a 1 that have exactly two pairs of consecutive 0s and two pairs of consecutive 1s. - Jeremy Dover, Jul 04 2016

Consider the partitions of n into two parts (p,q). Then 2*a(n+2) represents the total volume of the family of rectangular prisms with dimensions p, q and |q - p|. - Wesley Ivan Hurt, Apr 12 2018

REFERENCES

Martin Gardner, Knotted Doughnuts, Chapter 11, pages 133-144.

C. Thomassen, Embeddings and Minors, in Handbook of Combinatorics, p. 314.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Bernardo M. Abrego, Oswin Aichholzer, Silvia Fernandez-Merchant, Pedro Ramos, Gelasio Salazar, The 2-Page Crossing Number of K_n, arXiv:1206.5669 [math.CO], 2012.

Bernardo M. Abrego, Oswin Aichholzer, Silvia Fernandez-Merchant, Pedro Ramos, Gelasio Salazar, The 2-Page Crossing Number of K_n Discrete Comput. Geom. 49 (2013), no. 4, 747--777. MR3068573.

J. Dolan et al., su(3) and Zarankiewicz's conjecture

Dhruv Mubayi, Counting substructures II: Hypergraphs, Combinatorica 33 (2013), no. 5, 591--612. MR3132928

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

FORMULA

If n even, n*(n-2)^2 *(n-4)/64; if n odd, (n-1)^2 *(n-3)^2/64.

G.f.: x^5*(1+x+x^2)/((1-x)^5*(1+x)^3). - Emeric Deutsch, Jan 12 2004

For n>2, a(n) = A007590(n-3)*A007590(n-1)/16. - Richard R. Forberg, Dec 03 2013

a(n) = (n^4-8*n^3+18*n^2-12*n+2*n*(n-2)*((1+(-1)^n)/2)+(2*n-3)^2*((1-(-1)^n)/2))/64. - Luce ETIENNE, Mar 22 2014

Euler transform of length 3 sequence [3, 3, -1]. - Michael Somos, Nov 02 2014

a(n) = a(4-n) for all n in Z. - Michael Somos, Nov 02 2014

0 = -3 + a(n) - a(n+1) - 3*a(n+2) + 3*a(n+3) + 3*a(n+4) - 3*a(n+5) - a(n+6) + a(n+7) for all n in Z. - Michael Somos, Nov 02 2014

0 = a(n)*(+a(n+2) + a(n+3)) + a(n+1)*(-3*a(n+2) +a(n+3)) for all n in Z. - Michael Somos, Nov 02 2014

a(n+1)^2 - a(n)*a(n+2) = binomial(n/2, 2)^3 for all even n in Z ( = 0 if n odd). - Michael Somos, Nov 02 2014

a(n)*(a(n+1) + a(n+2)) +a(n+1)*(-3*a(n+1) + a(n+2)) = 0 for all even n in Z ( = k^4 * (k^2 - 1) / 4 if n = 2*k + 1). - Michael Somos, Nov 02 2014

EXAMPLE

G.f. = x^5 + 3*x^6 + 9*x^7 + 18*x^8 + 36*x^9 + 60*x^10 + 100*x^11 + ...

MAPLE

A028723:=n->(1/4)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2); seq(A028723(n), n=0..100); # Wesley Ivan Hurt, Nov 01 2013

MATHEMATICA

Table[ If[ EvenQ[n], n(n - 2)^2(n - 4)/64, (n - 1)^2(n - 3)^2/64], {n, 0, 50} ]

Table[(n^4 - 8 n^3 + 18 n^2 - 12 n + 2 n (n - 2) ((1 + (- 1)^n)/2) + (2 n - 3)^2 ((1 - (-1)^n)/2))/64, {n, 0, 50}] (* Vincenzo Librandi, Mar 23 2014 *)

LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 0, 0, 0, 0, 1, 3, 9}, 50] (* Harvey P. Dale, Sep 13 2018 *)

PROG

(PARI) a(n) = if (n % 2, (n-1)^2 *(n-3)^2/64, n*(n-2)^2 *(n-4)/64); \\ Michel Marcus, Nov 02 2013

(PARI) {a(n) = prod(k=0, 3, (n - k) \ 2) / 4}; /* Michael Somos, Nov 02 2014 */

(MAGMA) [(n^4-8*n^3+18*n^2-12*n+2*n*(n-2)*((1+(-1)^n)/2)+(2*n-3)^2*((1-(-1)^n)/2))/64: n in [0..50]]; // Vincenzo Librandi, Mar 23 2014

CROSSREFS

Cf. A000241, A006918, A028723.

Sequence in context: A132920 A127645 A000241 * A213291 A264365 A291143

Adjacent sequences:  A028720 A028721 A028722 * A028724 A028725 A028726

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vincenzo Librandi, Mar 23 2014

STATUS

approved

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Last modified February 15 20:07 EST 2019. Contains 320138 sequences. (Running on oeis4.)