

A079908


Solution to the Dancing School Problem with 3 girls and n+3 boys: f(3,n).


35



1, 4, 14, 36, 76, 140, 234, 364, 536, 756, 1030, 1364, 1764, 2236, 2786, 3420, 4144, 4964, 5886, 6916, 8060, 9324, 10714, 12236, 13896, 15700, 17654, 19764, 22036, 24476, 27090, 29884, 32864, 36036, 39406, 42980, 46764, 50764, 54986, 59436
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OFFSET

0,2


COMMENTS

The Dancing School Problem: a line of g girls (g>0) with integer heights ranging from m to m+g1 cm and a line of g+h boys (h>=0) ranging in height from m to m+g+h1 cm are facing each other in a dancing school (m is the minimal height of both girls and boys).
A girl of height l can choose a boy of her own height or taller with a maximum of l+h cm. We call the number of possible matchings f(g,h).
This problem is equivalent to a rooks problem: The number of possible placings of g nonattacking rooks on a g X g+h chessboard with the restriction i <= j <= i+h for the placement of a rook on square (i,j): f(g,h) = per(B), the permanent of the corresponding (0,1)matrix B, b(i, j)=1 if and only if i <= j <= i+h
f(g,h) = per(B), the permanent of the (0,1)matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h.
For fixed g, f(g,n) is polynomial in n for n >= g2. See reference.


LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Lute Kamstra, Problem 29 in Vol. 5/3, No. 1, Mar 2002, p. 96. See also Vol. 5/3, Nos. 3 and 4.
Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283285.
Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.
Jaap Spies, Sage program to compute f(g,h).
Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).
Index entries for linear recurrences with constant coefficients, signature (4,6,4,1).


FORMULA

a(n) = max(1, n^3 + 3*n), found by elementary counting.
G.f.: 1+2*x*(2x+2*x^2)/(1x)^4.  R. J. Mathar, Nov 19 2007


MATHEMATICA

Join[{1}, Array[#^3+3*#&, 5!, 1]] (* Vladimir Joseph Stephan Orlovsky, Jan 08 2011 *)


PROG

(PARI) concat(1, vector(30, n, n^3+3*n)) \\ Derek Orr, Jun 18 2015


CROSSREFS

Cf. A061989, A079909A079928.
Cf. Essentially the same as A061989.
Sequence in context: A177110 A213045 A061989 * A038164 A327382 A193522
Adjacent sequences: A079905 A079906 A079907 * A079909 A079910 A079911


KEYWORD

nonn,easy


AUTHOR

Jaap Spies, Jan 28 2003


STATUS

approved



