A107350 -id:A107350 - cp's OEIS frontend

A060005 Number of ways of partitioning the integers {1,2,..,4n} into two (unordered) sets such that the sums of parts are equal in both sets (parts in either set will add up to (4n)*(4n+1)/4). Number of solutions to {1 +- 2 +- 3 +- ... +- 4n=0}.

1, 1, 7, 62, 657, 7636, 93846, 1199892, 15796439, 212681976, 2915017360, 40536016030, 570497115729, 8110661588734, 116307527411482, 1680341334827514, 24435006625667338, 357366669614512168, 5253165510907071170

Offset: 0

Roland Bacher, Mar 15 2001

nonn

			a(1)=1 since there is only one way of partitioning {1,2,3,4} into two sets of equal sum, namely {1,4}, {2,3}.

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 0..835 (terms < 10^1000, first 251 terms from Alois P. Heinz)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
L. Sallows, M. Gardner, R. K. Guy and D. E. Knuth, Serial isogons of 90 degrees, Math. Mag. 64 (1991), 315-324.

Cf. A060468, A007219, A107350, a(n)=A058377(4n), A227850.

b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
      `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i), i-1) +b(n+i, i-1)))
    end:
a:= n-> b(4*n, 4*n-1):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 30 2011

b[n_, i_] := b[n, i] = Module[{m = i*(i+1)/2}, If[n > m, 0, If[n == m, 1, b[Abs[n-i], i-1] + b[n+i, i-1]]]]; a[n_] := b[4*n, 4*n-1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Sep 26 2013, translated from Alois P. Heinz's Maple program *)

a(0)=1 and a(n) is half the coefficient of q^0 in product((q^(-k)+q^k), k=1..4*n) for n >= 1.

For n>=1, a(n) = (1/Pi)*16^n*J(4n) where J(n) = integral(t=0, Pi/2, cos(t)cos(2t)...cos(nt)dt). - Benoit Cloitre, Sep 24 2006

More terms from Alois P. Heinz, Oct 30 2011

A007219 Number of golygons of order 8n (or serial isogons of order 8n).

Original entry on oeis.org

1, 28, 2108, 227322, 30276740, 4541771016, 739092675672, 127674038970623, 23085759901610016, 4327973308197103600, 835531767841066680300, 165266721954751746697155, 33364181616540879268092840

Offset: 1

Simon Plouffe

A golygon of order N is a closed path along the streets of the Manhattan grid with successive edge lengths of 1,2,3,...,N (returning to the starting point after the edge of length N), and which makes a 90-degree turn (left or right) after each edge.

It is known that the order N must be a multiple of 8.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 92.

Vaclav Kotesovec, Table of n, a(n) for n = 1..100
A. K. Dewdney, An odd journey along even roads leads to home in Golygon City, Mathematical Recreations Column, Scientific American, July 1990, pp. 118-121.
A. K. Dewdney, Illustration of the unique golygon of order 8, from the article "An odd journey along even roads leads to home in Golygon City", Mathematical Recreations Column, Scientific American, July 1990, pp. 118-121.
A. K. Dewdney, Illustration of the 28 golygons of order 16, from the article "An odd journey along even roads leads to home in Golygon City", Mathematical Recreations Column, Scientific American, July 1990, pp. 118-121.
Adam P. Goucher, Golygons and golyhedra
L. Sallows, M. Gardner, R. K. Guy and D. E. Knuth, Serial isogons of 90 degrees, Math. Mag. 64 (1991), 315-324.
Eric Weisstein's World of Mathematics, Golygon

Cf. A060005, A107350.

cp's OEIS Frontend

A060005 Number of ways of partitioning the integers {1,2,..,4n} into two (unordered) sets such that the sums of parts are equal in both sets (parts in either set will add up to (4n)*(4n+1)/4). Number of solutions to {1 +- 2 +- 3 +- ... +- 4n=0}.

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