A007219 -id:A007219 - 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

A006718 Number of golygons of length 8n.

Original entry on oeis.org

1, 4, 112, 8432, 909288, 121106960, 18167084064, 2956370702688, 510696155882492, 92343039606440064, 17311893232788414400, 3342127071364266721200, 661066887819006986788620, 133456726466163517072371360

Offset: 0

N. J. A. Sloane

nonn

A007219 is the main entry for golygons.

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.

Seiichi Manyama, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Golygon

Cf. A060468, A063865, A292476.

See A007219 for much more information about golygons.

p1[n_] := Product[x^k + 1, {k, 1, n - 1, 2}] // Expand; p2[n_] := Product[x^k + 1, {k, 1, n/2}] // Expand; c[n_] := Coefficient[p1[n], x, n^2/8] * Coefficient[p2[n], x, n (n/2 + 1)/8]; a[n_] := c[8*n]; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Jul 24 2013, after Eric W. Weisstein *)

a(n) = 4 * A007219(n) for n > 0. - Charles R Greathouse IV, Apr 29 2012

a(n) = A060468(n) * A292476(2*n) = A063865(4*n) * A292476(2*n). - Seiichi Manyama, Sep 18 2017

a(0) = 1 prepended by Seiichi Manyama, Sep 18 2017

A107350 Number of isogons with a certain property.

Original entry on oeis.org

1, 4, 34, 346, 3965, 48396, 615966, 8082457, 108545916, 1484716135, 20612084010, 289688970195, 4113620233260, 58930127470164, 850641610106596, 12360278974175769, 180648953113093368, 2653875476976308643, 39167191622334514398, 580439539153823110678, 8633956582855204662785

Offset: 1

N. J. A. Sloane, May 23 2005

This and A060005 appear in the reference as incidental sequences when computing A007219.

Seiichi Manyama, Table of n, a(n) for n = 1..200
L. Sallows, M. Gardner, R. K. Guy and D. E. Knuth, Serial isogons of 90 degrees, Math. Mag. 64 (1991), 315-324.

Cf. A007219, A060005, A292476.

A107350 := proc(n) res := 1 ; for i from 0 to 4*n-1 do res := taylor(res*(1+x^(2*i+1)),x=0,8*n^2+1) ; od ; coeftayl(res,x=0,8*n^2)/2 ; end: for n from 1 to 25 do printf("%d, ",A107350(n)) ; od ; # R. J. Mathar, May 08 2007

a[n_] := SeriesCoefficient[Product[x^(2k - 1) + 1/x^(2k - 1), {k, 1, 4n}], {x, 0, 0}]/2;
Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Mar 10 2023 *)

product[ {1+x^(2i+1)},i=0,1,...,4n-1] = 1+...+2*a(n)*x^(8n^2)+.... (g.f.). - R. J. Mathar, May 08 2007

a(n) = A292476(2*n)/2. - Seiichi Manyama, Sep 18 2017

More terms from R. J. Mathar, May 08 2007

A292700 1/4 of the number of paths that are made of alternate vertical and horizontal 4n consecutive prime (> 2 and < prime(4n+2)) steps. (Each path begins and ends at the same point but is allowed to intersect itself.)

Original entry on oeis.org

0, 0, 0, 2, 4, 40, 756, 8466, 70664, 788240, 11086299, 114208465, 1536869680, 19129734630, 228346732944, 3059857239120, 38601632542723, 509554928113867, 6793745301391293, 92409631618813044, 1240139159309482820, 16995549322569645324, 236256012348427662180

Offset: 1

Seiichi Manyama, Sep 21 2017

nonn

			U, D, L and R express up, down, left and right.
n = 4:
3R, 5U, 7L, 11D, 13L, 17U, 19R, 23U, 29L, 31U, 37R, 41U, 43R, 47D, 53L, 59D.
3R, 5U, 7L, 11U, 13L, 17D, 19R, 23U, 29L, 31U, 37R, 41D, 43R, 47U, 53L, 59D.

Cf. A000040, A007219, A292698, A292699.

Constant term in the expansion of (1/4) * Product_{k=1..2*n} (x^prime(2*k) + 1/x^prime(2*k)) * Product_{k=1..2*n} (y^prime(2*k+1) + 1/y^prime(2*k+1)).

a(n) = A292698(n) * A292699(n).

A273089 Number of lattice n-gons with ordered sides 1, 2, 3, ..., n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 5, 6, 0, 0, 584, 882, 0, 0, 18026, 194741, 0, 0, 644414, 960834, 0, 0, 229910636

Offset: 1

Bernardo Recamán, May 14 2016

First 16 terms calculated by Stefan Kohl.

			a(8) = 3:
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a(11) = 5:
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. - _Hugo Pfoertner_, Mar 20 2020

Stefan Kohl, Lattice n-gons with ordered side lengths 1,2,3,…,n, Answer on MathOverflow, May 4, 2016.
MathOverflow, Lattice n-gons with ordered side lengths 1,2,3,...,n
Giovanni Resta, Examples of n-gons with minimal and maximal area
L. Sallows, M. Gardner, R. K. Guy and D. E. Knuth, Serial isogons of 90 degrees, Math. Mag. 64 (1991), 315-324.
Wikipedia, Golygon

Cf. A007219.

Typo in a(15) corrected and a(17)-a(27) added by Giovanni Resta, Mar 26 2020

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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Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A006718 Number of golygons of length 8n.

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Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A107350 Number of isogons with a certain property.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A292700 1/4 of the number of paths that are made of alternate vertical and horizontal 4*n consecutive prime (> 2 and < prime(4*n+2)) steps. (Each path begins and ends at the same point but is allowed to intersect itself.)

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Author

Keywords

Examples

Crossrefs

Formula

A273089 Number of lattice n-gons with ordered sides 1, 2, 3, ..., n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A292700 1/4 of the number of paths that are made of alternate vertical and horizontal 4n consecutive prime (> 2 and < prime(4n+2)) steps. (Each path begins and ends at the same point but is allowed to intersect itself.)