

A062890


Number of quadrilaterals that can be formed with perimeter n. In other words, number of partitions of n into four parts such that the sum of any three is more than the fourth.


14



0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 12, 16, 18, 23, 24, 31, 33, 41, 43, 53, 55, 67, 69, 83, 86, 102, 104, 123, 126, 147, 150, 174, 177, 204, 207, 237, 241, 274, 277, 314, 318, 358, 362, 406, 410, 458, 462, 514, 519, 575, 579, 640, 645, 710
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,8


COMMENTS

Partition sets of n into four parts (sides) such that the sum of any three is more than the fourth do not uniquely define a quadrilateral, even if it is further constrained to be cyclic. This is because the order of adjacent sides is important. E.g. the partition set [1,1,2,2] for a perimeter n=6 can be reordered to generate two noncongruent cyclic quadrilaterals, [1,2,1,2] and [1,1,2,2], where the first is a rectangle and the second a kite.  Frank M Jackson, Jun 29 2012


LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 19.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: kgon partitions, Bull. Austral Math. Soc., 64 (2001), 321329.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,0,0,1,1,1,1,1,1).


FORMULA

G.f.: x^4*(1+x+x^5)/((1x^2)*(1x^3)*(1x^4)*(1x^6)).
a(2*n+6) = A026810(2*n+6)  A000601(n), a(2*n+7) = A026810(2*n+7)  A000601(n) for n >= 0.  Seiichi Manyama, Jun 08 2017
From Wesley Ivan Hurt, Jan 01 2021: (Start)
a(n) = a(n1) + a(n2)  a(n3) + a(n4)  a(n5)  a(n8) + a(n9)  a(n10) + a(n11) + a(n12)  a(n13).
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((nk)/3)} Sum_{i=j..floor((njk)/2)} sign(floor((i+j+k)/(nijk+1))). (End)


EXAMPLE

a(7) = 2 as the two partitions are (1,2,2,2), (1,1,2,3) and in each sum of any three is more than the fourth.


MATHEMATICA

CoefficientList[Series[x^4*(1+x+x^5)/((1x^2)*(1x^3)*(1x^4)*(1x^6)), {x, 0, 60}], x] (* Frank M Jackson, Jun 09 2017 *)


CROSSREFS

Number of kgons that can be formed with perimeter n: A005044 (k=3), this sequence (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).
Sequence in context: A169693 A180121 A331418 * A259626 A058586 A090467
Adjacent sequences: A062887 A062888 A062889 * A062891 A062892 A062893


KEYWORD

nonn,easy


AUTHOR

Amarnath Murthy, Jun 29 2001


EXTENSIONS

More terms from Vladeta Jovovic and Dean Hickerson, Jul 01 2001


STATUS

approved



