A005044 -id:A005044 - cp's OEIS frontend

A070109 Number of right integer triangles with perimeter n and relatively prime side lengths.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

Right integer triangles have integer areas: see A070142, A051516.

a(n) is nonzero iff n is in A024364.

			For n=30 there are A005044(30) = 19 integer triangles; only one is right: 5+12+13 = 30, 5^2+12^2 = 13^2; therefore a(30) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (obtained from the b-file of A078926)
Eric Weisstein's World of Mathematics, Right Triangle.
Eric Weisstein's World of Mathematics, Pythagorean Triples.
Reinhard Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A051493, A070093, A070101, A070138, A070084, A070137.

unitaryDivisors[n_] := Cases[Divisors[n], d_ /; GCD[d, n/d] == 1];
A078926[n_] := Count[unitaryDivisors[n], d_ /; OddQ[d] && Sqrt[n] < d < Sqrt[2n]];
a[n_] := If[EvenQ[n], A078926[n/2], 0];
Table[a[n], {n, 1, 1716}] (* Jean-François Alcover, Oct 04 2021 *)

a(n) = A078926(n/2) if n is even; a(n)=0 if n is odd.
a(n) = A051493(n) - A070094(n) - A070102(n).
a(n) <= A024155(n).

Secondary offset added by Antti Karttunen, Oct 07 2017

A127002 Number of partitions of n that have the form a+a+b+c where a,b,c are distinct.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 4, 3, 7, 8, 11, 11, 17, 17, 23, 23, 30, 31, 39, 38, 48, 49, 58, 58, 70, 70, 82, 82, 95, 96, 110, 109, 125, 126, 141, 141, 159, 159, 177, 177, 196, 197, 217, 216, 238, 239, 260, 260, 284, 284, 308, 308, 333, 334, 360, 359, 387, 388, 415, 415, 445

Offset: 1

Clark Kimberling, Jan 01 2007

From Gus Wiseman, Apr 19 2019: (Start)
Also the number of integer partitions of n - 4 of the form a+b, a+a+b, or a+a+b+c, ignoring ordering. A bijection can be constructed from the partitions described in the name by subtracting one from all parts and deleting zeros. These are also partitions with adjusted frequency depth (A323014, A325280) equal to their length plus one, and their Heinz numbers are given by A325281. For example, the a(7) = 1 through a(13) = 11 partitions are:
  (21)  (31)   (32)   (42)   (43)    (53)    (54)
        (211)  (41)   (51)   (52)    (62)    (63)
               (221)  (411)  (61)    (71)    (72)
               (311)         (322)   (332)   (81)
                             (331)   (422)   (441)
                             (511)   (611)   (522)
                             (3211)  (3221)  (711)
                                     (4211)  (3321)
                                             (4221)
                                             (4311)
                                             (5211)
(End)

			a(10) counts these partitions: {1,1,2,6}, (1,1,3,5), {2,2,1,5}.
a(11) counts {1,1,2,7}, {1,1,3,6}, {1,1,4,5}, {2,2,1,6}, {2,2,3,4}, {3,3,1,4}, {4,4,1,2}
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(7) = 1 through a(13) = 11 partitions of the form a+a+b+c are the following. The Heinz numbers of these partitions are given by A085987.
  (3211)  (3221)  (3321)  (5221)  (4322)  (4332)  (4432)
          (4211)  (4221)  (5311)  (4331)  (4431)  (5332)
                  (4311)  (6211)  (4421)  (5322)  (5422)
                  (5211)          (5411)  (5331)  (5521)
                                  (6221)  (6411)  (6322)
                                  (6311)  (7221)  (6331)
                                  (7211)  (7311)  (6511)
                                          (8211)  (7411)
                                                  (8221)
                                                  (8311)
                                                  (9211)
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1)

Cf. A000041, A008284, A085987, A090858, A116608, A117571, A183558, A325242, A325244, A325280, A325281.

R:=PowerSeriesRing(Integers(), 70); [0,0,0,0,0,0] cat Coefficients(R!( x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) )); // G. C. Greubel, May 30 2019

g:=sum(sum(sum(x^(i+j+k)*(x^i+x^j+x^k),i=1..j-1),j=2..k-1),k=3..80): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..65); # Emeric Deutsch, Jan 05 2007
isA127002 := proc(p) local s; if nops(p) = 4 then s := convert(p,set) ; if nops(s) = 3 then RETURN(1) ; else RETURN(0) ; fi ; else RETURN(0) ; fi ; end:
A127002 := proc(n) local part,res,p; part := combinat[partition](n) ; res := 0 ; for p from 1 to nops(part) do res := res+isA127002(op(p,part)) ; od ; RETURN(res) ; end:
for n from 1 to 200 do print(A127002(n)) ; od ; # R. J. Mathar, Jan 07 2007

Table[Length[Select[IntegerPartitions[n],Sort[Length/@Split[#]]=={1,1,2}&]],{n,70}] (* Gus Wiseman, Apr 19 2019 *)
Rest[CoefficientList[Series[x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)), {x,0,70}], x]] (* G. C. Greubel, May 30 2019 *)

my(x='x+O('x^70)); concat(vector(6), Vec(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)))) \\ G. C. Greubel, May 30 2019

a=(x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 70).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 30 2019

G.f.: x^7*(1+2*x+3*x^2)/((1-x^2)*(1-x^3)*(1-x^4)) - Vladeta Jovovic, Jan 03 2007
G.f.: Sum_{k>=3} Sum_{j=2..k-1} Sum_{m=1..j-1} x^(m+j+k)*(x^m +x^j +x^k). - Emeric Deutsch, Jan 05 2007
a(n) = binomial(floor((n-1)/2),2) - floor((n-1)/3) - floor((n-1)/4) + floor(n/4). - Mircea Merca, Nov 23 2013
a(n) = A005044(n-4) + 2*A005044(n-3) + 3*A005044(n-2). - R. J. Mathar, Nov 23 2013

A325691 Number of length-3 integer partitions of n whose largest part is not greater than the sum of the other two.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 7, 9, 8, 11, 10, 13, 12, 15, 14, 18, 16, 20, 19, 23, 21, 26, 24, 29, 27, 32, 30, 36, 33, 39, 37, 43, 40, 47, 44, 51, 48, 55, 52, 60, 56, 64, 61, 69, 65, 74, 70, 79, 75, 84, 80, 90, 85, 95, 91, 101, 96, 107, 102, 113

Offset: 0

Gus Wiseman, May 15 2019

Also the number of possible triples of edge-lengths of a triangle with perimeter n, where degenerate (self-intersecting) triangles are allowed.

The number of triples (a,b,c) for 1 <= a <= b <= c <= a+b and a+b+c = n. - Yuchun Ji, Oct 15 2020

			The a(3) = 1 through a(12) = 6 partitions:
  (111)  (211)  (221)  (222)  (322)  (332)  (333)  (433)  (443)  (444)
                       (321)  (331)  (422)  (432)  (442)  (533)  (543)
                                     (431)  (441)  (532)  (542)  (552)
                                                   (541)  (551)  (633)
                                                                 (642)
                                                                 (651)

Stefano Spezia, Table of n, a(n) for n = 0..10000
Colin Defant, Michael Joseph, Matthew Macauley, and Alex McDonough, Torsors and tilings from toric toggling, arXiv:2305.07627 [math.CO], 2023. See g.f. at p. 20.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1).

Cf. A001399, A005044 (nondegenerate triangles), A008642, A069905, A124278.

Cf. A325688, A325690, A325694.

Table[Length[Select[IntegerPartitions[n,{3}],#[[1]]<=#[[2]]+#[[3]]&]],{n,0,30}]

Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 + x - x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8. (End)
a(n) = A005044(n+3) - A000035(n+3). i.e., remove the only one triple (a=0,b,b) if n is even from the A005044 which is the number of triples (a,b,c) for 0 <= a <= b <= c <= a+b and a+b+c = n. - Yuchun Ji, Oct 15 2020
The above conjectured formulas are true. - Stefano Spezia, May 19 2023

A069906 Number of pentagons that can be formed with perimeter n. In other words, number of partitions of n into five parts such that the sum of any four is more than the fifth.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 16, 23, 25, 35, 39, 52, 57, 74, 81, 103, 111, 139, 150, 184, 197, 239, 256, 306, 325, 385, 409, 480, 507, 590, 623, 719, 756, 867, 911, 1038, 1087, 1232, 1289, 1453, 1516, 1701, 1774, 1981, 2061, 2293

Offset: 0

N. J. A. Sloane, May 05 2002

From Frank M Jackson, Jul 10 2012: (Start)
I recently commented on A062890 that:
"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 non-congruent cyclic quadrilaterals, [1,2,1,2] and [1,1,2,2], where the first is a rectangle and the second a kite."
This comment applies to all integer polygons (other than triangles) that are generated from a perimeter of length n. Not sure how best to correct for the above observation but my suggestion would be to change the definition of the present sequence to read:
"The number of cyclic integer pentagons differing only in circumradius that can be generated from an integer perimeter n." (End)

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: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 1, 1, 0, -1, 0, -1, -2, 0, 0, 0, 0, 2, 1, 0, 1, 0, -1, -1, 0, -1, 0, 1).

Number of k-gons that can be formed with perimeter n: A005044 (k=3), A062890 (k=4), this sequence (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).

CoefficientList[Series[x^5(1-x^11)/((1-x)(1-x^2)(1-x^4)(1-x^5)(1-x^6) (1-x^8)),{x,0,60}],x] (* Harvey P. Dale, Dec 16 2011 *)

G.f.: x^5*(1-x^11)/((1-x)*(1-x^2)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^8)).

a(2*n+8) = A026811(2*n+8) - A002621(n), a(2*n+9) = A026811(2*n+9) - A002621(n) for n >= 0. - Seiichi Manyama, Jun 08 2017

A069907 Number of hexagons that can be formed with perimeter n. In other words, partitions of n into six parts such that the sum of any 5 is more than the sixth.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 28, 37, 46, 59, 71, 91, 107, 134, 157, 193, 222, 271, 308, 371, 419, 499, 559, 661, 734, 860, 952, 1106, 1216, 1405, 1537, 1764, 1923, 2193, 2381, 2703, 2923, 3301, 3561, 4002, 4302, 4817, 5164

Offset: 0

N. J. A. Sloane, May 05 2002

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, European Journal of Combinatorics, Volume 22, Issue 7, October 2001, Pages 887-904.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, -1, 0, -1, 0, 0, -1, 0, -1, 1, -1, 1, 1, 1, 1, -1, 1, -1, 0, -1, 0, 0, -1, 0, -1, 1, 1, 1, 0, -1).

Number of k-gons that can be formed with perimeter n: A005044 (k=3), A062890 (k=4), A069906 (k=5), this sequence (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).

concat(vector(6), Vec(x^6*(1-x^4+x^5+x^7-x^8-x^13)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10)) + O(x^80))) \\ Michel Marcus, Jun 24 2017

G.f.: x^6*(1-x^4+x^5+x^7-x^8-x^13)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10)).

a(2*n+10) = A026812(2*n+10) - A002622(n), a(2*n+11) = A026812(2*n+11) - A002622(n) for n >= 0. - Seiichi Manyama, Jun 08 2017

A070201 Number of integer triangles with perimeter n having integral inradius.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 8, 0, 0, 0, 1, 0, 3

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = #{k | A070083(k) = n and A070200(k) = exact inradius};
a(n) = A070203(n) + A070204(n);
a(n) = A070205(n) + A070206(n) + A024155(n);
a(odd) = 0.

			a(36)=2, as there are two integer triangles with integer inradius having perimeter=32:
First: [A070080(368), A070081(368), A070082(368)] = [9,10,17], for s = A070083(368)/2 = (9+10+17)/2 = 18: inradius = sqrt((s-9)*(s-10)*(s-17)/s) = sqrt(9*8*1/18) = sqrt(4) = 2; therefore A070200(368) = 2.
2nd: [A070080(370), A070081(370), A070082(370)] = [9,12,15], for s = A070083(370)/2 = (9+12+15)/2 = 18: inradius = sqrt((s-9)*(s-12)*(s-15)/s) = sqrt(9*6*3/18) = sqrt(9) = 3; therefore A070200(370) = 3.

Seiichi Manyama, Table of n, a(n) for n = 1..5000
Mohammad K. Azarian, Solution to Problem S125: Circumradius and Inradius, Math Horizons, Vol. 16, Issue 2, November 2008, p. 32.
Eric Weisstein's World of Mathematics, Incircle.
Eric Weisstein's World of Mathematics, Heron's Formula.
Reinhard Zumkeller, Integer-sided triangles

Cf. A070209, A070202, A070208, A005044, A070140.

Cf. A120062, A120572, A331040.

def A(n)
  cnt = 0
  (1..n / 3).each{|a|
    (a..(n - a) / 2).each{|b|
      c = n - a - b
      if a + b > c
        s = n / 2r
        t = (s - a) * (s - b) * (s - c) / s
        if t.denominator == 1
          t = t.to_i
          cnt += 1 if Math.sqrt(t).to_i ** 2 == t
        end
      end
    }
  }
  cnt
end
def A070201(n)
  (1..n).map{|i| A(i)}
end
p A070201(100) # Seiichi Manyama, Oct 06 2017

A288253 Number of heptagons that can be formed with perimeter n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 13, 19, 24, 34, 42, 58, 70, 93, 112, 145, 171, 218, 256, 320, 372, 458, 528, 643, 735, 884, 1006, 1198, 1352, 1597, 1795, 2102, 2350, 2732, 3041, 3513, 3892, 4468, 4934, 5633, 6194, 7037, 7715, 8722, 9531, 10728, 11690

Offset: 7

Seiichi Manyama, Jun 07 2017

Number of (a1, a2, ... , a7) where 1 <= a1 <= ... <= a7 and a1 + a2 + ... + a6 > a7.

Seiichi Manyama, Table of n, a(n) for n = 7..10000
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018. [This thesis cites this sequence entry, but it's just a typo: the intended sequence entry is A288853.]
Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 1, 0, 0, 1, 0, -1, -1, -1, 0, 0, -2, 0, 0, 1, 1, 0, 1, 2, 1, 0, 1, -1, 0, -1, -2, -1, 0, -1, -1, 0, 0, 2, 0, 0, 1, 1, 1, 0, -1, 0, 0, -1, 0, -1, 0, 1).

Number of k-gons that can be formed with perimeter n: A005044 (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), this sequence (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).

G.f.: x^7/((1-x)*(1-x^2)* ... *(1-x^7)) - x^12/(1-x) * 1/((1-x^2)*(1-x^4)* ... *(1-x^12)).

a(2*n+12) = A026813(2*n+12) - A288341(n), a(2*n+13) = A026813(2*n+13) - A288341(n) for n >= 0. - Seiichi Manyama, Jun 08 2017

A288254 Number of octagons that can be formed with perimeter n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 104, 134, 167, 211, 258, 322, 389, 480, 572, 698, 825, 996, 1165, 1395, 1620, 1923, 2216, 2611, 2991, 3500, 3984, 4633, 5248, 6066, 6836, 7860, 8820, 10089, 11273, 12835, 14288, 16197

Offset: 8

Seiichi Manyama, Jun 07 2017

Number of (a1, a2, ... , a8) where 1 <= a1 <= ... <= a8 and a1 + a2 + ... + a7 > a8.

Seiichi Manyama, Table of n, a(n) for n = 8..10000
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1, 1, -1, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, -1, 1, 0, 0, 1, -1, 1, -1, 2, -2, 0, 0, 0, 0, 0, 0, -2, 2, -1, 1, -1, 1, 0, 0, 1, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, -1, 1, -1, 1, 1, -1).

Number of k-gons that can be formed with perimeter n: A005044 (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), this sequence (k=8), A288255 (k=9), A288256 (k=10).

G.f.: x^8/((1-x)*(1-x^2)* ... *(1-x^8)) - x^14/(1-x) * 1/((1-x^2)*(1-x^4)* ... *(1-x^14)).

a(2*n+14) = A026814(2*n+14) - A288342(n), a(2*n+15) = A026814(2*n+15) - A288342(n) for n >= 0. - Seiichi Manyama, Jun 08 2017

A288255 Number of nonagons that can be formed with perimeter n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 50, 69, 87, 116, 145, 189, 233, 299, 363, 458, 553, 687, 820, 1009, 1195, 1453, 1709, 2058, 2404, 2872, 3331, 3948, 4557, 5361, 6152, 7194, 8215, 9547, 10853, 12543, 14199, 16329, 18407, 21067, 23666, 26964, 30179, 34248, 38207

Offset: 9

Seiichi Manyama, Jun 07 2017

Number of (a1, a2, ... , a9) where 1 <= a1 <= ... <= a9 and a1 + a2 + ... + a8 > a9.

Seiichi Manyama, Table of n, a(n) for n = 9..10000
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 1, 0, 0, 0, 0, 1, -1, -1, 0, -1, -1, 0, 0, 0, -1, 1, 0, 0, 1, 1, 2, 0, 1, 1, 0, 0, 1, -1, -1, -2, -1, -1, -2, 0, -1, -1, -1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 1, -1, 0, 0, -1, -1, 0, -2, -1, -1, 0, 0, -1, 1, 0, 0, 0, 1, 1, 0, 1, 1, -1, 0, 0, 0, 0, -1, 0, -1, 0, 1).

Number of k-gons that can be formed with perimeter n: A005044 (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), this sequence (k=9), A288256 (k=10).

G.f.: x^9/((1-x)*(1-x^2)* ... *(1-x^9)) - x^16/(1-x) * 1/((1-x^2)*(1-x^4)* ... *(1-x^16)).

a(2*n+16) = A026815(2*n+16) - A288343(n), a(2*n+17) = A026815(2*n+17) - A288343(n) for n >= 0. - Seiichi Manyama, Jun 08 2017

A288256 Number of decagons that can be formed with perimeter n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 71, 93, 121, 157, 200, 255, 321, 404, 500, 623, 762, 939, 1137, 1388, 1664, 2015, 2396, 2877, 3398, 4050, 4748, 5623, 6553, 7711, 8936, 10454, 12051, 14024, 16088, 18626, 21275, 24516, 27882, 31991, 36244, 41411, 46746

Offset: 10

Seiichi Manyama, Jun 07 2017

Number of (a1, a2, ... , a10) where 1 <= a1 <= ... <= a10 and a1 + a2 + ... + a9 > a10.

Seiichi Manyama, Table of n, a(n) for n = 10..10000
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 1, 1, 0, -1, 0, -1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, 2, 1, 1, 0, 1, -2, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -2, 1, -1, 1, -1, 2, 1, 1, 1, 1, 2, -1, 1, -1, 1, -2, 1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -2, 1, 0, 1, 1, 2, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 0, -1, -1, 0, -1, 0, 1, 1, 0, 1, 0, -1).

Number of k-gons that can be formed with perimeter n: A005044 (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), this sequence (k=10).

G.f.: x^10/((1-x)*(1-x^2)* ... *(1-x^10)) - x^18/(1-x) * 1/((1-x^2)*(1-x^4)* ... *(1-x^18)).

a(2*n+18) = A026816(2*n+18) - A288344(n), a(2*n+19) = A026816(2*n+19) - A288344(n) for n >= 0.

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A069906 Number of pentagons that can be formed with perimeter n. In other words, number of partitions of n into five parts such that the sum of any four is more than the fifth.

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