A288256 -id:A288256 - cp's OEIS frontend

A005044 Alcuin's sequence: expansion of x^3/((1-x^2)(1-x^3)(1-x^4)).

0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, 24, 21, 27, 24, 30, 27, 33, 30, 37, 33, 40, 37, 44, 40, 48, 44, 52, 48, 56, 52, 61, 56, 65, 61, 70, 65, 75, 70, 80, 75, 85, 80, 91, 85, 96, 91, 102, 96, 108, 102, 114, 108, 120

Offset: 0

Robert G. Wilson v

a(n) is the number of triangles with integer sides and perimeter n.
Also a(n) is the number of triangles with distinct integer sides and perimeter n+6, i.e., number of triples (a, b, c) such that 1 < a < b < c < a+b, a+b+c = n+6. - Roger Cuculière
With a different offset (i.e., without the three leading zeros, as in A266755), the number of ways in which n empty casks, n casks half-full of wine and n full casks can be distributed to 3 persons in such a way that each one gets the same number of casks and the same amount of wine [Alcuin]. E.g., for n=2 one can give 2 people one full and one empty and the 3rd gets two half-full. (Comment corrected by Franklin T. Adams-Watters, Oct 23 2006)
For m >= 2, the sequence {a(n) mod m} is periodic with period 12*m. - Martin J. Erickson (erickson(AT)truman.edu), Jun 06 2008
Number of partitions of n into parts 2, 3, and 4, with at least one part 3. - Joerg Arndt, Feb 03 2013
For several values of p and q the sequence (A005044(n+p) - A005044(n-q)) leads to known sequences, see the crossrefs. - Johannes W. Meijer, Oct 12 2013
For n>=3, number of partitions of n-3 into parts 2, 3, and 4. - David Neil McGrath, Aug 30 2014
Also, a(n) is the number of partitions mu of n of length 3 such that mu_1-mu_2 is even and mu_2-mu_3 is even (see below example). - John M. Campbell, Jan 29 2016
For n > 1, number of triangles with odd side lengths and perimeter 2*n-3. - Wesley Ivan Hurt, May 13 2019
Number of partitions of n+1 into 4 parts whose largest two parts are equal. - Wesley Ivan Hurt, Jan 06 2021
For n>=3, number of weak partitions of n-3 (that is, allowing parts of size 0) into three parts with no part exceeding (n-3)/2. Also, number of weak partitions of n-3 into three parts, all of the same parity as n-3. - Kevin Long, Feb 20 2021
Also, a(n) is the number of incongruent acute triangles formed from the vertices of a regular n-gon. - Frank M Jackson, Nov 04 2022

			There are 4 triangles of perimeter 11, with sides 1,5,5; 2,4,5; 3,3,5; 3,4,4. So a(11) = 4.
G.f. = x^3 + x^5 + x^6 + 2*x^7 + x^8 + 3*x^9 + 2*x^10 + 4*x^11 + 3*x^12 + ...
From _John M. Campbell_, Jan 29 2016: (Start)
Letting n = 15, there are a(n)=7 partitions mu |- 15 of length 3 such that mu_1-mu_2 is even and mu_2-mu_3 is even:
(13,1,1) |- 15
(11,3,1) |- 15
(9,5,1) |- 15
(9,3,3) |- 15
(7,7,1) |- 15
(7,5,3) |- 15
(5,5,5) |- 15
(End)

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. Wiley, NY, Chap.10, Section 10.2, Problems 5 and 6, pp. 451-2.
D. Olivastro: Ancient Puzzles. Classic Brainteasers and Other Timeless Mathematical Games of the Last 10 Centuries. New York: Bantam Books, 1993. See p. 158.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 8, #30 (First published: San Francisco: Holden-Day, Inc., 1964)

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Alcuin of York, Propositiones ad acuendos juvenes, [Latin with English translation] - see Problem 12.
G. E. Andrews, A note on partitions and triangles with integer sides, Amer. Math. Monthly, 86 (1979), 477-478.
G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 19.
Donald J. Bindner and Martin Erickson, Alcuin's Sequence, Amer. Math. Monthly, 119, February 2012, pp. 115-121.
P. Bürgisser and C. Ikenmeyer, Fundamental invariants of orbit closures, arXiv preprint arXiv:1511.02927 [math.AG], 2015. See Section 5.5.
James East and Ron Niles, Integer polygons of given perimeter, Bull. Aust. Math. Soc. 100 (2019), no. 1, 131-147.
James East and Ron Niles, Integer Triangles of Given Perimeter: A New Approach via Group Theory., Amer. Math. Monthly 126 (2019), no. 8, 735-739.
Wulf-Dieter Geyer, Lecture on history of medieval mathematics [broken link]
M. D. Hirschhorn, Triangles With Integer Sides
M. D. Hirschhorn, Triangles With Integer Sides, Revisited
R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39. [Annotated scanned copy]
T. Jenkyns and E. Muller, Triangular triples from ceilings to floors, Amer. Math. Monthly, 107 (Aug. 2000), 634-639.
J. H. Jordan, R. Walch and R. J. Wisner, Triangles with integer sides, Amer. Math. Monthly, 86 (1979), 686-689.
Hermann Kremer, Posting to de.sci.mathematik (1), (2), and (3). [Dead links]
Hermann Kremer, Posting to alt.math.recreational, June 2004.
N. Krier and B. Manvel, Counting integer triangles, Math. Mag., 71 (1998), 291-295.
Mathforum, Triangle Perimeters
Augustine O. Munagi, Computation of q-partial fractions, INTEGERS: Electronic Journal Of Combinatorial Number Theory, 7 (2007), #A25.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
S. A. Shirali, Case Studies in Experimental Mathematics, 2013.
David Singmaster, Triangles with Integer Sides and Sharing Barrels, College Math J, 21:4 (1990) 278-285.
James Tanton, Young students approach integer triangles, FOCUS 22 no. 5 (2002), 4 - 6.
James Tanton, Integer Triangles, Chapter 11 in "Mathematics Galore!" (MAA, 2012).
Eric Weisstein's World of Mathematics, Alcuin's Sequence, Integer Triangle, and Triangle.
Wikipedia, Propositiones ad acuendos juvenes.
R. G. Wilson v, Letter to N. J. A. Sloane, date unknown.
Index entries for two-way infinite sequences
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,-1,-1,-1,0,1).

See A266755 for a version without the three leading zeros.
Cf. A002620, A070083, A008795.
Both bisections give (essentially) A001399.
(See the comments.) Cf. A008615 (p=1, q=3, offset=0), A008624 (3, 3, 0), A008679 (3, -1, 0), A026922 (1, 5, 1), A028242 (5, 7, 0), A030451 (6, 6, 0), A051274 (3, 5, 0), A052938 (8, 4, 0), A059169 (0, 6, 1), A106466 (5, 4, 0), A130722 (2, 7, 0)
Cf. this sequence (k=3), A288165 (k=4), A288166 (k=5).
Number of k-gons that can be formed with perimeter n: this sequence (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).

a005044 = p [2,3,4] . (subtract 3) where
p _ 0 = 1
p [] _ = 0
p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 28 2013

A005044 := n-> floor((1/48)*(n^2+3*n+21+(-1)^(n-1)*3*n)): seq(A005044(n), n=0..73);
A005044 := -1/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**3; # Simon Plouffe in his 1992 dissertation

a[n_] := Round[If[EvenQ[n], n^2, (n + 3)^2]/48] (* Peter Bertok, Jan 09 2002 *)
CoefficientList[Series[x^3/((1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 0, 105}], x] (* Robert G. Wilson v, Jun 02 2004 *)
me[n_] := Module[{i, j, sum = 0}, For[i = Ceiling[(n - 3)/3], i <= Floor[(n - 3)/2], i = i + 1, For[j = Ceiling[(n - i - 3)/2], j <= i, j = j + 1, sum = sum + 1] ]; Return[sum]; ] mine = Table[me[n], {n, 1, 11}]; (* Srikanth (sriperso(AT)gmail.com), Aug 02 2008 *)
LinearRecurrence[{0,1,1,1,-1,-1,-1,0,1},{0,0,0,1,0,1,1,2,1},80] (* Harvey P. Dale, Sep 22 2014 *)
Table[Length@Select[IntegerPartitions[n, {3}], Max[#]*180 < 90 n &], {n, 1, 100}] (* Frank M Jackson, Nov 04 2022 *)

PARI
```
a(n) = round(n^2 / 12) - (n\2)^2 \ 4
```
PARI
```
a(n) = (n^2 + 6*n * (n%2) + 24) \ 48
```

a(n)=if(n%2,n+3,n)^2\/48 \\ Charles R Greathouse IV, May 02 2016

concat(vector(3), Vec((x^3)/((1-x^2)*(1-x^3)*(1-x^4)) + O(x^70))) \\ Felix Fröhlich, Jun 07 2017

a(n) = a(n-6) + A059169(n) = A070093(n) + A070101(n) + A024155(n).
For odd indices we have a(2*n-3) = a(2*n). For even indices, a(2*n) = nearest integer to n^2/12 = A001399(n).
For all n, a(n) = round(n^2/12) - floor(n/4)*floor((n+2)/4) = a(-3-n) = A069905(n) - A002265(n)*A002265(n+2).
For n = 0..11 (mod 12), a(n) is respectively n^2/48, (n^2 + 6*n - 7)/48, (n^2 - 4)/48, (n^2 + 6*n + 21)/48, (n^2 - 16)/48, (n^2 + 6*n - 7)/48, (n^2 + 12)/48, (n^2 + 6*n + 5)/48, (n^2 - 16)/48, (n^2 + 6*n + 9)/48, (n^2 - 4)/48, (n^2 + 6*n + 5)/48.
Euler transform of length 4 sequence [ 0, 1, 1, 1]. - Michael Somos, Sep 04 2006
a(-3 - n) = a(n). - Michael Somos, Sep 04 2006
a(n) = sum(ceiling((n-3)/3) <= i <= floor((n-3)/2), sum(ceiling((n-i-3)/2) <= j <= i, 1 ) ) for n >= 1. - Srikanth K S, Aug 02 2008
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 >= 9. - David Neil McGrath, Aug 30 2014
a(n+3) = a(n) if n is odd; a(n+3) = a(n) + floor(n/4) + 1 if n is even. Sketch of proof: There is an obvious injective map from perimeter-n triangles to perimeter-(n+3) triangles defined by f(a,b,c) = (a+1,b+1,c+1). It is easy to show f is surjective for odd n, while for n=2k the image of f is only missing the triangles (a,k+2-a,k+1) for 1 <= a <= floor(k/2)+1. - James East, May 01 2016
a(n) = round(n^2/48) if n is even; a(n) = round((n+3)^2/48) if n is odd. - James East, May 01 2016
a(n) = (6*n^2 + 18*n - 9*(-1)^n*(2*n + 3) - 36*sin(Pi*n/2) - 36*cos(Pi*n/2) + 64*cos(2*Pi*n/3) - 1)/288. - Ilya Gutkovskiy, May 01 2016
a(n) = A325691(n-3) + A000035(n) for n>=3. The bijection between partition(n,[2,3,4]) and not-over-half partition(n,3,n/2) + partition(n,2,n/2) can be built by a Ferrers(part)[0+3,1,2] map. And the last partition(n,2,n/2) is unique [n/2,n/2] if n is even, it is given by A000035. - Yuchun Ji, Sep 24 2020
a(4n+3) = a(4n) + n+1, a(4n+4) = a(4n+1) = A000212(n+1), a(4n+5) = a(4n+2) + n+1, a(4n+6) = a(4n+3) = A007980(n). - Yuchun Ji, Oct 10 2020
a(n)-a(n-4) = A008615(n-1). - R. J. Mathar, Jun 23 2021
a(n)-a(n-2) = A008679(n-3). - R. J. Mathar, Jun 23 2021

Additional comments from Reinhard Zumkeller, May 11 2002
Yaglom reference and mod formulas from Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 27 2000
The reference to Alcuin of York (735-804) was provided by Hermann Kremer (hermann.kremer(AT)onlinehome.de), Jun 18 2004

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.

Original entry on oeis.org

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

Offset: 0

Amarnath Murthy, Jun 29 2001

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. - Frank M Jackson, Jun 29 2012

			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.

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 (1,1,-1,1,-1,0,0,-1,1,-1,1,1,-1).

Number of k-gons 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).

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

G.f.: x^4*(1+x+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^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(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-8) + a(n-9) - a(n-10) + a(n-11) + a(n-12) - a(n-13).
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} sign(floor((i+j+k)/(n-i-j-k+1))). (End)

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

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

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

A124278 Triangle of the number of nondegenerate k-gons having perimeter n and whose sides are nondecreasing, for k=3..n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 3, 4, 4, 3, 2, 1, 1, 2, 5, 5, 4, 3, 2, 1, 1, 4, 7, 8, 6, 5, 3, 2, 1, 1, 3, 8, 9, 9, 6, 5, 3, 2, 1, 1, 5, 11, 14, 12, 10, 7, 5, 3, 2, 1, 1, 4, 12, 16, 16, 13, 10, 7, 5, 3, 2, 1, 1, 7, 16, 23, 22, 19, 14, 11, 7, 5, 3, 2, 1, 1, 5, 18, 25, 28, 24, 20, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 3

T. D. Noe, Oct 24 2006

For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides.

T(n,k) = number of partitions of n into k parts (k >= 3) in which all parts are less than n/2. Also T(n,k) = number of partitions of 2*n into k parts in which all parts are even and less than n. - L. Edson Jeffery, Mar 19 2012

			For polygons having perimeter 7: 2 triangles, 2 quadrilaterals, 2 pentagons, 1 hexagon and 1 heptagon. The triangle begins
1
0 1
1 1 1
1 1 1 1
2 2 2 1 1
1 3 2 2 1 1

T. D. Noe, Rows n=3..102 of triangle, flattened
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-Gon Partitions, Bull. Austral. Math. Soc. 64 (2001), 321-329.
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Cf. A124287 (similar, but with no restriction on the sides).
Cf. A210249 (gives row sums of this sequence for n >= 3).
Cf. A005044 (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).

b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [],
      zip((x, y)-> x+y, b(n, i-1), `if`(i>n, [],
                    [0, b(n-i, i)[]]), 0)))
    end:
T:= n-> b(n, ceil(n/2)-1)[4..n+1][]:
seq(T(n), n=3..20);  # Alois P. Heinz, Jul 15 2013

Flatten[Table[p=IntegerPartitions[n]; Length[Select[p, Length[ # ]==k && #[[1]] < Total[Rest[ # ]]&]], {n,3,30}, {k,3,n}]]
(* second program: *)
QP = QPochhammer; T[n_, k_] := SeriesCoefficient[x^k*(1/QP[x, x, k] + x^(k - 2)/((x-1)*QP[x^2, x^2, k-1])), {x, 0, n}]; Table[T[n, k], {n, 3, 16}, {k, 3, n}] // Flatten (* Jean-François Alcover, Jan 08 2016 *)

G.f. for column k is x^k/(product_{i=1..k} 1-x^i) - x^(2k-2)/(1-x)/(product_{i=1..k-1} 1-x^(2i)).

cp's OEIS Frontend

A005044 Alcuin's sequence: expansion of x^3/((1-x^2)*(1-x^3)*(1-x^4)).

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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.

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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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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.

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

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

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

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A124278 Triangle of the number of nondegenerate k-gons having perimeter n and whose sides are nondecreasing, for k=3..n.

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A005044 Alcuin's sequence: expansion of x^3/((1-x^2)(1-x^3)(1-x^4)).

A288344 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^9)).