0,8

a(n) = number of triangles with integer sides and perimeter n.

Also a(n) = 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 CUCULIERE (cuculier(AT)sophocle.imaginet.fr).

With a different offset (i.e. without the three leading zeros), also 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 12m. - Martin J. Erickson (erickson(AT)truman.edu), Jun 06 2008

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.

R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.

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.

N. Krier and B. Manvel, Counting integer triangles, Math. Mag., 71 (1998), 291-295.

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.

David Singmaster, Triangles with integer sides and sharing barrels, College Math J, 21:4 (1990) 278-285.

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)

T. D. Noe, Table of n, a(n) for n=0..1000

Index entries for two-way infinite sequences

Alcuin of York, Propositiones ad acuendos juvenes, [Latin with English translation] - see Problem 12.

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 19.

Wulf-Dieter Geyer, Lecture on history of medieval mathematics

M. D. Hirschhorn, Triangles With Integer Sides

M. D. Hirschhorn, Triangles With Integer Sides, Revisited

Hermann Kremer, Posting to de.sci.mathematik (1)

Hermann Kremer, Posting to de.sci.mathematik (2)

Hermann Kremer, Posting to de.sci.mathematik (3)

Hermann Kremer, Posting to alt.math.recreational

Mathforum, Triangle Perimeters

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

D. Singmaster, Triangles with Integer Sides and Sharing Barrels.

J. Tanton, Young students approach integer triangles

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Integer Triangle

For odd indices we have a(2n-3)=a(2n). For even indices, a(2n) = 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 + 6n - 7)/48, (n^2 - 4)/48, (n^2 + 6n + 21)/48, (n^2 - 16)/48, (n^2 + 6n - 7)/48, (n^2 + 12)/48, (n^2 + 6n + 5)/48, (n^2 - 16)/48, (n^2 + 6n + 9)/48, (n^2 - 4)/48, (n^2 + 6n + 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 (sriperso(AT)gmail.com), Aug 02 2008

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.

A005044 := n-> floor((1/48)*(n^2+3*n+21+(-1)^(n-1)*3*n));

A005044:=-1/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**3; [S. Plouffe in his 1992 dissertation.]

a[n_] := Round[If[EvenQ[n], n^2, (n + 3)^2]/48] - from Peter Bertok Jan 09 2002

CoefficientList[Series[x^3/((1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 0, 105}], x] (from 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

(PARI) a(n)=round(n^2/12)-(n\2)^2\4

(PARI) a(n)=(n^2+6*n*(n%2)+24)\48

a(n) = a(n-6) + A059169(n) = A070093(n) + A070101(n) + A024155(n).

Cf. A002620, A001399, A062890, A069906, A069907, A070083.

Cf. A008795.

Sequence in context: A028242 A030451 A029162 this_sequence A029142 A054685 A143618

Adjacent sequences: A005041 A005042 A005043 this_sequence A005045 A005046 A005047

easy,nonn,nice

Robert G. Wilson v (rgwv(AT)rgwv.com)

More terms from Erich Friedman (erich.friedman(AT)stetson.edu). Additional comments from reinhard.zumkeller(AT)gmail.com, May 11 2002

Yaglom reference and mod formulae 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