%I A005044 M0146
%S A005044 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,
%T A005044 19,24,21,27,24,30,27,33,30,37,33,40,37,44,40,48,44,52,48,56,52,61,56,
%U A005044 65,61,70,65,75,70,80,75,85,80,91,85,96,91,102,96,108,102,114,108,120
%N A005044 Alcuin's sequence: expansion of x^3/((1-x^2)*(1-x^3)*(1-x^4)).
%C A005044 a(n) = number of triangles with integer sides and perimeter n.
%C A005044 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).
%C A005044 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)
%C A005044 For m >= 2, the sequence {a(n) mod m} is periodic with period 12m. - 
               Martin J. Erickson (erickson(AT)truman.edu), Jun 06 2008
%D A005044 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005044 G. E. Andrews, A note on partitions and triangles with integer sides, 
               Amer. Math. Monthly, 86 (1979), 477-478.
%D A005044 G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, 
               Annals Combinatorics, 4 (2000), 327-338.
%D A005044 G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: 
               k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
%D A005044 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
%D A005044 R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.
%D A005044 T. Jenkyns and E. Muller, Triangular triples from ceilings to floors, 
               Amer. Math. Monthly, 107 (Aug. 2000), 634-639.
%D A005044 J. H. Jordan, R. Walch and R. J. Wisner, Triangles with integer sides, 
               Amer. Math. Monthly, 86 (1979), 686-689.
%D A005044 N. Krier and B. Manvel, Counting integer triangles, Math. Mag., 71 (1998), 
               291-295.
%D A005044 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 A005044 D. Olivastro: Ancient Puzzles. Classic Brainteasers and Other Timeless 
               Mathematical Games of the Last 10 Centuries. New York: Bantam Books, 
               1993. See p. 158.
%D A005044 David Singmaster, Triangles with integer sides and sharing barrels, College 
               Math J, 21:4 (1990) 278-285.
%D A005044 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)
%H A005044 T. D. Noe, <a href="b005044.txt">Table of n, a(n) for n=0..1000</a>
%H A005044 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A005044 Alcuin of York, <a href="http://www.beyond-the-illusion.com/files/History/
               Science/host1-2.txt">Propositiones ad acuendos juvenes</a>, [Latin 
               with English translation] - see Problem 12.
%H A005044 G. E. Andrews, P. Paule and A. Riese, <a href="http://www.risc.uni-linz.ac.at/
               research/combinat/risc/publications/#ppaule">MacMahon's partition 
               analysis III. The Omega package</a>, p. 19.
%H A005044 Wulf-Dieter Geyer, <a href="http://www.mi.uni-erlangen.de/~geyer/geschima">
               Lecture on history of medieval mathematics</a>
%H A005044 M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/
               paper98.pdf">Triangles With Integer Sides</a>
%H A005044 M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/
               paper72.pdf">Triangles With Integer Sides, Revisited</a>
%H A005044 Hermann Kremer, <a href="http://groups.google.de/groups?selm=cacqdf%24f0q%241%40online.de">
               Posting to de.sci.mathematik (1)</a>
%H A005044 Hermann Kremer, <a href="http://groups.google.de/groups?selm=canh34%24444%241%40online.de">
               Posting to de.sci.mathematik (2)</a>
%H A005044 Hermann Kremer, <a href="http://groups.google.de/groups?selm=cankur%24km3%241%40online.de">
               Posting to de.sci.mathematik (3)</a>
%H A005044 Hermann Kremer, <a href="http://groups.google.de/groups?selm=cavdfh$l8a$1@online.de">
               Posting to alt.math.recreational</a>
%H A005044 Mathforum, <a href="http://mathforum.org/library/drmath/view/51547.html">
               Triangle Perimeters</a>
%H A005044 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005044 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005044 D. Singmaster, <a href="http://www2.edc.org/makingmath/handbook/Teacher/
               GettingInformation/TrianglesAndBarrels.pdf">Triangles with Integer 
               Sides and Sharing Barrels</a>.
%H A005044 J. Tanton, <a href="http://www.maa.org/features/integertriangles.pdf">
               Young students approach integer triangles</a>
%H A005044 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AlcuinsSequence.html">Link to a section of The World of Mathematics.</
               a>
%H A005044 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Triangle.html">Link to a section of The World of Mathematics.</a>
%H A005044 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               IntegerTriangle.html">Integer Triangle</a>
%F A005044 For odd indices we have a(2n-3)=a(2n). For even indices, a(2n) = nearest 
               integer to n^2/12 = A001399(n).
%F A005044 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).
%F A005044 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
%F A005044 Euler transform of length 4 sequence [ 0, 1, 1, 1]. - Michael Somos Sep 
               04 2006
%F A005044 a(-3-n)=a(n). - Michael Somos Sep 04 2006
%F A005044 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
%e A005044 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.
%p A005044 A005044 := n-> floor((1/48)*(n^2+3*n+21+(-1)^(n-1)*3*n));
%p A005044 A005044:=-1/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**3; [S. Plouffe in his 
               1992 dissertation.]
%t A005044 a[n_] := Round[If[EvenQ[n], n^2, (n + 3)^2]/48] - from Peter Bertok Jan 
               09 2002
%t A005044 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)
%t A005044 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
%o A005044 (PARI) a(n)=round(n^2/12)-(n\2)^2\4
%o A005044 (PARI) a(n)=(n^2+6*n*(n%2)+24)\48
%Y A005044 a(n) = a(n-6) + A059169(n) = A070093(n) + A070101(n) + A024155(n).
%Y A005044 Cf. A002620, A001399, A062890, A069906, A069907, A070083.
%Y A005044 Cf. A008795.
%Y A005044 Sequence in context: A028242 A030451 A029162 this_sequence A029142 A054685 
               A143618
%Y A005044 Adjacent sequences: A005041 A005042 A005043 this_sequence A005045 A005046 
               A005047
%K A005044 easy,nonn,nice
%O A005044 0,8
%A A005044 Robert G. Wilson v (rgwv(AT)rgwv.com)
%E A005044 More terms from Erich Friedman (erich.friedman(AT)stetson.edu). Additional 
               comments from reinhard.zumkeller(AT)gmail.com, May 11 2002
%E A005044 Yaglom reference and mod formulae from Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), 
               May 27 2000
%E A005044 The reference to Alcuin of York (735-804) was provided by Hermann Kremer 
               (hermann.kremer(AT)onlinehome.de), Jun 18 2004