%I M0146 #216 Nov 09 2022 19:31:15

%S 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,

%T 21,19,24,21,27,24,30,27,33,30,37,33,40,37,44,40,48,44,52,48,56,52,61,

%U 56,65,61,70,65,75,70,80,75,85,80,91,85,96,91,102,96,108,102,114,108,120

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

%C a(n) is the number of triangles with integer sides and perimeter n.

%C 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_

%C 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)

%C 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

%C Number of partitions of n into parts 2, 3, and 4, with at least one part 3. - _Joerg Arndt_, Feb 03 2013

%C 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

%C For n>=3, number of partitions of n-3 into parts 2, 3, and 4. - _David Neil McGrath_, Aug 30 2014

%C 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

%C For n > 1, number of triangles with odd side lengths and perimeter 2*n-3. - _Wesley Ivan Hurt_, May 13 2019

%C Number of partitions of n+1 into 4 parts whose largest two parts are equal. - _Wesley Ivan Hurt_, Jan 06 2021

%C 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

%C 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

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

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

%D 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 Seiichi Manyama, <a href="/A005044/b005044.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)

%H Alcuin of York, <a href="http://web.archive.org/web/20081029065718/http://beyond-the-illusion.com/files/History/Science/host1-2.txt">Propositiones ad acuendos juvenes</a>, [Latin with English translation] - see Problem 12.

%H G. E. Andrews, <a href="http://www.jstor.org/stable/2320420">A note on partitions and triangles with integer sides</a>, Amer. Math. Monthly, 86 (1979), 477-478.

%H G. E. Andrews, <a href="https://dx.doi.org/10.1007/PL00001284">MacMahon's Partition Analysis II: Fundamental Theorems</a>, Annals Combinatorics, 4 (2000), 327-338.

%H G. E. Andrews, P. Paule and A. Riese, <a href="http://www.risc.jku.at/publications/download/risc_163/PAIX.pdf">MacMahon's Partition Analysis IX: k-gon partitions</a>, Bull. Austral Math. Soc., 64 (2001), 321-329.

%H 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 Donald J. Bindner and Martin Erickson, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.119.02.115">Alcuin's Sequence</a>, Amer. Math. Monthly, 119, February 2012, pp. 115-121.

%H P. Bürgisser and C. Ikenmeyer, <a href="http://arxiv.org/abs/1511.02927">Fundamental invariants of orbit closures</a>, arXiv preprint arXiv:1511.02927 [math.AG], 2015. See Section 5.5.

%H James East and Ron Niles, <a href="https://www.cambridge.org/core/journals/bulletin-of-the-australian-mathematical-society/article/integer-polygons-of-given-perimeter/DE5B2A1D7E038E2E68D182E424C6070F">Integer polygons of given perimeter</a>, Bull. Aust. Math. Soc. 100 (2019), no. 1, 131-147.

%H James East and Ron Niles, <a href="https://www.tandfonline.com/doi/full/10.1080/00029890.2019.1632632">Integer Triangles of Given Perimeter: A New Approach via Group Theory.</a>, Amer. Math. Monthly 126 (2019), no. 8, 735-739.

%H Wulf-Dieter Geyer, <a href="http://www.mi.uni-erlangen.de/~geyer/geschima">Lecture on history of medieval mathematics</a> [broken link]

%H M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper98.pdf">Triangles With Integer Sides</a>

%H M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper74.pdf">Triangles With Integer Sides, Revisited</a>

%H R. Honsberger, <a href="/A005044/a005044_1.pdf">Mathematical Gems III</a>, Math. Assoc. Amer., 1985, p. 39. [Annotated scanned copy]

%H T. Jenkyns and E. Muller, <a href="http://www.jstor.org/stable/2589119">Triangular triples from ceilings to floors</a>, Amer. Math. Monthly, 107 (Aug. 2000), 634-639.

%H J. H. Jordan, R. Walch and R. J. Wisner, <a href="http://www.jstor.org/stable/2321300">Triangles with integer sides</a>, Amer. Math. Monthly, 86 (1979), 686-689.

%H Hermann Kremer, Posting to de.sci.mathematik <a href="http://groups.google.de/groups?selm=cacqdf%24f0q%241%40online.de">(1)</a>, <a href="http://groups.google.de/groups?selm=canh34%24444%241%40online.de">(2)</a>, and <a href="http://groups.google.de/groups?selm=cankur%24km3%241%40online.de">(3)</a>. [Dead links]

%H Hermann Kremer, <a href="http://groups.google.de/groups?selm=cavdfh$l8a$1@online.de">Posting to alt.math.recreational</a> [Dead link]

%H N. Krier and B. Manvel, <a href="http://www.jstor.org/stable/2690701">Counting integer triangles</a>, Math. Mag., 71 (1998), 291-295.

%H Mathforum, <a href="http://mathforum.org/library/drmath/view/51547.html">Triangle Perimeters</a>

%H Augustine O. Munagi, <a href="http://www.emis.de/journals/INTEGERS/papers/h25/h25.Abstract.html">Computation of q-partial fractions</a>, INTEGERS: Electronic Journal Of Combinatorial Number Theory, 7 (2007), #A25.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H S. A. Shirali, <a href="http://atcm.mathandtech.org/EP2013/invited_papers/3612013_20372.pdf">Case Studies in Experimental Mathematics</a>, 2013.

%H David Singmaster, <a href="http://www2.edc.org/makingmath/handbook/Teacher/GettingInformation/TrianglesAndBarrels.pdf">Triangles with Integer Sides and Sharing Barrels</a>, College Math J, 21:4 (1990) 278-285.

%H James Tanton, <a href="http://www.themathcircle.org/integertriangles.pdf">Young students approach integer triangles</a>, FOCUS 22 no. 5 (2002), 4 - 6.

%H James Tanton, <a href="http://www.jamestanton.com/?p=1413">Integer Triangles</a>, Chapter 11 in “Mathematics Galore!” (MAA, 2012).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlcuinsSequence.html">Alcuin's Sequence</a>, <a href="http://mathworld.wolfram.com/IntegerTriangle.html">Integer Triangle</a>, and <a href="http://mathworld.wolfram.com/Triangle.html">Triangle</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Propositiones_ad_Acuendos_Juvenes">Propositiones ad acuendos juvenes</a>.

%H R. G. Wilson v, <a href="/A005044/a005044.pdf">Letter to N. J. A. Sloane</a>, date unknown.

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Mo#Molien">Index entries for Molien series</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1,-1,-1,-1,0,1).

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

%F 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).

%F 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 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.

%F Euler transform of length 4 sequence [ 0, 1, 1, 1]. - _Michael Somos_, Sep 04 2006

%F a(-3 - n) = a(n). - _Michael Somos_, Sep 04 2006

%F 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

%F 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

%F 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

%F 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

%F 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

%F 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

%F 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

%F a(n)-a(n-4) = A008615(n-1). - _R. J. Mathar_, Jun 23 2021

%F a(n)-a(n-2) = A008679(n-3). - _R. J. Mathar_, Jun 23 2021

%e 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.

%e 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 + ...

%e From _John M. Campbell_, Jan 29 2016: (Start)

%e 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:

%e (13,1,1) |- 15

%e (11,3,1) |- 15

%e (9,5,1) |- 15

%e (9,3,3) |- 15

%e (7,7,1) |- 15

%e (7,5,3) |- 15

%e (5,5,5) |- 15

%e (End)

%p A005044 := n-> floor((1/48)*(n^2+3*n+21+(-1)^(n-1)*3*n)): seq(A005044(n), n=0..73);

%p A005044 := -1/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**3; # _Simon Plouffe_ in his 1992 dissertation

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

%t CoefficientList[Series[x^3/((1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 0, 105}], x] (* _Robert G. Wilson v_, Jun 02 2004 *)

%t 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 *)

%t 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 *)

%t Table[Length@Select[IntegerPartitions[n, {3}], Max[#]*180 < 90 n &], {n, 1, 100}] (* _Frank M Jackson_, Nov 04 2022 *)

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

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

%o (PARI) a(n)=if(n%2,n+3,n)^2\/48 \\ _Charles R Greathouse IV_, May 02 2016

%o (PARI) concat(vector(3), Vec((x^3)/((1-x^2)*(1-x^3)*(1-x^4)) + O(x^70))) \\ _Felix Fröhlich_, Jun 07 2017

%o (Haskell)

%o a005044 = p [2,3,4] . (subtract 3) where

%o p _ 0 = 1

%o p [] _ = 0

%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

%o -- _Reinhard Zumkeller_, Feb 28 2013

%Y See A266755 for a version without the three leading zeros.

%Y Cf. A002620, A070083, A008795.

%Y Both bisections give (essentially) A001399.

%Y (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)

%Y Cf. this sequence (k=3), A288165 (k=4), A288166 (k=5).

%Y 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).

%K easy,nonn,nice

%O 0,8

%A _Robert G. Wilson v_

%E Additional comments from _Reinhard Zumkeller_, May 11 2002

%E Yaglom reference and mod formulas from Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 27 2000

%E The reference to Alcuin of York (735-804) was provided by Hermann Kremer (hermann.kremer(AT)onlinehome.de), Jun 18 2004