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A005044 Alcuin's sequence: expansion of x^3/((1-x^2)*(1-x^3)*(1-x^4)).
(Formerly M0146)
54
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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 Cuculière.

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 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

REFERENCES

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.

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

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

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)

LINKS

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

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, 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.

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

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)

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

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

Hermann Kremer, Posting to alt.math.recreational

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. - N. J. A. Sloane, Apr 16 2011

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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.

Eric Weisstein's World of Mathematics, Triangle.

Eric Weisstein's World of Mathematics, Integer Triangle

Wikipedia, Propositiones ad acuendos juvenes.

Index entries for two-way infinite sequences

FORMULA

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(ceil((n-3)/3) <= i <= floor((n-3)/2), sum(ceil((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

EXAMPLE

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 + ...

MAPLE

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.]

MATHEMATICA

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 *)

PROG

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

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

(Haskell)

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

CROSSREFS

Cf. A002620, A001399, A062890, A069906, A069907, A070083, A008795.

(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)

Sequence in context: A241825 A029162 A225854 * A029142 A054685 A246581

Adjacent sequences:  A005041 A005042 A005043 * A005045 A005046 A005047

KEYWORD

easy,nonn,nice

AUTHOR

Robert G. Wilson v

EXTENSIONS

More terms from Erich Friedman

Additional comments from Reinhard Zumkeller, 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

STATUS

approved

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Last modified November 20 21:15 EST 2014. Contains 249754 sequences.