Alcuin's sequence

Last updated

In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of: [1]

The sequence begins with these integers:

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 (sequence A005044 in the OEIS )

The nth term indexed from zero, i.e., the coefficient of in the power series, is the number of triangles with integer sides and perimeter n. [1] It is also 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.

If one deletes the three leading zeros, then it is the number of ways in which n empty casks, n casks half-full of wine and n full casks can be distributed to three persons in such a way that each one gets the same number of casks and the same amount of wine. This is the generalization of problem 12 appearing in Propositiones ad Acuendos Juvenes ("Problems to Sharpen the Young") usually attributed to Alcuin. That problem is given as,

Problem 12: A certain father died and left as an inheritance to his three sons 30 glass flasks, of which 10 were full of oil, another 10 were half full, while another 10 were empty. Divide the oil and flasks so that an equal share of the commodities should equally come down to the three sons, both of oil and glass. [2]
In Latin: "XII. Propositio de quodam paterfamilias et tribus filius eius : Quidam paterfamilias moriens dimisit haereditatem tribus filiis suis, XXX ampullas uitreas, quarum decem fuerunt plenae oleo. Aliae decem dimidiae. Tertiae decem uacuae. Diuidat, qui potest, oleum et ampullas ut unicuique eorum de tribus filiis aequaliter obueniat tam de uitro, quam de oleo." [3]

The term "Alcuin's sequence" may be traced back to D. Olivastro's 1993 book on mathematical games, Ancient Puzzles: Classical Brainteasers and Other Timeless Mathematical Games of the Last 10 Centuries (Bantam, New York). [4]

The sequence with the three leading zeros deleted is equivalently the sequence of coefficients of the power-series expansion of [5] [6]

This sequence has also been called Alcuin's sequence by some authors. [6]

Related Research Articles

In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: For example, The value of 0! is 1, according to the convention for an empty product.

In mathematics, a polynomial is a mathematical expression consisting of indeterminates and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1.

<span class="mw-page-title-main">Square-free integer</span> Number without repeated prime factors

In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 32. The smallest positive square-free numbers are

The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. The conjecture has been shown to hold for all positive integers up to 2.95×1020, but no general proof has been found.

19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.

27 is the natural number following 26 and preceding 28.

84 (eighty-four) is the natural number following 83 and preceding 85. It is seven dozens.

<span class="mw-page-title-main">37 (number)</span> Natural number

37 (thirty-seven) is the natural number following 36 and preceding 38.

105 is the natural number following 104 and preceding 106.

300 is the natural number following 299 and preceding 301.

The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is the chairman of the OEIS Foundation.

126 is the natural number following 125 and preceding 127.

<span class="mw-page-title-main">Look-and-say sequence</span> Integer sequence

In mathematics, the look-and-say sequence is the sequence of integers beginning as follows:

In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like for a nonnegative integer . Specifically, the binomial series is the MacLaurin series for the function , where and . Explicitly,

In mathematics, the nth Motzkin number is the number of different ways of drawing non-intersecting chords between n points on a circle. The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, combinatorics and number theory.

252 is the natural number following 251 and preceding 253.

<span class="mw-page-title-main">Kempner function</span>

In number theory, the Kempner function is defined for a given positive integer to be the smallest number such that divides the factorial . For example, the number does not divide , , or , but does divide ,so .

Backhouse's constant is a mathematical constant named after Nigel Backhouse. Its value is approximately 1.456 074 948.

14 (fourteen) is the natural number following 13 and preceding 15.

<span class="mw-page-title-main">Constant-recursive sequence</span> Infinite sequence of numbers satisfying a linear equation

In mathematics, an infinite sequence of numbers is called constant-recursive if it satisfies an equation of the form

References

  1. 1 2 Sloane, N. J. A. (ed.). "SequenceA005044(Alcuin's sequence)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  2. ↑ Problems to Sharpen the Young, John Hadley and David Singmaster, The Mathematical Gazette, 76, #475 (March 1992), p. 109
  3. ↑ https://www.documentacatholicaomnia.eu/02m/0735-0804,_Alcuinus,_Propositiones_Alcuini_Karoli_Magni_Imperatoris_Ad_Acuendos_Juvenes,_MLT.pdf
  4. ↑ Binder, Donald J.; Erickson, Martin (2012), "Alcuin's Sequence", American Mathematical Monthly, 119 (2): 115–121, doi:10.4169/amer.math.monthly.119.02.115, S2CID   207521021
  5. ↑ Sloane, N. J. A. (ed.). "SequenceA266755". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
  6. 1 2 Weisstein, Eric W. "Alcuin's Sequence". MathWorld .
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.