Number of paths between grid corners, allowing diagonal steps
Delannoy number
Named after
Henri–Auguste Delannoy
No. of known terms
infinity
Formula
OEIS index
A008288
Square array of Delannoy
In mathematics, a Delannoy number counts the paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east. The Delannoy numbers are named after French army officer and amateur mathematician Henri Delannoy.[1]
The Delannoy number also counts the global alignments of two sequences of lengths and ,[2] the points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin,[3] and, in cellular automata, the cells in an m-dimensional von Neumann neighborhood of radius n.[4]
^Covington, Michael A. (2004), "The number of distinct alignments of two strings", Journal of Quantitative Linguistics, 11 (3): 173–182, doi:10.1080/0929617042000314921, S2CID 40549706
^Luther, Sebastian; Mertens, Stephan (2011), "Counting lattice animals in high dimensions", Journal of Statistical Mechanics: Theory and Experiment, 2011 (9): P09026, arXiv:1106.1078, Bibcode:2011JSMTE..09..026L, doi:10.1088/1742-5468/2011/09/P09026, S2CID 119308823
^Breukelaar, R.; Bäck, Th. (2005), "Using a Genetic Algorithm to Evolve Behavior in Multi Dimensional Cellular Automata: Emergence of Behavior", Proceedings of the 7th Annual Conference on Genetic and Evolutionary Computation (GECCO '05), New York, NY, USA: ACM, pp. 107–114, doi:10.1145/1068009.1068024, ISBN 1-59593-010-8, S2CID 207157009
In mathematics, a Delannoynumber D {\displaystyle D} counts the paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner...
Catalan–Mersenne numberDelannoynumber Fuss–Catalan number List of factorial and binomial topics Lobb numbers Motzkin number Narayana number Narayana polynomials...
0),(1,1)} . In this formulation, we see that the number of possible matches is the Delannoynumber. The optimal match is the match that satisfies all...
the number 1 differently than larger numbers, sometimes even not as a number at all. Euclid, for example, defined a unit first and then a number as a...
transform Delannoynumber Eulerian number Hypergeometric function List of factorial and binomial topics Macaulay representation of an integer Motzkin number Multiplicities...
numberDelannoynumber Motzkin number Narayana polynomials Schröder number Pascal's triangle Learning materials related to Partition related number triangles...
Telephone number which represent the number of ways of drawing chords if intersections are allowed Delannoynumber Narayana number Schröder number Yi Wang...
In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci...
r^{2}+(r+1)^{2}} . The number of cells in a d-dimensional von Neumann neighborhood of range r is the Delannoynumber D(d,r). The number of cells on a surface...
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has...
A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16461) that remains the same when its digits are...
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that...
In number theory, a perfect number is a positive integer that is equal to the sum of its positive proper divisors, that is, divisors excluding the number...
composite number is a positive integer that has more divisors than any smaller positive integer. A related concept is that of a largely composite number, a positive...
triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples...
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n ( n + 1 ) {\displaystyle n(n+1)} . The study...
rows in the middle rather than 2, the number of tilings drops to the much smaller number D(n,n), a Delannoynumber, which has only exponential rather than...
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron...
In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy...
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns...
In number theory, a pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1, either from...
In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition...
frequently arise in combinatorics. Moreover, all three can be defined as the number of partitions of n elements into k non-empty subsets, where each subset...
In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes...
2,5,15,52,203,877,4140,\dots } (sequence A000110 in the OEIS). The Bell number B n {\displaystyle B_{n}} counts the different ways to partition a set that...
Cullen number is a member of the integer sequence C n = n ⋅ 2 n + 1 {\displaystyle C_{n}=n\cdot 2^{n}+1} (where n {\displaystyle n} is a natural number). Cullen...
mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer...
Global Information
