Search results
Results from the WOW.Com Content Network
0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ... "subtract if possible, otherwise add" : a (0) = 0; for n > 0, a ( n ) = a ( n − 1) − n if that number is positive and not already in the sequence, otherwise a ( n ) = a ( n − 1) + n , whether or not that number is already in the sequence.
Suppose one wants to determine the 5-combination at position 72. The successive values of () for n = 4, 5, 6, ... are 0, 1, 6, 21, 56, 126, 252, ..., of which the largest one not exceeding 72 is 56, for n = 8. Therefore c 5 = 8, and the remaining elements form the 4-combination at position 72 − 56 = 16. The successive values of () for n = 3 ...
The numbers of compositions of n +1 into k +1 ordered partitions form Pascal's triangle Using the Fibonacci sequence to count the {1, 2}-restricted compositions of n, for example, the number of ways one can ascend a staircase of length n, taking one or two steps at a time
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
The number of such strings is the number of ways to place 10 stars in 13 positions, () = =, which is the number of 10-multisubsets of a set with 4 elements. Bijection between 3-subsets of a 7-set (left) and 3-multisets with elements from a 5-set (right).
Two lists count as the same if it is possible to both reorder and relabel them as above and produce the same result. For example, (3, 5, 3) and (2, 9, 9) count as the same because they can be reordered as (3, 3, 5) and (9, 9, 2) and then relabeling both produces the same list (1, 1, 2).
In this sequence, the positions at which the numbers in the sequence are divisible by a prime p form an arithmetic progression; for instance, the even numbers in the sequence are the numbers a i where i is congruent to 1 mod 3. The progressions divisible by different primes form a covering system, showing that every number in the sequence is ...
G(3) is at least 4 (since cubes are congruent to 0, 1 or −1 mod 9); for numbers less than 1.3 × 10 9, 1 290 740 is the last to require 6 cubes, and the number of numbers between N and 2N requiring 5 cubes drops off with increasing N at sufficient speed to have people believe that G(3) = 4; [17] the largest number now known not to be a sum of ...