more A number that can be shown as a pentagonal pattern of dots. The first five pentagonal numbers are 1, 5, 12, 22 and 35.

1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187… (sequence A000326 in the OEIS).

Every pentagonal number is 1/3 of a triangular number. ., the first few of which are 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, (OEIS A001318).

Since in the visual representation of pn, the pentagon has n+1 dots on each side, counting the number of dots on each side and multiplying by 5, we get, 5(n+1). However, we have counted the dots at the vertices twice, so we subtract 5: which gives us pn=5n.

No odd perfect numbers are known, hence all known perfect numbers are hexagonal. For example, the 2nd hexagonal number is 2×3 = 6; the 4th is 4×7 = 28; the 16th is 16×31 = 496; and the 64th is 64×127 = 8128. The largest number that cannot be written as a sum of at most four hexagonal numbers is 130.

951 is the 20th Centered Pentagonal Number | Find the Factors.

Table of formulae and values

Pattern Rules. A numerical pattern is a sequence of numbers that has been created based on a formula or rule called a pattern rule. Pattern rules can use one or more mathematical operations to describe the relationship between consecutive numbers in the pattern. There are two primary categories of numerical patterns.

The definition of a pattern is someone or something used as a model to make a copy, a design, or an expected action. An example of a pattern is the paper sections a seamstress uses to make a dress; a dress pattern. An example of a pattern is polka dots. An example of a pattern is rush hour traffic; a traffic pattern.