Hexagonal number

Article summary

that a hexagonal number (middle column) can be rearranged as rectangular and odd-sided triangular numbers A hexagonal number is a figurate number. The nth hexagonal number h n is the number of distinct dots in a pattern of dots consisting of the outlines of regular hexagons with sides up to n dots, when the hexagons are overlaid so that they share one vertex. The formula for the nth hexagonal number : h_n= 2n^2-n = n(2n-1) = \frac{2n(2n-1)}{2}. The first few hexagonal numbers are: :1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946... Every hexagonal number is a triangular number, but only every other triangular number (the 1st, 3rd, 5th, 7th, etc.) is a hexagonal number. Like a triangular number, the digital root in base 10 of a hexagonal number can only be 1, 3, 6, or 9. The digital root pattern, repeating every nine terms, is "1 6 6 1 9 3 1 3 9". Every even perfect number is hexagonal, given by the formula : M_p 2^{p-1} = M_p \frac{M_p + 1}{2} = h_{(M_p+1)/2}=h_{2^{p-1}} :where M p is a Mersenne prime. 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.…

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