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centeredn
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There are 9 letters in CENTEREDN ( C3D2E1N1R1T1 )
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A Centered nonagonal number is a centered figurate number that represents a nonagon with a dot in the center and all other dots surrounding the center dot in successive nonagonal layers. The centered nonagonal number for n is given by the formula* * * * N * c * ( * n * ) * = * * * * ( * 3 * n * − * 2 * ) * ( * 3 * n * − * 1 * ) * * 2 * * * . * * * {\displaystyle Nc(n)={\frac {(3n-2)(3n-1)}{2}}.} * Multiplying the (n - 1)th triangular number by 9 and then adding 1 yields the nth centered nonagonal number, but centered nonagonal numbers have an even simpler relation to triangular numbers: every third triangular number (the 1st, 4th, 7th, etc.) is also a centered nonagonal number.Thus, the first few centered nonagonal numbers are * 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946.The list above includes the perfect numbers 28 and 496. * All even perfect numbers are triangular numbers whose index is an odd Mersenne prime. Since every Mersenne prime greater than 3 is congruent to 1 modulo 3, it follows that every even perfect number greater than 6 is a centered nonagonal number. * In 1850, Sir Frederick Pollock conjectured that every natural number is the sum of at most eleven centered nonagonal numbers, which has been neither proven nor disproven. |