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A051866
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14-gonal (or tetradecagonal) numbers: a(n) = n*(6*n-5).
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28
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0, 1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, 804, 949, 1106, 1275, 1456, 1649, 1854, 2071, 2300, 2541, 2794, 3059, 3336, 3625, 3926, 4239, 4564, 4901, 5250, 5611, 5984, 6369, 6766, 7175, 7596, 8029, 8474, 8931, 9400, 9881, 10374
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OFFSET
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0,3
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COMMENTS
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Sequence found by reading the line from 0, in the direction 0, 14, ... and the parallel line from 1, in the direction 1, 39, ..., in the square spiral whose vertices are the generalized 14-gonal numbers A195818. Also sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 14, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012
Starting with offset 1, this is the binomial transform of (1, 13, 12, 0, 0, 0, ...). - Gary W. Adamson, Jul 29 2015
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REFERENCES
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Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing, 2012, page 6.
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LINKS
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FORMULA
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a(12*a(n)+67*n+1) = a(12*a(n) + 67*n) + a(12*n + 1). - Vladimir Shevelev, Jan 24 2014
Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi + log(432))/10.
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + 2*sqrt(3)*arccoth(sqrt(3)) - log(2))/5. (End)
Product_{n>=2} (1 - 1/a(n)) = 6/7. - Amiram Eldar, Jan 21 2021
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MAPLE
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MATHEMATICA
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Table[n*(6*n - 5), {n, 0, 100}] (* Robert Price, Oct 11 2018 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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