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Round[Surd[#, 3]]&/@Differences[Range[0, 70]^3] (* Harvey P. Dale, Aug 01 2020 *)
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1, 2, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24
1,1
0,2
Table[Round[(3*n^2 + 3*n + 1)^(1/3)], {n, 0, 100}] (* T. D. Noe, Oct 22 2013 *)
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2, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 27, 28, 28, 28, 28, 28, 29, 29, 29, 29, 29, 30, 30, 30, 30, 31, 31, 31, 31, 31, 32, 32, 32
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cube Cube roots of difference of consecutive cubes , rounded to whole number.
3x^2+3x+1 is the difference of two adjacent cubes, taking the cube root and rounding to a whole number yields an element of the series. 3 cubes is 27, inserting 3 into the formulat =s 37, 37 plus 27 is 64 the next cube after 27; the cube root of 37 is 3.33222etc rounded to 3 is the element in the series.
~3x3n^2+3x3n+1 is the difference of two adjacent cubes, taking the cube root and rounding to a whole number yields an element of the series. 3 cubes is 27, inserting 3 into the formulat formula =s 37, 37 plus 27 is 64 the next cube after 27; the cube root of 37 is 3.33222etc 33222... rounded to 3 is the element in the series.
(PARI) a(n)=round((3*n*(n+1)+1)^(1/3)) \\ Charles R Greathouse IV, Oct 22 2013
Cf. A003215.
nonn,changed,easy
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