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%I #43 Nov 23 2022 11:43:01
%S 0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T 4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,
%U 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7
%N First tetrahedral coordinate; repeat m (m+1)*(m+2)/2 times.
%C If {(X,Y,Z)} are triples of nonnegative integers with X >= Y >= Z ordered by X, Y and Z, then X=A056556(n), Y=A056557(n) and Z=A056558(n)
%C From _Gus Wiseman_, Jul 03 2019: (Start)
%C Also the maximum number of distinct multiplicities among integer partitions of n. For example, random partitions of 56 realizing each number of distinct multiplicities are:
%C 1: (24,17,6,5,3,1)
%C 2: (10,9,9,5,5,4,4,3,3,2,1,1)
%C 3: (6,5,5,5,4,4,4,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)
%C 4: (28,5,5,3,3,3,2,2,1,1,1,1,1)
%C 5: (13,4,4,4,4,4,3,3,3,2,2,2,2,2,2,1,1)
%C 6: (6,5,5,4,4,4,3,3,3,3,2,2,2,2,2,1,1,1,1,1,1)
%C The maximum number of distinct multiplicities is 6, so a(56) = 6.
%C (End)
%F a(n) = floor(x) where x is the (largest real) solution to x^3 + 3x^2 + 2x - 6n = 0; a(A000292(n)) = n+1.
%F a(n+1) = a(n)+1 if a(n) = A056558(n), otherwise a(n). - _Graeme McRae_, Jan 09 2007
%F a(n) = floor(t/3 + 1/t - 1), where t = (81*n + 3*sqrt(729*n^2 - 3))^(1/3). - _Ridouane Oudra_, Mar 21 2021
%F a(n) = floor(t + 1/(3*t) - 1), where t = (6*n)^(1/3), for n>=1. - _Ridouane Oudra_, Nov 04 2022
%e 3 is (3+1) * (3+2)/2 = 10 times in the sequence all these occurrences are in consecutive places. The first 3 is at position binomial(3 + 2, 3) = 10, the last one at binomial((3 + 1) + 2, 3) - 1. - _David A. Corneth_, Oct 14 2022
%t Table[Table[m, {(m+1)(m+2)/2}], {m, 0, 7}] // Flatten (* _Jean-François Alcover_, Feb 28 2019 *)
%o (PARI) a(n)=my(t=polrootsreal(x^3+3*x^2+2*x-6*n)); t[#t]\1 \\ _Charles R Greathouse IV_, Feb 22 2017
%Y Cf. A000292, A003056, A056557, A056558, A056559, A056560.
%Y See also A194847.
%Y Cf. A088880, A088881, A116608, A325242.
%K nonn,easy
%O 0,5
%A _Henry Bottomley_, Jun 26 2000
%E Incorrect formula deleted by _Ridouane Oudra_, Nov 04 2022
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