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%I #65 Sep 22 2025 16:01:04
%S 0,0,1,0,1,2,2,3,6,6,10,14,18,24,35,42,58,76,97,124,164,202,261,329,
%T 412,514,649,795,992,1223,1503,1839,2262,2741,3346,4056,4908,5919,
%U 7150,8568,10297,12320,14721,17542,20911,24808,29456,34870,41232,48652,57389
%N Number of 3's in the last section of the set of partitions of n.
%C Also number of 3's in all partitions of n that do not contain 1 as a part.
%C Also 0 together with the first differences of A024787. - _Omar E. Pol_, Nov 13 2011
%C a(n) is the number of partitions of n having fewer 1s than 2s; e.g., a(7) counts these 3 partitions: [5, 2], [3, 2, 2], [2, 2, 2, 1]. - _Clark Kimberling_, Mar 31 2014
%C The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - _Omar E. Pol_, Apr 07 2014
%H Alois P. Heinz, <a href="/A182713/b182713.txt">Table of n, a(n) for n = 1..1000</a>
%F It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3), n >= 0. - _Omar E. Pol_, Feb 04 2012
%F a(n) ~ A000041(n)/3 ~ exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n). - _Vaclav Kotesovec_, Jan 03 2019
%e a(7) = 2 counts the 3's in 7 = 4+3 = 3+2+2. The 3's in 7 = 3+3+1 = 3+2+1+1 = 3+1+1+1+1 do not count.
%e From _Omar E. Pol_, Oct 27 2012: (Start)
%e --------------------------------------
%e Last section Number
%e of the set of of
%e partitions of 7 3's
%e --------------------------------------
%e 7 .............................. 0
%e 4 + 3 .......................... 1
%e 5 + 2 .......................... 0
%e 3 + 2 + 2 ...................... 1
%e . 1 .......................... 0
%e . 1 ...................... 0
%e . 1 ...................... 0
%e . 1 .................. 0
%e . 1 ...................... 0
%e . 1 .................. 0
%e . 1 .................. 0
%e . 1 .............. 0
%e . 1 .............. 0
%e . 1 .......... 0
%e . 1 ...... 0
%e ------------------------------------
%e . 5 - 3 = 2
%e .
%e In the last section of the set of partitions of 7 the difference between the sum of the third column and the sum of the fourth column is 5 - 3 = 2 equaling the number of 3's, so a(7) = 2 (see also A024787).
%e (End)
%p b:= proc(n, i) option remember; local g, h;
%p if n=0 then [1, 0]
%p elif i<2 then [0, 0]
%p else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
%p [g[1]+h[1], g[2]+h[2]+`if`(i=3, h[1], 0)]
%p fi
%p end:
%p a:= n-> b(n, n)[2]:
%p seq(a(n), n=1..70); # _Alois P. Heinz_, Mar 18 2012
%t z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 2]], {n, 0, z}] (* _Clark Kimberling_, Mar 31 2014 *)
%t b[n_, i_] := b[n, i] = Module[{g, h}, If[n == 0, {1, 0}, If[i<2, {0, 0}, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; Join[g[[1]] + h[[1]], g[[2]] + h[[2]] + If[i == 3, h[[1]], 0]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 70}] (* _Jean-François Alcover_, Nov 30 2015, after _Alois P. Heinz_ *)
%t Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 3], {n, 51}] (* _Robert Price_, May 15 2020 *)
%o (SageMath) A182713 = lambda n: sum(list(p).count(3) for p in Partitions(n) if 1 not in p) # _D. S. McNeil_, Nov 29 2010
%Y Column 3 of A194812.
%Y Cf. A135010, A138121, A174455, A182703, A182712, A182714, A240056.
%K nonn
%O 1,6
%A _Omar E. Pol_, Nov 28 2010