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A182713 Number of 3's in the last section of the set of partitions of n. 7
0, 0, 1, 0, 1, 2, 2, 3, 6, 6, 10, 14, 18, 24, 35, 42, 58, 76, 97, 124, 164, 202, 261, 329, 412, 514, 649, 795, 992, 1223, 1503, 1839, 2262, 2741, 3346, 4056, 4908, 5919, 7150, 8568, 10297, 12320, 14721, 17542, 20911, 24808, 29456, 34870, 41232, 48652, 57389 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Also number of 3's in all partitions of n that do not contain 1 as a part.

Also 0 together with the first differences of A024787. - Omar E. Pol, Nov 13 2011

a(n) = 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

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3), n >= 0. - Omar E. Pol, Feb 04 2012

Conjecture: a(n) ~ A000041(n)/3 ~ exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n). - Vaclav Kotesovec, Jan 03 2019

EXAMPLE

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.

From Omar E. Pol, Oct 27 2012: (Start)

--------------------------------------

Last section                   Number

of the set of                    of

partitions of 7                 3's

--------------------------------------

7 .............................. 0

4 + 3 .......................... 1

5 + 2 .......................... 0

3 + 2 + 2 ...................... 1

.   1 .......................... 0

.       1 ...................... 0

.       1 ...................... 0

.           1 .................. 0

.       1 ...................... 0

.           1 .................. 0

.           1 .................. 0

.               1 .............. 0

.               1 .............. 0

.                   1 .......... 0

.                       1 ...... 0

------------------------------------

.       5 - 3 =                  2

.

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).

(End)

MAPLE

b:= proc(n, i) option remember; local g, h;

      if n=0 then [1, 0]

    elif i<2 then [0, 0]

    else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));

         [g[1]+h[1], g[2]+h[2]+`if`(i=3, h[1], 0)]

      fi

    end:

a:= n-> b(n, n)[2]:

seq(a(n), n=1..70);  # Alois P. Heinz, Mar 18 2012

MATHEMATICA

(See A240056.) - Clark Kimberling, Mar 31 2014

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 *)

PROG

(Sage) A182713 = lambda n: sum(list(p).count(3) for p in Partitions(n) if 1 not in p) # D. S. McNeil, Nov 29 2010

CROSSREFS

Column 3 of A194812.

Cf. A135010, A138121, A182703, A182712, A182714, A240056.

Sequence in context: A077012 A078921 A011961 * A229626 A091770 A032058

Adjacent sequences:  A182710 A182711 A182712 * A182714 A182715 A182716

KEYWORD

nonn

AUTHOR

Omar E. Pol, Nov 28 2010

STATUS

approved

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Last modified April 3 19:43 EDT 2020. Contains 333198 sequences. (Running on oeis4.)