A309830 Sum of the odd parts appearing among the smallest parts of the partitions of n into 5 parts.

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 26, 30, 40, 48, 62, 72, 91, 105, 129, 148, 182, 206, 248, 282, 335, 377, 443, 496, 576, 642, 743, 823, 943, 1044, 1188, 1308, 1479, 1623, 1823, 1994, 2233, 2433, 2709, 2948, 3268, 3544, 3913, 4233, 4654, 5023

Offset

0,8

Links

Table of n, a(n) for n=0..54.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1,0,2,-2,-4,2,4,2,-4,-2,2,0,-1,1,2,-1,-2,-1,2,1,-1).

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} l * (l mod 2).

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8) + 2*a(n-10) - 2*a(n-11) - 4*a(n-12) + 2*a(n-13) + 4*a(n-14) + 2*a(n-15) - 4*a(n-16) - 2*a(n-17) + 2*a(n-18) - a(n-20) + a(n-21) + 2*a(n-22) - a(n-23) - 2*a(n-24) - a(n-25) + 2*a(n-26) + a(n-27) - a(n-28) for n > 27.

Example

Figure 1: The partitions of n into 5 parts for n = 5, 6, ...
                                                        1+1+1+1+5
                                                        1+1+1+2+4
                                            1+1+1+1+4   1+1+1+3+3
                                1+1+1+1+3   1+1+1+2+3   1+1+2+2+3
        1+1+1+1+1   1+1+1+1+2   1+1+1+2+2   1+1+2+2+2   1+2+2+2+2
-----------------------------------------------------------------------
  n  |      5           6           7           8           9       ...
-----------------------------------------------------------------------
a(n) |      1           1           2           3           5       ...
-----------------------------------------------------------------------

Mathematica

LinearRecurrence[{1, 2, -1, -2, -1, 2, 1, -1, 0, 2, -2, -4, 2, 4,
  2, -4, -2, 2, 0, -1, 1, 2, -1, -2, -1, 2, 1, -1}, {0, 0, 0, 0, 0, 1,
   1, 2, 3, 5, 6, 9, 11, 15, 18, 26, 30, 40, 48, 62, 72, 91, 105, 129,
   148, 182, 206, 248}, 50]

Crossrefs

Cf. A309787, A309831, A309834.

Sequence in context: A351003 A008771 A309831 * A008770 A347548 A230515

Adjacent sequences: A309827 A309828 A309829 * A309831 A309832 A309833

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 19 2019

Status

approved