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

%I A309830 #29 Nov 07 2019 08:13:15
%S A309830 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,
%T A309830 182,206,248,282,335,377,443,496,576,642,743,823,943,1044,1188,1308,
%U A309830 1479,1623,1823,1994,2233,2433,2709,2948,3268,3544,3913,4233,4654,5023
%N A309830 Sum of the odd parts appearing among the smallest parts of the partitions of n into 5 parts.
%H A309830 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A309830 <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, 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).
%F A309830 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).
%F A309830 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.
%e A309830 Figure 1: The partitions of n into 5 parts for n = 5, 6, ...
%e A309830                                                         1+1+1+1+5
%e A309830                                                         1+1+1+2+4
%e A309830                                             1+1+1+1+4   1+1+1+3+3
%e A309830                                 1+1+1+1+3   1+1+1+2+3   1+1+2+2+3
%e A309830         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
%e A309830 -----------------------------------------------------------------------
%e A309830   n  |      5           6           7           8           9       ...
%e A309830 -----------------------------------------------------------------------
%e A309830 a(n) |      1           1           2           3           5       ...
%e A309830 -----------------------------------------------------------------------
%t A309830 LinearRecurrence[{1, 2, -1, -2, -1, 2, 1, -1, 0, 2, -2, -4, 2, 4,
%t A309830   2, -4, -2, 2, 0, -1, 1, 2, -1, -2, -1, 2, 1, -1}, {0, 0, 0, 0, 0, 1,
%t A309830    1, 2, 3, 5, 6, 9, 11, 15, 18, 26, 30, 40, 48, 62, 72, 91, 105, 129,
%t A309830    148, 182, 206, 248}, 50]
%Y A309830 Cf. A309787, A309831, A309834.
%K A309830 nonn
%O A309830 0,8
%A A309830 _Wesley Ivan Hurt_, Aug 19 2019