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%I #16 Nov 07 2019 07:01:52
%S 0,0,0,0,0,0,0,2,4,6,8,10,12,18,24,34,44,54,64,80,96,118,140,168,196,
%T 232,268,312,356,408,460,530,600,680,760,850,940,1052,1164,1298,1432,
%U 1578,1724,1896,2068,2266,2464,2688,2912,3166,3420,3704,3988,4302
%N Sum of the even parts appearing among the third largest parts of the partitions of n into 4 parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_28">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,0,-2,4,-6,6,-6,5,-4,0,4,-5,6,-6,6,-4,2,0,-2,2,-2,2,-2,1).
%F a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} j * ((j-1) mod 2).
%F a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - 2*a(n-7) + 4*a(n-8) - 6*a(n-9) + 6*a(n-10) - 6*a(n-11) + 5*a(n-12) - 4*a(n-13) + 4*a(n-15) - 5*a(n-16) + 6*a(n-17) - 6*a(n-18) + 6*a(n-19) - 4*a(n-20) + 2*a(n-21) - 2*a(n-23) + 2*a(n-24) - 2*a(n-25) + 2*a(n-26) - 2*a(n-27) + a(n-28) for n > 27. - _Wesley Ivan Hurt_, Sep 04 2019
%e Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
%e 1+1+1+9
%e 1+1+2+8
%e 1+1+3+7
%e 1+1+4+6
%e 1+1+1+8 1+1+5+5
%e 1+1+2+7 1+2+2+7
%e 1+1+1+7 1+1+3+6 1+2+3+6
%e 1+1+2+6 1+1+4+5 1+2+4+5
%e 1+1+3+5 1+2+2+6 1+3+3+5
%e 1+1+1+6 1+1+4+4 1+2+3+5 1+3+4+4
%e 1+1+1+5 1+1+2+5 1+2+2+5 1+2+4+4 2+2+2+6
%e 1+1+2+4 1+1+3+4 1+2+3+4 1+3+3+4 2+2+3+5
%e 1+1+3+3 1+2+2+4 1+3+3+3 2+2+2+5 2+2+4+4
%e 1+2+2+3 1+2+3+3 2+2+2+4 2+2+3+4 2+3+3+4
%e 2+2+2+2 2+2+2+3 2+2+3+3 2+3+3+3 3+3+3+3
%e --------------------------------------------------------------------------
%e n | 8 9 10 11 12 ...
%e --------------------------------------------------------------------------
%e a(n) | 4 6 8 10 12 ...
%e --------------------------------------------------------------------------
%e - _Wesley Ivan Hurt_, Sep 04 2019
%t Table[Sum[Sum[Sum[j * Mod[j - 1, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
%t LinearRecurrence[{2, -2, 2, -2, 2, 0, -2, 4, -6, 6, -6, 5, -4, 0, 4, -5, 6, -6, 6, -4, 2, 0, -2, 2, -2, 2, -2, 1}, {0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 8, 10, 12, 18, 24, 34, 44, 54, 64, 80, 96, 118, 140, 168, 196, 232, 268, 312}, 60] (* _Wesley Ivan Hurt_, Sep 04 2019 *)
%Y Cf. A309711, A309715.
%K nonn
%O 0,8
%A _Wesley Ivan Hurt_, Aug 13 2019
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