A309690 - OEIS

%I #14 Sep 03 2019 23:23:58

%S 0,0,0,0,0,2,4,4,4,8,12,16,20,26,32,38,44,58,72,80,88,106,124,142,160,

%T 182,204,226,248,284,320,346,372,414,456,498,540,588,636,684,732,800,

%U 868,922,976,1052,1128,1204,1280,1364,1448,1532,1616,1726,1836,1928

%N Sum of the even parts appearing among the second largest parts of the partitions of n into 3 parts.

%H Colin Barker, <a href="/A309690/b309690.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (2,-3,4,-3,2,1,-4,6,-8,6,-4,1,2,-3,4,-3,2,-1).

%F a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} i * ((i-1) mod 2).

%F From _Colin Barker_, Aug 23 2019: (Start)

%F G.f.: 2*x^5*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2).

%F a(n) = 2*a(n-1) - 3*a(n-2) + 4*a(n-3) - 3*a(n-4) + 2*a(n-5) + a(n-6) - 4*a(n-7) + 6*a(n-8) - 8*a(n-9) + 6*a(n-10) - 4*a(n-11) + a(n-12) + 2*a(n-13) - 3*a(n-14) + 4*a(n-15) - 3*a(n-16) + 2*a(n-17) - a(n-18) for n>17.

%F (End)

%e Figure 1: The partitions of n into 3 parts for n = 3, 4, ...

%e                                                           1+1+8

%e                                                    1+1+7  1+2+7

%e                                                    1+2+6  1+3+6

%e                                             1+1+6  1+3+5  1+4+5

%e                                      1+1+5  1+2+5  1+4+4  2+2+6

%e                               1+1+4  1+2+4  1+3+4  2+2+5  2+3+5

%e                        1+1+3  1+2+3  1+3+3  2+2+4  2+3+4  2+4+4

%e          1+1+1  1+1+2  1+2+2  2+2+2  2+2+3  2+3+3  3+3+3  3+3+4    ...

%e -----------------------------------------------------------------------

%e   n  |     3      4      5      6      7      8      9     10      ...

%e -----------------------------------------------------------------------

%e a(n) |     0      0      2      4      4      4      8     12      ...

%e -----------------------------------------------------------------------

%t Table[Sum[Sum[i * Mod[i - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]

%t LinearRecurrence[{2, -3, 4, -3, 2, 1, -4, 6, -8, 6, -4, 1, 2, -3, 4, -3, 2, -1}, {0, 0, 0, 0, 0, 2, 4, 4, 4, 8, 12, 16, 20, 26, 32, 38, 44, 58}, 80]

%o (PARI) concat([0,0,0,0,0], Vec(2*x^5*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4) / ((1 - x)^4*(1 + x)^2*(1 - x + x^2)^2*(1 + x^2)^2*(1 + x + x^2)^2) + O(x^60))) \\ _Colin Barker_, Aug 23 2019

%Y Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309687, A309688, A309689, A309692, A309694.

%K nonn,easy

%O 0,6

%A _Wesley Ivan Hurt_, Aug 12 2019