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Number of odd parts appearing among the second largest parts of the partitions of n into 3 parts.
11

%I #21 Sep 01 2019 09:23:11

%S 0,0,0,1,1,1,1,2,3,4,4,5,6,7,8,10,11,12,13,15,17,19,20,22,24,26,28,31,

%T 33,35,37,40,43,46,48,51,54,57,60,64,67,70,73,77,81,85,88,92,96,100,

%U 104,109,113,117,121,126,131,136,140,145,150,155,160,166

%N Number of odd parts appearing among the second largest parts of the partitions of n into 3 parts.

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

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1,0,1,-2,2,-2,1).

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

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

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

%F a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) - 2*a(n-7) + 2*a(n-8) - 2*a(n-9) + a(n-10) for n>9.

%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) | 1 1 1 1 2 3 4 4 ...

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

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

%t LinearRecurrence[{2, -2, 2, -1, 0, 1, -2, 2, -2, 1}, {0, 0, 0, 1, 1, 1, 1, 2, 3, 4}, 80]

%o (PARI) a(n) = sum(j=1, n\3, sum(i=j, (n-j)\2, i % 2)); \\ _Michel Marcus_, Aug 23 2019

%Y Cf. A026923, A026927, A309683, A309684, A309685, A309686, A309688, A309689, A309690, A309692, A309694.

%K nonn,easy

%O 0,8

%A _Wesley Ivan Hurt_, Aug 12 2019