A309882 - OEIS

A309882

Sum of the even parts appearing among the fourth largest parts of the partitions of n into 5 parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6, 10, 14, 18, 24, 30, 40, 52, 68, 88, 110, 136, 166, 198, 240, 286, 340, 404, 478, 560, 652, 754, 872, 1000, 1146, 1308, 1488, 1686, 1908, 2148, 2416, 2708, 3028, 3376, 3758, 4168, 4616, 5098, 5630, 6200, 6816, 7482, 8198

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OFFSET

0,10

LINKS

Table of n, a(n) for n=0..53.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12, 12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2, 4, -4, 4, -4, 4, -4, 3, -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)} k * ((k-1) mod 2).

a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 4*a(n-4) + 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - 2*a(n-8) - 2*a(n-9) + 6*a(n-10) - 10*a(n-11) + 12*a(n-12) - 12*a(n-13) + 12*a(n-14) - 12*a(n-15) + 11*a(n-16) - 9*a(n-17) + 4*a(n-18) + 4*a(n-19) - 9*a(n-20) + 11*a(n-21) - 12*a(n-22) + 12*a(n-23) - 12*a(n-24) + 12*a(n-25) - 10*a(n-26) + 6*a(n-27) - 2*a(n-28) - 2*a(n-29) + 4*a(n-30) - 4*a(n-31) + 4*a(n-32) - 4*a(n-33) + 4*a(n-34) - 4*a(n-35) + 3*a(n-36) - a(n-37) for n > 36.

EXAMPLE

Figure 1: The partitions of n into 5 parts for n = 10, 11, ..

1+1+1+1+10

1+1+1+2+9

1+1+1+3+8

1+1+1+4+7

1+1+1+5+6

1+1+1+1+9 1+1+2+2+8

1+1+1+2+8 1+1+2+3+7

1+1+1+3+7 1+1+2+4+6

1+1+1+4+6 1+1+2+5+5

1+1+1+5+5 1+1+3+3+6

1+1+1+1+8 1+1+2+2+7 1+1+3+4+5

1+1+1+2+7 1+1+2+3+6 1+1+4+4+4

1+1+1+3+6 1+1+2+4+5 1+2+2+2+7

1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6

1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5

1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5

1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4

1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4

1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6

1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5

1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4

1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4

2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3

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n | 10 11 12 13 14 ...

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a(n) | 4 6 10 14 18 ...

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MATHEMATICA

Table[Sum[Sum[Sum[Sum[k * Mod[k - 1, 2], {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]

LinearRecurrence[{3, -4, 4, -4, 4, -4, 4, -2, -2, 6, -10, 12, -12,

12, -12, 11, -9, 4, 4, -9, 11, -12, 12, -12, 12, -10, 6, -2, -2,

4, -4, 4, -4, 4, -4, 3, -1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 6,

10, 14, 18, 24, 30, 40, 52, 68, 88, 110, 136, 166, 198, 240, 286,

340, 404, 478, 560, 652, 754, 872, 1000, 1146, 1308}, 50]

CROSSREFS

Cf. A309879, A309880, A309881.

Sequence in context: A059254 A024518 A128422 * A098380 A007782 A035501

Adjacent sequences: A309879 A309880 A309881 * A309883 A309884 A309885

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Aug 21 2019

STATUS

approved