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A309690 Sum of the even parts appearing among the second largest parts of the partitions of n into 3 parts. 12
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, 182, 204, 226, 248, 284, 320, 346, 372, 414, 456, 498, 540, 588, 636, 684, 732, 800, 868, 922, 976, 1052, 1128, 1204, 1280, 1364, 1448, 1532, 1616, 1726, 1836, 1928 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,1,-4,6,-8,6,-4,1,2,-3,4,-3,2,-1).
FORMULA
a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} i * ((i-1) mod 2).
From Colin Barker, Aug 23 2019: (Start)
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).
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.
(End)
EXAMPLE
Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
-----------------------------------------------------------------------
n | 3 4 5 6 7 8 9 10 ...
-----------------------------------------------------------------------
a(n) | 0 0 2 4 4 4 8 12 ...
-----------------------------------------------------------------------
MATHEMATICA
Table[Sum[Sum[i * Mod[i - 1, 2], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 80}]
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]
PROG
(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
CROSSREFS
Sequence in context: A342272 A182635 A188346 * A173531 A122788 A078003
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Aug 12 2019
STATUS
approved

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Last modified June 17 11:50 EDT 2024. Contains 373445 sequences. (Running on oeis4.)