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A236364
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Sum of all the middle parts in the partitions of 3n into 3 parts.
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6
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1, 5, 18, 40, 80, 135, 217, 320, 459, 625, 836, 1080, 1378, 1715, 2115, 2560, 3077, 3645, 4294, 5000, 5796, 6655, 7613, 8640, 9775, 10985, 12312, 13720, 15254, 16875, 18631, 20480, 22473, 24565, 26810, 29160, 31672, 34295, 37089, 40000, 43091, 46305, 49708
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history;
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OFFSET
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1,2
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).
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FORMULA
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a(n) = A000330(n) + Sum_{i=1..floor((n-1)/2)} (n + i)*(n - 2i).
a(n) = n*(n+1)*(2*n+1)/6 - floor((n-1)/2) * (4*floor((n-1)/2)^2 + 3*(n+2)*floor((n-1)/2) - 6*n^2 + 3*n + 2)/6.
G.f.: x*(x^4+3*x^3+7*x^2+3*x+1)/((x-1)^4*(x+1)^2). - Joerg Arndt, Jan 23 2014
a(n) = (n*(3-3*(-1)^n+10*n^2))/16. - Colin Barker, Jan 23 2014
a(n) = (n^3 + ceiling(n/2)^3 + floor(n/2)^3)/2. - Wesley Ivan Hurt, Apr 15 2016
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Wesley Ivan Hurt, Nov 19 2021
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EXAMPLE
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Add second columns for a(n):
13 + 1 + 1
12 + 2 + 1
11 + 3 + 1
10 + 4 + 1
9 + 5 + 1
8 + 6 + 1
7 + 7 + 1
10 + 1 + 1 11 + 2 + 2
9 + 2 + 1 10 + 3 + 2
8 + 3 + 1 9 + 4 + 2
7 + 4 + 1 8 + 5 + 2
6 + 5 + 1 7 + 6 + 2
7 + 1 + 1 8 + 2 + 2 9 + 3 + 3
6 + 2 + 1 7 + 3 + 2 8 + 4 + 3
5 + 3 + 1 6 + 4 + 2 7 + 5 + 3
4 + 4 + 1 5 + 5 + 2 6 + 6 + 3
4 + 1 + 1 5 + 2 + 2 6 + 3 + 3 7 + 4 + 4
3 + 2 + 1 4 + 3 + 2 5 + 4 + 3 6 + 5 + 4
1 + 1 + 1 2 + 2 + 2 3 + 3 + 3 4 + 4 + 4 5 + 5 + 5
3(1) 3(2) 3(3) 3(4) 3(5) .. 3n
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1 5 18 40 80 .. a(n)
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MAPLE
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A236364:=n->n*(n+1)*(2*n+1)/6 - floor((n-1)/2) * (4*floor((n-1)/2)^2 + (3*n+6)*floor((n-1)/2) - 6*n^2 + 3*n + 2)/6; seq(A236364(n), n=1..100);
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MATHEMATICA
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Table[Sum[i^2, {i, n}] + Sum[(n + i) (n - 2 i), {i, Floor[(n - 1)/2]}], {n, 100}]
CoefficientList[Series[(x^4 + 3 x^3 + 7 x^2 + 3 x + 1)/ ((x - 1)^4 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 18 2014 *)
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PROG
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(PARI) Vec(x*(x^4+3*x^3+7*x^2+3*x+1)/((x-1)^4*(x+1)^2) + O(x^100)) \\ Colin Barker, Jan 23 2014
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CROSSREFS
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Cf. A000330, A019298, A235988.
Sequence in context: A007742 A225272 A276819 * A352368 A000338 A212343
Adjacent sequences: A236361 A236362 A236363 * A236365 A236366 A236367
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KEYWORD
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nonn,easy
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AUTHOR
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Wesley Ivan Hurt, Jan 23 2014
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EXTENSIONS
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Name clarified by Wesley Ivan Hurt, Apr 16 2016
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STATUS
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approved
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