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 A236364 Sum of all the middle parts in the partitions of 3n into 3 parts. 6
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1). FORMULA 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 EXAMPLE 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 ------------------------------------------------------------------------ 1 5 18 40 80 .. a(n) MAPLE 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); MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A000330, A019298, A235988. Sequence in context: A007742 A225272 A276819 * A352368 A000338 A212343 Adjacent sequences: A236361 A236362 A236363 * A236365 A236366 A236367 KEYWORD nonn,easy AUTHOR Wesley Ivan Hurt, Jan 23 2014 EXTENSIONS Name clarified by Wesley Ivan Hurt, Apr 16 2016 STATUS approved

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Last modified March 22 18:49 EDT 2023. Contains 361433 sequences. (Running on oeis4.)