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%I #55 Feb 09 2023 21:58:50
%S 3,18,63,144,285,486,777,1152,1647,2250,3003,3888,4953,6174,7605,9216,
%T 11067,13122,15447,18000,20853,23958,27393,31104,35175,39546,44307,
%U 49392,54897,60750,67053,73728,80883,88434,96495,104976,113997,123462,133497,144000
%N Sum of the partition parts of 3n into 3 parts.
%H G. C. Greubel, <a href="/A235988/b235988.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F a(n) = 3*n^3 - 3*n*floor(n^2/4).
%F a(n) = 3n * A077043(n).
%F a(n) = a(n-1) + 3*A077043(n-1) + A001651(n) + A093353(3n-2).
%F From _Colin Barker_, Jan 18 2014: (Start)
%F a(n) = (3*n*(1-(-1)^n+6*n^2))/8.
%F G.f.: 3*x*(x^4+4*x^3+8*x^2+4*x+1) / ((x-1)^4*(x+1)^2). (End)
%F a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 6. - _Wesley Ivan Hurt_, Nov 15 2015
%F E.g.f.: 3*x*((4 + 9*x + 3*x^2)*cosh(x) + 3*(1 + 3*x + x^2)*sinh(x))/4. - _Stefano Spezia_, Feb 09 2023
%e a(2) = 18; 3(2) = 6 has 3 partitions into 3 parts: (4, 1, 1), (3, 2, 1), and (2, 2, 2). The sum of the parts is 18.
%e Figure 1: The partitions of 3n into 3 parts for n = 1, 2, 3, ...
%e 13 + 1 + 1
%e 12 + 2 + 1
%e 11 + 3 + 1
%e 10 + 4 + 1
%e 9 + 5 + 1
%e 8 + 6 + 1
%e 7 + 7 + 1
%e 10 + 1 + 1 11 + 2 + 2
%e 9 + 2 + 1 10 + 3 + 2
%e 8 + 3 + 1 9 + 4 + 2
%e 7 + 4 + 1 8 + 5 + 2
%e 6 + 5 + 1 7 + 6 + 2
%e 7 + 1 + 1 8 + 2 + 2 9 + 3 + 3
%e 6 + 2 + 1 7 + 3 + 2 8 + 4 + 3
%e 5 + 3 + 1 6 + 4 + 2 7 + 5 + 3
%e 4 + 4 + 1 5 + 5 + 2 6 + 6 + 3
%e 4 + 1 + 1 5 + 2 + 2 6 + 3 + 3 7 + 4 + 4
%e 3 + 2 + 1 4 + 3 + 2 5 + 4 + 3 6 + 5 + 4
%e 1 + 1 + 1 2 + 2 + 2 3 + 3 + 3 4 + 4 + 4 5 + 5 + 5
%e 3(1) 3(2) 3(3) 3(4) 3(5) .. 3n
%e ------------------------------------------------------------------------
%e 3 18 63 144 285 .. a(n)
%e - _Wesley Ivan Hurt_, Sep 07 2019
%p A235988:=n->3*n^3 - 3*n*floor(n^2/4); seq(A235988(n), n=1..100);
%t Table[3 n^3 - 3 n*Floor[n^2/4], {n, 100}] (* or *) CoefficientList[ Series[3*x*(x^4 + 4*x^3 + 8*x^2 + 4*x + 1)/((x - 1)^4*(x + 1)^2), {x, 0, 30}], x]
%t LinearRecurrence[{2,1,-4,1,2,-1},{3,18,63,144,285,486},40] (* _Harvey P. Dale_, May 17 2018 *)
%o (PARI) a(n)=3*n^3 - n^2\4*3*n \\ _Charles R Greathouse IV_, Oct 07 2015
%o (PARI) x='x+O('x^50); Vec(3*x*(x^4+4*x^3+8*x^2+4*x+1)/((x-1)^4*(x+1)^2)) \\ _Altug Alkan_, Nov 01 2015
%o (Magma) [3*n^3-3*n*Floor(n^2/4): n in [1..100]]; // _Wesley Ivan Hurt_, Nov 01 2015
%o (Magma) [3*n*(1-(-1)^n+6*n^2)/8: n in [1..40]]; // _Vincenzo Librandi_, Nov 18 2015
%Y Cf. A001651, A077043, A093353.
%K nonn,easy
%O 1,1
%A _Wesley Ivan Hurt_, Jan 17 2014
%E a(165) in b-file corrected by _Andrew Howroyd_, Feb 21 2018
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