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%I #33 May 04 2021 15:23:59
%S 2,2,6,14,20,44,52,100,110,190,202,322,336,504,520,744,762,1050,1070,
%T 1430,1452,1892,1916,2444,2470,3094,3122,3850,3880,4720,4752,5712,
%U 5746,6834,6870,8094,8132,9500,9540,11060,11102,12782,12826,14674,14720,16744
%N a(n) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 + 12*11 + ... + (up to the n-th term).
%C For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = Sum_{j=1..k-1} (floor((n-j)/k)-floor((n-j-1)/k)) * (Product_{i=1..j} n-i-j+k+1) + Sum_{j=1..n} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=2.
%C The denominators of the generating functions for these sequences are (1 + x)*(1 - x^k)^(k+1). - _Georg Fischer_ and _Andrew Howroyd_, Mar 07 2020
%H Colin Barker, <a href="/A319866/b319866.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F G.f.: 2*x/((-1 + x)^2*(1 + x)^2) + 2*(x^2 + 3*x^4)/((-1 + x)^4 (1 + x)^3). - _Stefano Spezia_, Sep 30 2018
%F From _Colin Barker_, Sep 30 2018: (Start)
%F a(n) = (4*n - 6*n + 3*n^2 + 2*n^3) / 12 for n even.
%F a(n) = (15 + 4*n + 6*n - 3*n^2 + 2*n^3) / 12 for n odd.
%F a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>7.
%F (End)
%e a(1) = 2;
%e a(2) = 2*1 = 2;
%e a(3) = 2*1 + 4 = 6;
%e a(4) = 2*1 + 4*3 = 14;
%e a(5) = 2*1 + 4*3 + 6 = 20;
%e a(6) = 2*1 + 4*3 + 6*5 = 44;
%e a(7) = 2*1 + 4*3 + 6*5 + 8 = 52;
%e a(8) = 2*1 + 4*3 + 6*5 + 8*7 = 100;
%e a(9) = 2*1 + 4*3 + 6*5 + 8*7 + 10 = 110;
%e a(10) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 = 190;
%e a(11) = 2*1 + 4*3 + 6*5 + 8*7 + 10*9 + 12 = 202;
%e etc.
%p a:=(n,k)->add((floor((n-j)/k)-floor((n-j-1)/k))*(mul(n-i-j+k+1,i=1..j)),j=1..k-1) + add((floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,2),n=1..50); # _Muniru A Asiru_, Sep 30 2018
%t k:=2; a[n_]:= Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])* Product[n-i-j+k+1, {i,1,j }] , {j,1,k-1} ] + Sum[(Floor[j/k]-Floor[(j-1)/k])* Product[j-i+1, {i,1,k} ], {j,1,n}]; Array[a, 50] (* _Stefano Spezia_, Sep 30 2018 *)
%t CoefficientList[Series[2/((-1 + x)^2 (1 + x)^2) + ( 2 (x + 3 x^3))/((-1 + x)^4 (1 + x)^3), {x, 0, 50}], x] (* _Stefano Spezia_, Sep 30 2018 *)
%o (PARI) Vec(2*x*(1 - x^2 + 4*x^3) / ((1 - x)^4*(1 + x)^3) + O(x^50)) \\ _Colin Barker_, Sep 30 2018
%Y For similar sequences, see: A000217 (k=1), this sequence (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), A319872 (k=8), A319873 (k=9), A319874 (k=10).
%K nonn,easy
%O 1,1
%A _Wesley Ivan Hurt_, Sep 29 2018
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