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%I #19 Oct 18 2018 10:34:34
%S 6,30,120,360,720,720,732,852,2040,12600,95760,666000,666018,666306,
%T 670896,739440,1694160,14032080,14032104,14032632,14044224,14287104,
%U 19132560,110941200,110941230,110942070,110965560,111598920,128041920,538459200,538459236
%N a(n) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15*14*13 + ... + (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=6.
%H Colin Barker, <a href="/A319870/b319870.txt">Table of n, a(n) for n = 1..1000</a>
%e a(1) = 6;
%e a(2) = 6*5 = 30;
%e a(3) = 6*5*4 = 120;
%e a(4) = 6*5*4*3 = 360;
%e a(5) = 6*5*4*3*2 = 720;
%e a(6) = 6*5*4*3*2*1 = 720;
%e a(7) = 6*5*4*3*2*1 + 12 = 732;
%e a(8) = 6*5*4*3*2*1 + 12*11 = 852;
%e a(9) = 6*5*4*3*2*1 + 12*11*10 = 2040;
%e a(10) = 6*5*4*3*2*1 + 12*11*10*9 = 12600;
%e a(11) = 6*5*4*3*2*1 + 12*11*10*9*8 = 95760;
%e a(12) = 6*5*4*3*2*1 + 12*11*10*9*8*7 = 666000;
%e a(13) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18 = 666018;
%e a(14) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17 = 666306;
%e a(15) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16 = 670896;
%e a(16) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15 = 739440;
%e a(17) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15*14 = 1694160;
%e a(18) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15*14*13 = 14032080;
%e a(19) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15*14*13 + 24 = 14032104;
%e a(20) = 6*5*4*3*2*1 + 12*11*10*9*8*7 + 18*17*16*15*14*13 + 24*23 = 14032632;
%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,6),n=1..35); # _Muniru A Asiru_, Sep 30 2018
%t k:=6; 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 *)
%Y For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), this sequence (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 30 2018
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