A319887 - OEIS

%I #12 Oct 05 2018 08:06:01

%S 4,12,24,24,16,-32,-312,-1656,-1644,-1524,-336,10224,10208,9984,6864,

%T -33456,-33436,-33076,-26616,82824,82800,82272,70680,-172200,-172172,

%U -171444,-152544,319200,319168,318208,289440,-543840,-543804,-542580,-501000,869880

%N a(n) = 4*3*2*1 - 8*7*6*5 + 12*11*10*9 - 16*15*14*13 + ... - (up to the n-th term).

%C For similar sequences that alternate in descending blocks of k natural numbers, we have: a(n) = (-1)^floor(n/k) * 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} (-1)^(floor(j/k)+1) * (floor(j/k) - floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=4.

%e a(1) = 4;

%e a(2) = 4*3 = 12;

%e a(3) = 4*3*2 = 24;

%e a(4) = 4*3*2*1 = 24;

%e a(5) = 4*3*2*1 - 8 = 16;

%e a(6) = 4*3*2*1 - 8*7 = -32;

%e a(7) = 4*3*2*1 - 8*7*6 = -312;

%e a(8) = 4*3*2*1 - 8*7*6*5 = -1656;

%e a(9) = 4*3*2*1 - 8*7*6*5 + 12 = -1644;

%e a(10) = 4*3*2*1 - 8*7*6*5 + 12*11 = -1524;

%e a(11) = 4*3*2*1 - 8*7*6*5 + 12*11*10 = -336;

%e a(12) = 4*3*2*1 - 8*7*6*5 + 12*11*10*9 = 10224;

%e a(13) = 4*3*2*1 - 8*7*6*5 + 12*11*10*9 - 16 = 10208;

%e a(14) = 4*3*2*1 - 8*7*6*5 + 12*11*10*9 - 16*15 = 9984;

%e a(15) = 4*3*2*1 - 8*7*6*5 + 12*11*10*9 - 16*15*14 = 6864;

%e a(16) = 4*3*2*1 - 8*7*6*5 + 12*11*10*9 - 16*15*14*13 = -33456;

%e a(17) = 4*3*2*1 - 8*7*6*5 + 12*11*10*9 - 16*15*14*13 + 20 = -33436;

%e a(18) = 4*3*2*1 - 8*7*6*5 + 12*11*10*9 - 16*15*14*13 + 20*19 = -33076;

%e a(19) = 4*3*2*1 - 8*7*6*5 + 12*11*10*9 - 16*15*14*13 + 20*19*18 = -26616;

%e a(20) = 4*3*2*1 - 8*7*6*5 + 12*11*10*9 - 16*15*14*13 + 20*19*18*17 = 82824;

%e etc.

%p a:=(n,k)->(-1)^(floor(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( (-1)^(floor(j/k)+1)*(floor(j/k)-floor((j-1)/k))*(mul(j-i+1,i=1..k)),j=1..n): seq(a(n,4),n=1..40); # _Muniru A Asiru_, Sep 30 2018

%Y For similar sequences, see: A001057 (k=1), A319885 (k=2), A319886 (k=3), this sequence (k=4), A319888 (k=5), A319889 (k=6), A319890 (k=7), A319891 (k=8), A319892 (k=9), A319893 (k=10).

%K sign,easy

%O 1,1

%A _Wesley Ivan Hurt_, Sep 30 2018