OFFSET
1,1
COMMENTS
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=8.
EXAMPLE
a(1) = 8;
a(2) = 8*7 = 56;
a(3) = 8*7*6 = 336;
a(4) = 8*7*6*5 = 1680;
a(5) = 8*7*6*5*4 = 6720;
a(6) = 8*7*6*5*4*3 = 20160;
a(7) = 8*7*6*5*4*3*2 = 40320;
a(8) = 8*7*6*5*4*3*2*1 = 40320;
a(9) = 8*7*6*5*4*3*2*1 - 16 = 40304;
a(10) = 8*7*6*5*4*3*2*1 - 16*15 = 40080;
a(11) = 8*7*6*5*4*3*2*1 - 16*15*14 = 36960;
a(12) = 8*7*6*5*4*3*2*1 - 16*15*14*13 = -3360;
a(13) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12 = -483840;
a(14) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11 = -5725440;
a(15) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11*10 = -57617280;
a(16) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11*10*9 = -518878080;
a(17) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11*10*9 + 24 = -518878056;
a(18) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11*10*9 + 24*23 = -518877528;
a(19) = 8*7*6*5*4*3*2*1 - 16*15*14*13*12*11*10*9 + 24*23*22 = -518865936;
etc.
MAPLE
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, 8), n=1..30); # Muniru A Asiru, Sep 30 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Wesley Ivan Hurt, Sep 30 2018
STATUS
approved