OFFSET
1,2
COMMENTS
In general, for alternating sequences that multiply the first k natural numbers, and subtract/add the products of the next k natural numbers (preserving the order of operations up to n), we have a(n) = (-1)^floor(n/k) * Sum_{i=1..k-1} (1-sign((n-i) mod k)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/k)+1) * (1-sign(i mod k)) * (Product_{j=1..k} (i-j+1)). Here k=6.
An alternating version of A319207.
FORMULA
a(n) = (-1)^floor(n/6) * Sum_{i=1..5} (1-sign((n-i) mod 6)) * (Product_{j=1..i} (n-j+1)) + Sum_{i=1..n} (-1)^(floor(i/6)+1) * (1-sign(i mod 6)) * (Product_{j=1..6} (i-j+1)).
EXAMPLE
a(1) = 1;
a(2) = 1*2 = 2;
a(3) = 1*2*3 = 6;
a(4) = 1*2*3*4 = 24;
a(5) = 1*2*3*4*5 = 120;
a(6) = 1*2*3*4*5*6 = 720;
a(7) = 1*2*3*4*5*6 - 7 = 713;
a(8) = 1*2*3*4*5*6 - 7*8 = 664;
a(9) = 1*2*3*4*5*6 - 7*8*9 = 216;
a(10) = 1*2*3*4*5*6 - 7*8*9*10 = -4320;
a(11) = 1*2*3*4*5*6 - 7*8*9*10*11 = -54720;
a(12) = 1*2*3*4*5*6 - 7*8*9*10*11*12 = -664560;
a(13) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13 = -664547;
a(14) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14 = -664378;
a(15) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14*15 = -661830;
a(16) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14*15*16 = -620880;
a(17) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14*15*16*17 = 78000; etc.
MATHEMATICA
a[n_]:=(-1)^Floor[n/6]*Sum[(1-Sign[Mod[n-i, 6]])*Product[n-j+1, {j, 1, i}], {i, 1, 5}]+Sum[(-1)^(Floor[i/6]+1)*(1-Sign[Mod[i, 6]])*Product[i-j+1, {j, 1, 5}], {i, 1, n}]; Array[a, 30] (* Stefano Spezia, Sep 23 2018 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Wesley Ivan Hurt, Sep 22 2018
STATUS
approved