A319546 a(n) = 1*2*3*4*5*6 - 7*8*9*10*11*12 + 13*14*15*16*17*18 - ... + (up to n).

1, 2, 6, 24, 120, 720, 713, 664, 216, -4320, -54720, -664560, -664547, -664378, -661830, -620880, 78000, 12701520, 12701501, 12701140, 12693540, 12525960, 8663640, -84207600, -84207575, -84206950, -84190050, -83716200, -69957000, 343310400, 343310369

Offset: 1

Author

Wesley Ivan Hurt, Sep 22 2018

Keywords

• sign
• easy

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.

Examples

			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.
		

Crossrefs

For similar sequences, see: A001057 (k=1), A319373 (k=2), A319543 (k=3), A319544 (k=4), A319545 (k=5), this sequence (k=6), A319547 (k=7), A319549 (k=8), A319550 (k=9), A319551 (k=10).

Cf. A319207.

Programs

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 *)

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)).