A319889 a(n) = 6*5*4*3*2*1 - 12*11*10*9*8*7 + 18*17*16*15*14*13 - 24*23*22*21*20*19 + ... - (up to the n-th term).

6, 30, 120, 360, 720, 720, 708, 588, -600, -11160, -94320, -664560, -664542, -664254, -659664, -591120, 363600, 12701520, 12701496, 12700968, 12689376, 12446496, 7601040, -84207600, -84207570, -84206730, -84183240, -83549880, -67106880, 343310400, 343310364

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=6.

Links

Table of n, a(n) for n=1..31.

Example

a(1) = 6;
a(2) = 6*5 = 30;
a(3) = 6*5*4 = 120;
a(4) = 6*5*4*3 = 360;
a(5) = 6*5*4*3*2 = 720;
a(6) = 6*5*4*3*2*1 = 720;
a(7) = 6*5*4*3*2*1 - 12 = 708;
a(8) = 6*5*4*3*2*1 - 12*11 = 588;
a(9) = 6*5*4*3*2*1 - 12*11*10 = -600;
a(10) = 6*5*4*3*2*1 - 12*11*10*9 = -11160;
a(11) = 6*5*4*3*2*1 - 12*11*10*9*8 = -94320;
a(12) = 6*5*4*3*2*1 - 12*11*10*9*8*7 = -664560;
a(13) = 6*5*4*3*2*1 - 12*11*10*9*8*7 + 18 = -664542;
a(14) = 6*5*4*3*2*1 - 12*11*10*9*8*7 + 18*17 = -664254;
a(15) = 6*5*4*3*2*1 - 12*11*10*9*8*7 + 18*17*16 = -659664;
a(16) = 6*5*4*3*2*1 - 12*11*10*9*8*7 + 18*17*16*15 = -591120;
a(17) = 6*5*4*3*2*1 - 12*11*10*9*8*7 + 18*17*16*15*14 = 363600;
a(18) = 6*5*4*3*2*1 - 12*11*10*9*8*7 + 18*17*16*15*14*13 = 12701520;
a(19) = 6*5*4*3*2*1 - 12*11*10*9*8*7 + 18*17*16*15*14*13 - 24 = 12701496;
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, 6), n=1..35); # Muniru A Asiru, Sep 30 2018

Crossrefs

For similar sequences, see: A001057 (k=1), A319885 (k=2), A319886 (k=3), A319887 (k=4), A319888 (k=5), this sequence (k=6), A319890 (k=7), A319891 (k=8), A319892 (k=9), A319893 (k=10).

Sequence in context: A334331 A117489 A050510 * A319870 A055281 A346174

Adjacent sequences: A319886 A319887 A319888 * A319890 A319891 A319892

Keyword

sign,easy

Author

Wesley Ivan Hurt, Sep 30 2018

Status

approved