A319872 a(n) = 8*7*6*5*4*3*2*1 + 16*15*14*12*11*10*9 + ... + (up to the n-th term).

8, 56, 336, 1680, 6720, 20160, 40320, 40320, 40336, 40560, 43680, 84000, 564480, 5806080, 57697920, 518958720, 518958744, 518959272, 518970864, 519213744, 524059200, 615867840, 2263322880, 30173149440, 30173149472, 30173150432, 30173179200, 30174012480

Offset: 1

Author

Wesley Ivan Hurt, Sep 30 2018

Keywords

• nonn
• easy

Comments

For similar multiply/add sequences in descending blocks of k natural numbers, we have: a(n) = 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} (floor(j/k)-floor((j-1)/k)) * (Product_{i=1..k} j-i+1). Here, k=8.

Examples

			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 = 40336;
a(10) = 8*7*6*5*4*3*2*1 + 16*15 = 40560;
a(11) = 8*7*6*5*4*3*2*1 + 16*15*14 = 43680;
a(12) = 8*7*6*5*4*3*2*1 + 16*15*14*13 = 84000;
a(13) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12 = 564480;
a(14) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11 = 5806080;
a(15) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10 = 57697920;
a(16) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 = 518958720;
a(17) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24 = 518958744;
a(18) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24*23 = 518959272;
a(19) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24*23*22 = 518970864;
a(20) = 8*7*6*5*4*3*2*1 + 16*15*14*13*12*11*10*9 + 24*23*22*21 = 519213744;
etc.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000

Crossrefs

For similar sequences, see: A000217 (k=1), A319866 (k=2), A319867 (k=3), A319868 (k=4), A319869 (k=5), A319870 (k=6), A319871 (k=7), this sequence (k=8), A319873 (k=9), A319874 (k=10).

Programs

Maple

a:=(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((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

Mathematica

k:=8; a[n_]:=Sum[(Floor[(n-j)/k]-Floor[(n-j-1)/k])* Product[n-i-j+k+1, {i,1,j }], {j,1,k-1} ]  + Sum[(Floor[j/k]-Floor[(j-1)/k])* Product[j-i+1, {i,1,k} ], {j,1,n}]; Array[a, 50] (* Stefano Spezia, Sep 30 2018 *)