A319871 a(n) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + ... + (up to the n-th term).

7, 42, 210, 840, 2520, 5040, 5040, 5054, 5222, 7224, 29064, 245280, 2167200, 17302320, 17302341, 17302740, 17310300, 17445960, 19744200, 56372400, 603353520, 603353548, 603354276, 603373176, 603844920, 615147120, 874606320, 6570915120, 6570915155, 6570916310

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=7.

Examples

			a(1) = 7;
a(2) = 7*6 = 42;
a(3) = 7*6*5 = 210;
a(4) = 7*6*5*4 = 840;
a(5) = 7*6*5*4*3 = 2520;
a(6) = 7*6*5*4*3*2 = 5040;
a(7) = 7*6*5*4*3*2*1 = 5040;
a(8) = 7*6*5*4*3*2*1 + 14 = 5054;
a(9) = 7*6*5*4*3*2*1 + 14*13 = 5222;
a(10) = 7*6*5*4*3*2*1 + 14*13*12 = 7224;
a(11) = 7*6*5*4*3*2*1 + 14*13*12*11 = 29064;
a(12) = 7*6*5*4*3*2*1 + 14*13*12*11*10 = 245280;
a(13) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9 = 2167200;
a(14) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 = 17302320;
a(15) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21 = 17302341;
a(16) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21*20 = 17302740;
a(17) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21*20*19 = 17310300;
a(18) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21*20*19*18 = 17445960;
a(19) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21*20*19*18*17 = 19744200;
a(20) = 7*6*5*4*3*2*1 + 14*13*12*11*10*9*8 + 21*20*19*18*17*16 = 56372400;
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), this sequence (k=7), A319872 (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,7),n=1..30); # Muniru A Asiru, Sep 30 2018

Mathematica

k:=7; 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 *)