A319208 a(n) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 + 15*16*17*18*19*20*21 + ... + (up to n).

1, 2, 6, 24, 120, 720, 5040, 5048, 5112, 5760, 12960, 100080, 1240560, 17302320, 17302335, 17302560, 17306400, 17375760, 18697680, 45209520, 603353520, 603353542, 603354026, 603365664, 603657120, 611247120, 816480720, 6570915120, 6570915149, 6570915990

Offset: 1

Author

Wesley Ivan Hurt, Sep 13 2018

Keywords

• nonn
• easy

Comments

In general, for sequences that multiply the first k natural numbers, and then add the product of the next k natural numbers (preserving the order of operations up to n), we have a(n) = Sum_{i=1..floor(n/k)} (k*i)!/(k*i-k)! + Sum_{j=1..k-1} (1-sign((n-j) mod k)) * (Product_{i=1..j} n-i+1). Here, k=7.

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 = 5040;
a(8) = 1*2*3*4*5*6*7 + 8 = 5048;
a(9) = 1*2*3*4*5*6*7 + 8*9 = 5112;
a(10) = 1*2*3*4*5*6*7 + 8*9*10 = 5760;
a(11) = 1*2*3*4*5*6*7 + 8*9*10*11 = 12960;
a(12) = 1*2*3*4*5*6*7 + 8*9*10*11*12 = 100080;
a(13) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13 = 1240560;
a(14) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 = 17302320;
a(15) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 + 15 = 17302335;
a(16) = 1*2*3*4*5*6*7 + 8*9*10*11*12*13*14 + 15*16 = 17302560; etc.
		

Links

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

Crossrefs

Cf. A093361, (k=1) A000217, (k=2) A228958, (k=3) A319014, (k=4) A319205, (k=5) A319206, (k=6) A319207, (k=7) this sequence, (k=8) A319209, (k=9) A319211, (k=10) A319212.

Programs

Mathematica

Table[Total[Times@@@Partition[Range[n],UpTo[7]]],{n,30}] (* Harvey P. Dale, Aug 02 2020 *)

Formula

a(n) = Sum_{i=1..floor(n/7)} (7*i)!/(7*i-7)! + Sum_{j=1..6} (1-sign((n-j) mod 7)) * (Product_{i=1..j} n-i+1).