OFFSET
1,2
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.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
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).
EXAMPLE
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.
MATHEMATICA
Table[Total[Times@@@Partition[Range[n], UpTo[7]]], {n, 30}] (* Harvey P. Dale, Aug 02 2020 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Sep 13 2018
STATUS
approved