A319873 - OEIS

%I #19 Oct 18 2018 10:54:40

%S 9,72,504,3024,15120,60480,181440,362880,362880,362898,363186,367776,

%T 436320,1391040,13728960,160755840,1764685440,17643588480,17643588507,

%U 17643589182,17643606030,17644009680,17653276080,17856715680,22119259680,107157012480

%N a(n) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11*10 + ... + (up to the n-th term).

%C 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=9.

%H Colin Barker, <a href="/A319873/b319873.txt">Table of n, a(n) for n = 1..1000</a>

%e a(1) = 9;

%e a(2) = 9*8 = 72;

%e a(3) = 9*8*7 = 504;

%e a(4) = 9*8*7*6 = 3024;

%e a(5) = 9*8*7*6*5 = 15120;

%e a(6) = 9*8*7*6*5*4 = 60480;

%e a(7) = 9*8*7*6*5*4*3 = 181440;

%e a(8) = 9*8*7*6*5*4*3*2 = 362880;

%e a(9) = 9*8*7*6*5*4*3*2*1 = 362880;

%e a(10) = 9*8*7*6*5*4*3*2*1 + 18 = 362898;

%e a(11) = 9*8*7*6*5*4*3*2*1 + 18*17 = 363186;

%e a(12) = 9*8*7*6*5*4*3*2*1 + 18*17*16 = 367776

%e a(13) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15 = 436320;

%e a(14) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14 = 1391040;

%e a(15) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13 = 13728960;

%e a(16) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12 = 160755840;

%e a(17) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11 = 1764685440;

%e a(18) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11*10 = 17643588480;

%e a(19) = 9*8*7*6*5*4*3*2*1 + 18*17*16*15*14*13*12*11*10 + 27 = 17643588507;

%e etc.

%p 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,9),n=1..30); # _Muniru A Asiru_, Sep 30 2018

%t k:=9; 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 *)

%Y 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), A319872 (k=8), this sequence (k=9), A319874 (k=10).

%K nonn,easy

%O 1,1

%A _Wesley Ivan Hurt_, Sep 30 2018