|
%I #29 Jan 25 2024 18:47:58
%S 1,10,94,816,6872,57038,469238,3836430,31221874,253198806,2047761014,
%T 16526046182,133145419056,1071276327274,8610169465642,69143029079432,
%U 554860626424744,4450160058135914,35675446422203960,285892025190834636,2290356743575612582
%N a(n) is the sum of the numbers inside a square of side length 2n+1 located in Pascal's triangle at and below row n.
%C We will assume that the numbers of Pascal's triangle are written in the cells of a square lattice. Then row n has width 2n+1 and the square of cells starts there.
%H Alois P. Heinz, <a href="/A369335/b369335.txt">Table of n, a(n) for n = 0..1106</a>
%H Nicolay Avilov, <a href="/A369335/a369335_2.jpg">Illustration for terms a(0) - a(4)</a>.
%F Limit_{n->oo} a(n+1)/a(n) = 8.
%e a(0) = 1.
%e a(1) = 1 + 1 + 2 + 3 + 3 = 10.
%e -----------
%e | 1 2 1 |
%e | 3 3 |
%e a(2) = Sum | 4 6 4 | = 94.
%e | 10 10 |
%e | 15 20 15|
%e -----------
%t a[n_]:=Sum[Sum[Binomial[i,k],{k,Floor[(i+1-n)/2],Floor[(i+1-n)/2]+n-Mod[i-n,2]}],{i,n,3n}]; Array[a,21,0] (* _Stefano Spezia_, Jan 21 2024 *)
%Y Cf. A007318, A030662.
%K nonn
%O 0,2
%A _Nicolay Avilov_, Jan 20 2024
%E a(6)-a(20) from _Alois P. Heinz_, Jan 20 2024
|
