cp's OEIS Frontend

A369335 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.

%I A369335 #29 Jan 25 2024 18:47:58
%S A369335 1,10,94,816,6872,57038,469238,3836430,31221874,253198806,2047761014,
%T A369335 16526046182,133145419056,1071276327274,8610169465642,69143029079432,
%U A369335 554860626424744,4450160058135914,35675446422203960,285892025190834636,2290356743575612582
%N A369335 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 A369335 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 A369335 Alois P. Heinz, <a href="/A369335/b369335.txt">Table of n, a(n) for n = 0..1106</a>
%H A369335 Nicolay Avilov, <a href="/A369335/a369335_2.jpg">Illustration for terms a(0) - a(4)</a>.
%F A369335 Limit_{n->oo} a(n+1)/a(n) = 8.
%e A369335 a(0) = 1.
%e A369335 a(1) = 1 + 1 + 2 + 3 + 3 = 10.
%e A369335             -----------
%e A369335            | 1   2   1 |
%e A369335            |   3   3   |
%e A369335 a(2) = Sum | 4   6   4 | = 94.
%e A369335            |   10  10  |
%e A369335            | 15  20  15|
%e A369335             -----------
%t A369335 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 A369335 Cf. A007318, A030662.
%K A369335 nonn
%O A369335 0,2
%A A369335 _Nicolay Avilov_, Jan 20 2024
%E A369335 a(6)-a(20) from _Alois P. Heinz_, Jan 20 2024