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.

1, 10, 94, 816, 6872, 57038, 469238, 3836430, 31221874, 253198806, 2047761014, 16526046182, 133145419056, 1071276327274, 8610169465642, 69143029079432, 554860626424744, 4450160058135914, 35675446422203960, 285892025190834636, 2290356743575612582

Offset: 0

Author

Nicolay Avilov, Jan 20 2024

Keywords

• nonn

Comments

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.

Examples

			a(0) = 1.
a(1) = 1 + 1 + 2 + 3 + 3 = 10.
            -----------
           | 1   2   1 |
           |   3   3   |
a(2) = Sum | 4   6   4 | = 94.
           |   10  10  |
           | 15  20  15|
            -----------
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1106
• Nicolay Avilov, Illustration for terms a(0) - a(4).

Crossrefs

Cf. A007318, A030662.

Programs

Mathematica

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 *)

Formula

Limit_{n->oo} a(n+1)/a(n) = 8.

Extensions

a(6)-a(20) from Alois P. Heinz, Jan 20 2024