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

0,2

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1106

Nicolay Avilov, Illustration for terms a(0) - a(4).

Formula

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

Example

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|
            -----------

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 *)

Crossrefs

Cf. A007318, A030662.

Sequence in context: A192903 A050793 A281203 * A126633 A125422 A190988

Adjacent sequences: A369332 A369333 A369334 * A369336 A369337 A369338

Keyword

nonn

Author

Nicolay Avilov, Jan 20 2024

Extensions

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

Status

approved