A098359 Multiplication table of the square numbers read by antidiagonals.

1, 4, 4, 9, 16, 9, 16, 36, 36, 16, 25, 64, 81, 64, 25, 36, 100, 144, 144, 100, 36, 49, 144, 225, 256, 225, 144, 49, 64, 196, 324, 400, 400, 324, 196, 64, 81, 256, 441, 576, 625, 576, 441, 256, 81, 100, 324, 576, 784, 900, 900, 784, 576, 324, 100, 121, 400, 729, 1024, 1225, 1296, 1225, 1024, 729, 400, 121

Offset

1,2

Comments

sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k-1) = n. See A003991. - Charlie Marion, Apr 22 2013

Links

Table of n, a(n) for n=1..66.

Formula

A(n,k) = n^2*k^2.

G.f.: [xy(1+x)(1+y)] / [(1-x)^3 * (1-y)^3 ]. - Ralf Stephan, Oct 27 2004

Sum_{j=1..n} A(j,1+n-j)*j = A213547(n). - Alois P. Heinz, May 19 2025

Example

Square array A(n,k) begins:
   1,   4,   9,  16,   25,   36,   49, ...
   4,  16,  36,  64,  100,  144,  196, ...
   9,  36,  81, 144,  225,  324,  441, ...
  16,  64, 144, 256,  400,  576,  784, ...
  25, 100, 225, 400,  625,  900, 1225, ...
  36, 144, 324, 576,  900, 1296, 1764, ...
  49, 196, 441, 784, 1225, 1764, 2401, ...

Maple

A:= (n, k)-> (n*k)^2:
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, May 19 2025

Crossrefs

Cf. A000290, A003991, A098358, A098360, A098361, A213547.

Antidiagonal sums give A033455.

Main diagonal gives A000583.

Sequence in context: A284628 A262811 A294749 * A319435 A226096 A071567

Adjacent sequences: A098356 A098357 A098358 * A098360 A098361 A098362

Keyword

nonn,tabl

Author

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

Extensions

Offset corrected by Alois P. Heinz, May 19 2025

Status

approved