A124258 Triangle whose rows are sequences of increasing and decreasing squares: 1; 1,4,1; 1,4,9,4,1; ...

1, 1, 4, 1, 1, 4, 9, 4, 1, 1, 4, 9, 16, 9, 4, 1, 1, 4, 9, 16, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 64, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16, 25, 36, 49, 64, 81, 64, 49, 36, 25, 16, 9, 4, 1, 1, 4, 9, 16

Offset

1,3

Comments

The triangle A003983 with individual entries squared and each 2nd row skipped.

Analogous to A004737. - Peter Bala, Sep 25 2007

T(n,k) = min(n,k)^2. The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

Links

Table of n, a(n) for n=1..85.

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Formula

O.g.f.: (1+qx)^2/((1-x)(1-qx)^2(1-q^2x)) = 1 + x(1 + 4q + q^2) + x^2(1 + 4q + 9q^2 + 4q^3 + q^4) + ... . - Peter Bala, Sep 25 2007

From Boris Putievskiy, Jan 13 2013: (Start)

a(n) = (A004737(n))^2.

a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^2. (End)

Example

Triangle starts
  1;
  1, 4, 1;
  1, 4, 9, 4, 1:
  1, 4, 9, 16, 9, 4, 1:
From Boris Putievskiy, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1...1...
  1...4...4...4...4...4...
  1...4...9...9...9...9...
  1...4...9..16..16..16...
  1...4...9..16..25..25...
  1...4...9..16..25..36...
  ...
The start of the sequence as triangle array read by rows:
  1;
  1, 4, 1;
  1, 4, 9,  4,  1;
  1, 4, 9, 16,  9,  4,  1;
  1, 4, 9, 16, 25, 16,  9,  4, 1;
  1, 4, 9, 16, 25, 36, 25, 16, 9, 4, 1;
  ...
Row number k contains 2*k-1 numbers 1,4,...,(k-1)^2,k^2,(k-1)^2,...,4,1. (End)

Maple

A003983 := proc(n, k) min(n, k) ; end: A124258 := proc(n, k) A003983(n, k)^2 ; end: for d from 1 to 20 by 2 do for c from 1 to d do printf("%d, ", A124258(d+1-c, c)) ; od: od: # R. J. Mathar, Sep 21 2007
# second Maple program:
T:= n-> i^2$i=1..n, (n-i)^2$i=1..n-1:
seq(T(n), n=1..10);  # Alois P. Heinz, Feb 15 2022

Mathematica

Flatten[Table[Join[Range[n]^2, Range[n-1, 1, -1]^2], {n, 10}]] (* Harvey P. Dale, Jun 14 2015 *)

Crossrefs

Cf. A005900 (row sums), A004737, A133819, A133823, A133824, A133825, A003983.

Sequence in context: A246595 A209571 A269845 * A001638 A133826 A209565

Adjacent sequences: A124255 A124256 A124257 * A124259 A124260 A124261

Keyword

nonn,tabf,easy

Author

Jonathan Vos Post, Dec 16 2006

Extensions

More terms from R. J. Mathar, Sep 21 2007

Edited by N. J. A. Sloane, Jun 30 at the suggestion of R. J. Mathar

Status

approved