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
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
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