A055461 Square decrescendo subsequences: triangle T(n,k) = (n-k)^2, n >= 1, 0 <= k < n.

1, 4, 1, 9, 4, 1, 16, 9, 4, 1, 25, 16, 9, 4, 1, 36, 25, 16, 9, 4, 1, 49, 36, 25, 16, 9, 4, 1, 64, 49, 36, 25, 16, 9, 4, 1, 81, 64, 49, 36, 25, 16, 9, 4, 1, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1, 144, 121, 100, 81, 64, 49, 36, 25, 16, 9, 4, 1

Offset

1,2

Links

Robert Israel, Rows n = 1..141, flattened

Formula

a(n) = A004736(n)^2.

Sum_{k=0..n-1} T(n, k) = A000330(n) (row sums). - Michel Marcus, Dec 31 2012

G.f. as triangle: x*(1+x)/((1-x*y)*(1-x)^3). - Robert Israel, Jan 18 2018

Sum_{k=0..n-1} (-1)^k*T(n, k) = A000217(n) (alternating row sums). - Omar E. Pol, Jan 24 2014

From G. C. Greubel, Jan 31 2024: (Start)

T(2*n-1, n-1) = A000290(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A000292(n).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A194274(n).

Sum_{k=0..floor(n/2)} T(n, k) = A129371(n). (End)

Example

From Omar E. Pol, Jan 26 2014: (Start)
Triangle begins:
    1;
    4,  1;
    9,  4,  1;
   16,  9,  4,  1;
   25, 16,  9,  4,  1;
   36, 25, 16,  9,  4,  1;
   49, 36, 25, 16,  9,  4,  1;
   64, 49, 36, 25, 16,  9,  4,  1;
   81, 64, 49, 36, 25, 16,  9,  4,  1;
  100, 81, 64, 49, 36, 25, 16,  9,  4,  1;
  ...
For n = 7 the row sum is 49 + 36 + 25 + 16 + 9 + 4 + 1 = A000330(7) = 140.
The alternating row sum is 49 - 36 + 25 - 16 + 9 - 4 + 1 = A000217(7) = 28.
(End)

Maple

for n from 1 to 10 do
  seq((n-k)^2, k=0..n-1)
od; # Robert Israel, Jan 18 2018

Mathematica

Table[Range[n, 1, -1]^2, {n, 20}]//Flatten (* Harvey P. Dale, Apr 17 2020 *)

Prog

(Magma) [(n-k)^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Jan 31 2024
(SageMath) flatten([[(n-k)^2 for k in range(n)] for n in range(1, 16)]) # G. C. Greubel, Jan 31 2024

Crossrefs

Cf. A000290, A000292, A004736, A129371, A194274.

Cf. A000217 (alternating row sums), A000330 (row sums).

Sequence in context: A331145 A339635 A085691 * A324999 A104796 A132020

Adjacent sequences: A055458 A055459 A055460 * A055462 A055463 A055464

Keyword

easy,nonn,tabl

Author

Henry Bottomley, Jun 26 2000

Status

approved