A093995 n^2 appears n times, triangle read by rows.

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

Offset

1,2

Comments

Row sums give A000578.

Triangle sums give A000537.

Links

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Formula

T(n, k) = n^2, 1<=k<=n.

a(n) = floor(sqrt(2*n - 1) + 1/2)^2. - Ridouane Oudra, Jun 18 2019

From G. C. Greubel, Dec 27 2021: (Start)

T(n, n-k) = T(n, k).

Sum_{k=1..floor(n/2)} T(n, k) = [n=1] + A265645(n+1).

Sum_{k=1..floor(n/2)} T(n-k, k) = (1/48)*n*(n-1)*(7*(2*n-1) + 3*(-1)^n).

T(2*n-1, n) = A016754(n).

T(2*n, n) = A016742(n). (End)

Example

First few rows of the triangle are:
   1;
   4,  4;
   9,  9,  9;
  16, 16, 16, 16;
  25, 25, 25, 25, 25;
  36, 36, 36, 36, 36, 36;
  49, 49, 49, 49, 49, 49, 49;
  ...

Maple

seq(seq(n^2, i=1..n), n=1..20); # Ridouane Oudra, Jun 18 2019

Mathematica

Flatten[Table[Table[n^2, {n}], {n, 11}]]  (* Harvey P. Dale, Feb 18 2011 *)
Table[PadRight[{}, n, n^2], {n, 12}]//Flatten (* Harvey P. Dale, Jun 28 2023 *)

Prog

(Haskell)
a093995 n k = a093995_tabl !! (n-1) !! (k-1)
a093995_row n = a093995_tabl !! (n-1)
a093995_tabl = zipWith replicate [1..] $ tail a000290_list
a093995_list = concat a093995_tabl
-- Reinhard Zumkeller, Nov 11 2012, Mar 18 2011, Oct 17 2010
(Magma) [n^2: k in [1..n], n in [1..13]]; // G. C. Greubel, Dec 27 2021
(Sage) flatten([[n^2 for k in (1..n)] for n in (1..13)]) # G. C. Greubel, Dec 27 2021
(Python)
from math import isqrt
def A093995(n): return ((m:=isqrt(k:=n<<1))+(k>m*(m+1)))**2 # Chai Wah Wu, Nov 07 2024

Crossrefs

Cf. A000290, A000537, A000578, A016742, A016754, A127733, A199332, A265645.

Sequence in context: A016713 A343620 A168038 * A168280 A162564 A014694

Adjacent sequences: A093992 A093993 A093994 * A093996 A093997 A093998

Keyword

nonn,tabl

Author

Reinhard Zumkeller, May 24 2004

Extensions

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

Definition clarified by N. J. A. Sloane, Nov 09 2024

Status

approved