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
Examples
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; ...
Links
- Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Programs
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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
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Magma
[n^2: k in [1..n], n in [1..13]]; // G. C. Greubel, Dec 27 2021
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Maple
seq(seq(n^2, i=1..n), n=1..20); # Ridouane Oudra, Jun 18 2019
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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 *) -
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
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Sage
flatten([[n^2 for k in (1..n)] for n in (1..13)]) # G. C. Greubel, Dec 27 2021
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)
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
Comments