A140978 Repeat (n+1)^2 n times.

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

Offset: 1

Author

Paul Curtz, Aug 17 2008

Keywords

• nonn
• easy
• tabl

Comments

See A093995.

Frenicle writes the entries in the form a(n) = A055096(n)-A133819(n), with the flattened index view of A133819: 4=5-1, 9=10-1, 9=13-4, 16=17-1, 16=20-4, 16=25-9 etc.

Also triangle T(n, k) = (n+1)^2, 1<=k<=n. - Michel Marcus, Feb 03 2013

Links

• Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
• M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, in: Divers ouvrages des mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, table page 11.

Crossrefs

Cf. A000290.

Programs

Haskell

a140978 n k = a140978_tabl !! (n-1) !! (k-1)
a140978_row n = a140978_tabl !! (n-1)
a140978_tabl = map snd $ iterate
               (\(i, xs@(x:_)) -> (i + 2, map (+ i) (x:xs))) (5, [4])
-- Reinhard Zumkeller, Mar 23 2013

Mathematica

Table[PadRight[{},n,(n+1)^2],{n,10}]//Flatten (* Harvey P. Dale, Oct 10 2019 *)

Python

from math import isqrt
def A140978(n): return ((m:=isqrt(k:=n<<1))+(k>m*(m+1))+1)**2 # Chai Wah Wu, Nov 07 2024

Formula

a(n)=(A003057(n+1))^2. - R. J. Mathar, Aug 25 2008