A157454 Triangle read by rows: T(n, m) = min(2*m - 1, 2*(n - m) + 1).
1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 7, 5, 3, 1, 1, 3, 5, 7, 7, 5, 3, 1, 1, 3, 5, 7, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 9, 7, 5, 3, 1
Offset: 1
Examples
Triangle starts: 1; 1, 1; 1, 3, 1; 1, 3, 3, 1; 1, 3, 5, 3, 1; 1, 3, 5, 5, 3, 1; 1, 3, 5, 7, 5, 3, 1; 1, 3, 5, 7, 7, 5, 3, 1; 1, 3, 5, 7, 9, 7, 5, 3, 1; 1, 3, 5, 7, 9, 9, 7, 5, 3, 1; 1, 3, 5, 7, 9, 11, 9, 7, 5, 3, 1; ...
Links
- Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Programs
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GAP
Flat(List([1..12], n-> List([1..n], k-> Minimum(2*k-1, 2*(n-k)+1) ))); # G. C. Greubel, Jun 30 2019
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Haskell
import Data.List (inits) a157454 n k = a157454_tabl !! (n-1) !! (k-1) a157454_row n = a157454_tabl !! (n-1) a157454_tabl = concatMap h $ tail $ inits [1, 3 ..] where h xs = [xs ++ tail xs', xs ++ xs'] where xs' = reverse xs -- Reinhard Zumkeller, Dec 15 2013
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Magma
[[Min(2*k-1, 2*(n-k)+1): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jun 30 2019
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Mathematica
Table[Min[2*k-1, 2*(n-k)+1], {n,1,12}, {k,1,n}]//Flatten (* modified by G. C. Greubel, Jun 30 2019 *) -
PARI
{T(n,k) = min(2*k-1, 2*(n-k)+1)}; for(n=1,12, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jun 30 2019 -
Python
from math import isqrt def A157454(n): return (isqrt(n<<3)+1>>1)-abs((k:=n<<1)-((m:=isqrt(k))+(k>m*(m+1)))**2-1) # Chai Wah Wu, Jun 08 2025
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Sage
[[min(2*k-1, 2*(n-k)+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jun 30 2019
Formula
T(n,m) = T(n,n-m) = 2*m-1 for 0 <= m <= n/2, otherwise 2*(n-m)+1.
a(n) = 2*A003983(n) - 1.
From Ridouane Oudra, Jul 20 2019: (Start)
a(n) = t - abs(t^2-2n+1), where t = floor(sqrt(2n)+1/2). (End)
Extensions
Edited by the Associate Editors of the OEIS, Apr 10 2009