A049581 Table T(n,k) = |n-k| read by antidiagonals (n >= 0, k >= 0).
0, 1, 1, 2, 0, 2, 3, 1, 1, 3, 4, 2, 0, 2, 4, 5, 3, 1, 1, 3, 5, 6, 4, 2, 0, 2, 4, 6, 7, 5, 3, 1, 1, 3, 5, 7, 8, 6, 4, 2, 0, 2, 4, 6, 8, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 10, 8, 6, 4, 2, 0, 2, 4, 6, 8, 10, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 12, 10, 8, 6, 4, 2, 0, 2, 4, 6, 8, 10, 12
Offset: 0
Examples
Displayed as a triangle t(n, k): n\k 0 1 2 3 4 5 6 7 8 9 10 ... 0: 0 1: 1 1 2: 2 0 2 3: 3 1 1 3 4: 4 2 0 2 4 5: 5 3 1 1 3 5 6: 6 4 2 0 2 4 6 7: 7 5 3 1 1 3 5 7 8: 8 6 4 2 0 2 4 6 8 9: 9 7 5 3 1 1 3 5 7 9 10: 10 8 6 4 2 0 2 4 6 8 10 ... reformatted by _Wolfdieter Lang_, Feb 04 2018 Displayed as a table: 0 1 2 3 4 5 6 ... 1 0 1 2 3 4 5 ... 2 1 0 1 2 3 4 ... 3 2 1 0 1 2 3 ... 4 3 2 1 0 1 2 ... 5 4 3 2 1 0 1 ... 6 5 4 3 2 1 0 ... ...
Links
- Peter Kagey, Rows n = 0..125 of triangle, flattened
Crossrefs
Programs
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GAP
a := Flat(List([0..12],n->List([0..n],k->Maximum(k,n-k)-Minimum(k,n-k)))); # Muniru A Asiru, Jan 26 2018
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Magma
[[Abs(n-2*k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 07 2019
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Maple
seq(seq(abs(n-2*k),k=0..n),n=0..12); # Robert Israel, Sep 30 2015
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Mathematica
Table[Abs[(n-k) -k], {n,0,12}, {k,0,n}]//Flatten (* Michael De Vlieger, Sep 29 2015 *) Table[Join[Range[n,0,-2],Range[If[EvenQ[n],2,1],n,2]],{n,0,12}]//Flatten (* Harvey P. Dale, Sep 18 2023 *) -
PARI
a(n) = abs(2*(n+1)-binomial((sqrtint(8*(n+1))+1)\2, 2)-(binomial(1+floor(1/2 + sqrt(2*(n+1))), 2))-1); vector(100, n , a(n-1)) \\ Altug Alkan, Sep 29 2015
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PARI
{t(n,k) = abs(n-2*k)}; \\ G. C. Greubel, Jun 07 2019 -
Python
from math import isqrt def A049581(n): return abs((k:=n+1<<1)-((m:=isqrt(k))+(k>m*(m+1)))**2-1) # Chai Wah Wu, Nov 09 2024
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Sage
[[abs(n-2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 07 2019
Formula
G.f.: (x + y - 4*x*y + x^2*y + x*y^2)/((1-x)^2*(1-y)^2*(1-x*y)) = (x/(1-x)^2 + y/(1-y)^2)/(1-x*y). T(n,0) = T(0,n) = n; T(n+1,k+1) = T(n,k). - Franklin T. Adams-Watters, Feb 06 2006
a(n) = |A002260(n+1)-A004736(n+1)| or a(n) = |((n+1)-t*(t+1)/2) - ((t*t+3*t+4)/2-(n+1))| where t = floor((-1+sqrt(8*(n+1)-7))/2). - Boris Putievskiy, Dec 24 2012; corrected by Altug Alkan, Sep 30 2015
From Robert Israel, Sep 30 2015: (Start)
If b(n) = a(n+1) - 2*a(n) + a(n-1), then for n >= 3 we have
b(n) = -1 if n = (j^2+5j+4)/2 for some integer j >= 1
b(n) = -3 if n = (j^2+5j+6)/2 for some integer j >= 0
b(n) = 4 if n = 2j^2 + 6j + 4 for some integer j >= 0
b(n) = 2 if n = 2j^2 + 8j + 7 or 2j^2 + 8j + 8 for some integer j >= 0
b(n) = 0 otherwise. (End)
Triangle t(n,k) = max(k, n-k) - min(k, n-k). - Peter Luschny, Jan 26 2018
Triangle t(n, k) = |n - 2*k| for n >= 0, k = 0..n. See the Maple and Mathematica programs. Hence t(n, k)= t(n, n-k). - Wolfdieter Lang, Feb 04 2018
a(n) = |t^2 - 2*n - 1|, where t = floor(sqrt(2*n+1) + 1/2). - Ridouane Oudra, Jun 07 2019; Dec 11 2020
As a rectangle, T(n,k) = |n-k| = max(n,k) - min(n,k). - Clark Kimberling, May 11 2020
Comments