%I #113 Nov 09 2024 18:08:33

%S 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,

%T 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,

%U 9,7,5,3,1,1,3,5,7,9,11,12,10,8,6,4,2,0,2,4,6,8,10,12

%N Table T(n,k) = |n-k| read by antidiagonals (n >= 0, k >= 0).

%C Commutative non-associative operator with identity 0. T(nx,kx) = x T(n,k). A multiplicative analog is A089913. - _Marc LeBrun_, Nov 14 2003

%C For the characteristic polynomial of the n X n matrix M_n with entries M_n(i, j) = |i-j| see A203993. - _Wolfdieter Lang_, Feb 04 2018

%C For the determinant of the n X n matrix M_n with entries M_n(i, j) = |i-j| see A085750. - _Bernard Schott_, May 13 2020

%C a(n) = 0 iff n = 4 times triangular number (A046092). - _Bernard Schott_, May 13 2020

%H Peter Kagey, <a href="/A049581/b049581.txt">Rows n = 0..125 of triangle, flattened</a>

%F G.f.: (x + y - 4xy + x^2y + xy^2)/((1-x)^2 (1-y)^2) (1-xy)) = (x/(1-x)^2 + y/(1-y)^2)/(1-xy). T(n,0) = T(0,n) = n; T(n+1,k+1) = T(n,k). - _Franklin T. Adams-Watters_, Feb 06 2006

%F 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

%F From _Robert Israel_, Sep 30 2015: (Start)

%F If b(n) = a(n+1) - 2*a(n) + a(n-1), then for n >= 3 we have

%F b(n) = -1 if n = (j^2+5j+4)/2 for some integer j >= 1

%F b(n) = -3 if n = (j^2+5j+6)/2 for some integer j >= 0

%F b(n) = 4 if n = 2j^2 + 6j + 4 for some integer j >= 0

%F b(n) = 2 if n = 2j^2 + 8j + 7 or 2j^2 + 8j + 8 for some integer j >= 0

%F b(n) = 0 otherwise. (End)

%F Triangle t(n,k) = max(k, n-k) - min(k, n-k). - _Peter Luschny_, Jan 26 2018

%F 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

%F a(n) = |t^2 - 2*n - 1|, where t = floor(sqrt(2*n+1) + 1/2). - _Ridouane Oudra_, Jun 07 2019; Dec 11 2020

%F As a rectangle, T(n,k) = |n-k| = max(n,k) - min(n,k). - _Clark Kimberling_, May 11 2020

%e Displayed as a triangle t(n, k):

%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...

%e 0: 0

%e 1: 1 1

%e 2: 2 0 2

%e 3: 3 1 1 3

%e 4: 4 2 0 2 4

%e 5: 5 3 1 1 3 5

%e 6: 6 4 2 0 2 4 6

%e 7: 7 5 3 1 1 3 5 7

%e 8: 8 6 4 2 0 2 4 6 8

%e 9: 9 7 5 3 1 1 3 5 7 9

%e 10: 10 8 6 4 2 0 2 4 6 8 10

%e ... reformatted by _Wolfdieter Lang_, Feb 04 2018

%e Displayed as a table:

%e 0 1 2 3 4 5 6 ...

%e 1 0 1 2 3 4 5 ...

%e 2 1 0 1 2 3 4 ...

%e 3 2 1 0 1 2 3 ...

%e 4 3 2 1 0 1 2 ...

%e 5 4 3 2 1 0 1 ...

%e 6 5 4 3 2 1 0 ...

%e ...

%p seq(seq(abs(n-2*k),k=0..n),n=0..12); # _Robert Israel_, Sep 30 2015

%t Table[Abs[(n-k) -k], {n,0,12}, {k,0,n}]//Flatten (* _Michael De Vlieger_, Sep 29 2015 *)

%t Table[Join[Range[n,0,-2],Range[If[EvenQ[n],2,1],n,2]],{n,0,12}]//Flatten (* _Harvey P. Dale_, Sep 18 2023 *)

%o (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);

%o vector(100, n , a(n-1)) \\ _Altug Alkan_, Sep 29 2015

%o (PARI) {t(n,k) = abs(n-2*k)}; \\ _G. C. Greubel_, Jun 07 2019

%o (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

%o (Magma) [[Abs(n-2*k): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Jun 07 2019

%o (Sage) [[abs(n-2*k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jun 07 2019

%o (Python)

%o from math import isqrt

%o def A049581(n): return abs((k:=n+1<<1)-((m:=isqrt(k))+(k>m*(m+1)))**2-1) # _Chai Wah Wu_, Nov 09 2024

%Y Cf. A003989, A003990, A003991, A003056, A004247, A002260, A004736.

%Y Cf. A089913. Apart from signs, same as A114327. A203993.

%K nonn,tabl,easy,nice

%O 0,4

%A _N. J. A. Sloane_