This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A049581 #120 Apr 05 2025 01:20:23
%S A049581 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 A049581 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 A049581 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 A049581 Table T(n,k) = |n-k| read by antidiagonals (n >= 0, k >= 0).
%C A049581 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 A049581 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 A049581 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 A049581 a(n) = 0 iff n = 4 times triangular number (A046092). - _Bernard Schott_, May 13 2020
%H A049581 Peter Kagey, <a href="/A049581/b049581.txt">Rows n = 0..125 of triangle, flattened</a>
%F A049581 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
%F A049581 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 A049581 From _Robert Israel_, Sep 30 2015: (Start)
%F A049581 If b(n) = a(n+1) - 2*a(n) + a(n-1), then for n >= 3 we have
%F A049581 b(n) = -1 if n = (j^2+5j+4)/2 for some integer j >= 1
%F A049581 b(n) = -3 if n = (j^2+5j+6)/2 for some integer j >= 0
%F A049581 b(n) = 4 if n = 2j^2 + 6j + 4 for some integer j >= 0
%F A049581 b(n) = 2 if n = 2j^2 + 8j + 7 or 2j^2 + 8j + 8 for some integer j >= 0
%F A049581 b(n) = 0 otherwise. (End)
%F A049581 Triangle t(n,k) = max(k, n-k) - min(k, n-k). - _Peter Luschny_, Jan 26 2018
%F A049581 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 A049581 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 A049581 As a rectangle, T(n,k) = |n-k| = max(n,k) - min(n,k). - _Clark Kimberling_, May 11 2020
%e A049581 Displayed as a triangle t(n, k):
%e A049581 n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e A049581 0: 0
%e A049581 1: 1 1
%e A049581 2: 2 0 2
%e A049581 3: 3 1 1 3
%e A049581 4: 4 2 0 2 4
%e A049581 5: 5 3 1 1 3 5
%e A049581 6: 6 4 2 0 2 4 6
%e A049581 7: 7 5 3 1 1 3 5 7
%e A049581 8: 8 6 4 2 0 2 4 6 8
%e A049581 9: 9 7 5 3 1 1 3 5 7 9
%e A049581 10: 10 8 6 4 2 0 2 4 6 8 10
%e A049581 ... reformatted by _Wolfdieter Lang_, Feb 04 2018
%e A049581 Displayed as a table:
%e A049581 0 1 2 3 4 5 6 ...
%e A049581 1 0 1 2 3 4 5 ...
%e A049581 2 1 0 1 2 3 4 ...
%e A049581 3 2 1 0 1 2 3 ...
%e A049581 4 3 2 1 0 1 2 ...
%e A049581 5 4 3 2 1 0 1 ...
%e A049581 6 5 4 3 2 1 0 ...
%e A049581 ...
%p A049581 seq(seq(abs(n-2*k),k=0..n),n=0..12); # _Robert Israel_, Sep 30 2015
%t A049581 Table[Abs[(n-k) -k], {n,0,12}, {k,0,n}]//Flatten (* _Michael De Vlieger_, Sep 29 2015 *)
%t A049581 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 A049581 (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 A049581 vector(100, n , a(n-1)) \\ _Altug Alkan_, Sep 29 2015
%o A049581 (PARI) {t(n,k) = abs(n-2*k)}; \\ _G. C. Greubel_, Jun 07 2019
%o A049581 (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 A049581 (Magma) [[Abs(n-2*k): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Jun 07 2019
%o A049581 (Sage) [[abs(n-2*k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jun 07 2019
%o A049581 (Python)
%o A049581 from math import isqrt
%o A049581 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 A049581 Cf. A003989, A003990, A003991, A003056, A004247, A002260, A004736.
%Y A049581 Cf. A089913. Apart from signs, same as A114327. A203993.
%K A049581 nonn,tabl,easy,nice
%O A049581 0,4
%A A049581 _N. J. A. Sloane_