A049581 -id:A049581 - cp's OEIS frontend

A344839 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = abs(n * 2^max(0, w(k)-w(n)) - k * 2^max(0, w(n)-w(k))) (where w = A070939).

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

Offset: 0

Rémy Sigrist, May 29 2021

In other words, we right pad the binary expansion of the lesser of n and k with zeros (provided it is positive) so that both numbers have the same number of binary digits, and then take the absolute difference.

			Array T(n, k) begins:
  n\k|   0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
  ---+-------------------------------------------------------
    0|   0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
    1|   1  0  0  1  0  1  2  3  0  1   2   3   4   5   6   7
    2|   2  0  0  1  0  1  2  3  0  1   2   3   4   5   6   7
    3|   3  1  1  0  2  1  0  1  4  3   2   1   0   1   2   3
    4|   4  0  0  2  0  1  2  3  0  1   2   3   4   5   6   7
    5|   5  1  1  1  1  0  1  2  2  1   0   1   2   3   4   5
    6|   6  2  2  0  2  1  0  1  4  3   2   1   0   1   2   3
    7|   7  3  3  1  3  2  1  0  6  5   4   3   2   1   0   1
    8|   8  0  0  4  0  2  4  6  0  1   2   3   4   5   6   7
    9|   9  1  1  3  1  1  3  5  1  0   1   2   3   4   5   6
   10|  10  2  2  2  2  0  2  4  2  1   0   1   2   3   4   5
   11|  11  3  3  1  3  1  1  3  3  2   1   0   1   2   3   4
   12|  12  4  4  0  4  2  0  2  4  3   2   1   0   1   2   3
   13|  13  5  5  1  5  3  1  1  5  4   3   2   1   0   1   2
   14|  14  6  6  2  6  4  2  0  6  5   4   3   2   1   0   1
   15|  15  7  7  3  7  5  3  1  7  6   5   4   3   2   1   0

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10

Cf. A049581, A053645, A070939.

Cf. A344834 (AND), A344835 (OR), A344836 (XOR), A344837 (min), A344838 (max).

T(n,k,op=(x,y)->abs(x-y),w=m->#binary(m)) = { op(n*2^max(0, w(k)-w(n)), k*2^max(0, w(n)-w(k))) }

T(n, k) = T(k, n).
T(n, n) = 0.
T(n, 0) = n.
T(n, 1) = A053645(n) for any n > 0.

A330240 Square array T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros), read by antidiagonals, n, k >= 0.

Original entry on oeis.org

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, 11, 7, 5, 3, 1, 1, 3, 5, 7, 11, 11, 12, 10, 12, 6, 4, 2, 0, 2, 4, 6, 12, 10, 12, 13, 11, 11, 13, 5, 3, 1, 1, 3, 5, 13, 11, 11, 13, 14, 12, 10, 12, 14, 4, 2, 0, 2, 4, 14, 12, 10, 12, 14

Offset: 0

M. F. Hasler, Dec 06 2019

A digit-wise analog of A049581. Referred to as "box" operation by Eric Angelini.
The binary operator T: N x N -> N is commutative, so this table is symmetric: it does not matter in which direction the antidiagonals are read, and it would be sufficient to specify only the lower half of the square table: see A330238 for this triangle. Zero is the neutral element: T(x,0) = x for all x. Any x is its own inverse or opposite x', as shown by the zero diagonal T(x,x) = 0.
A measure of non-associativity is the "commutator" ((x T y) T x') T y' = ((x T y) T x) T y which would be zero in the associative case, given that x = x' for all x. Here it turns out to be given by 2*A053616, read as a triangle, and rows extended quasi-periodically with period 10, see example.

			The square array starts as follows:
   n |k=0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
  ---+-------------------------------------------------------------
   0 |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
   1 |  1  0  1  2  3  4  5  6  7  8 11 10 11 12 13 14 15 16 17 ...
   2 |  2  1  0  1  2  3  4  5  6  7 12 11 10 11 12 13 14 15 16 ...
   3 |  3  2  1  0  1  2  3  4  5  6 13 12 11 10 11 12 13 14 15 ...
   4 |  4  3  2  1  0  1  2  3  4  5 14 13 12 11 10 11 12 13 14 ...
   5 |  5  4  3  2  1  0  1  2  3  4 15 14 13 12 11 10 11 12 13 ...
   6 |  6  5  4  3  2  1  0  1  2  3 16 15 14 13 12 11 10 11 12 ...
   7 |  7  6  5  4  3  2  1  0  1  2 17 16 15 14 13 12 11 10 11 ...
   8 |  8  7  6  5  4  3  2  1  0  1 18 17 16 15 14 13 12 11 10 ...
   9 |  9  8  7  6  5  4  3  2  1  0 19 18 17 16 15 14 13 12 11 ...
  10 | 10 11 12 13 14 15 16 17 18 19  0  1  2  3  4  5  6  7  8 ...
  11 | 11 10 11 12 13 14 15 16 17 18  1  0  1  2  3  4  5  6  7 ...
  12 | 12 11 10 11 12 13 14 15 16 17  2  1  0  1  2  3  4  5  6 ...
   (...)
It differs from A049581 only if at least one index is > 9.
The table of commutators Comm(n,k) := T(T(T(n,k),n),k) reads as follows:
   n |k=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22...
  ---+---------------------------------------------------------------
   0 |  0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0...
   1 |  0 0 2 2 2 2 2 2 2 2  0  0  2  2  2  2  2  2  2  2  0  0  2...
   2 |  0 0 0 2 4 4 4 4 4 4  0  0  0  2  4  4  4  4  4  4  0  0  0...
   3 |  0 0 0 0 2 4 6 6 6 6  0  0  0  0  2  4  6  6  6  6  0  0  0...
   4 |  0 0 0 0 0 2 4 6 8 8  0  0  0  0  0  2  4  6  8  8  0  0  0...
   5 |  0 0 0 0 0 0 2 4 6 8  0  0  0  0  0  0  2  4  6  8  0  0  0...
   6 |  0 0 0 0 0 0 0 2 4 6  0  0  0  0  0  0  0  2  4  6  0  0  0...
   7 |  0 0 0 0 0 0 0 0 2 4  0  0  0  0  0  0  0  0  2  4  0  0  0...
   8 |  0 0 0 0 0 0 0 0 0 2  0  0  0  0  0  0  0  0  0  2  0  0  0...
   9 |  0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0...
  10 |  0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0 20 20 20...
  11 |  0 0 2 2 2 2 2 2 2 2  0  0  2  2  2  2  2  2  2  2 20 20 22...
  12 |  0 0 0 2 4 4 4 4 4 4  0  0  0  2  4  4  4  4  4  4 20 20 20...
   (...)
Up to row & column 10, the columns are twice the sequence A053616 written as triangle. The first 10 X 10 block repeats horizontally and vertically. Further away from the origin, the elements of this block multiplied by corresponding powers of 10 are added to the corresponding 10 X 10 blocks: e.g., the block Comm(130..139,270..279) = Comm(0..9,0..9) + 260, where 260 = 100*Comm(1,2) + 10*Comm(3,7).

Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019.
Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019. [Cached copy]

Cf. A330238 (variant excluding row & column 0), A330237 (lower left triangle), A049581 (T(n,k) = |n-k|).

A330240(a,b)=fromdigits(abs(Vec(digits(min(a,b)),if(a+b,-logint(a=max(a,b),10)-1))-digits(a)))

A353452 a(n) is the determinant of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

Original entry on oeis.org

1, 0, -1, 0, 1, -4, 12, 64, -172, -1348, 3456, 34240, -87084, 370640, -872336, -22639616, 52307088, -181323568, 399580288, 23627011200, -51305628400, -686160247552, 1545932859328, 68098264912128, -155370174372864, 6326621032802304, -13829529077133312, -1087288396552040448

Offset: 0

Stefano Spezia, Apr 19 2022

sign

			a(8) = -172:
    0,  1,  0,  0,  0,  0,  0,  0;
    1,  0,  1,  2,  0,  0,  0,  0;
    0,  1,  0,  1,  2,  3,  0,  0;
    0,  2,  1,  0,  1,  2,  3,  4;
    0,  0,  2,  1,  0,  1,  2,  3;
    0,  0,  3,  2,  1,  0,  1,  2;
    0,  0,  0,  3,  2,  1,  0,  1;
    0,  0,  0,  4,  3,  2,  1,  0.

Cf. A085750, A085807.

Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353453 (permanent).

Join[{1},Table[Det[Table[If[Min[i,j]

a(n) = matdet(matrix(n, n, i, j, if ((min(i,j) < max(i,j)) && (max(i,j) <= 2*min(i,j)), abs(i-j)))); \\ Michel Marcus, Apr 20 2022

from sympy import Matrix
def A353452(n): return Matrix(n, n, lambda i, j: abs(i-j) if min(i,j)Chai Wah Wu, Aug 29 2023

Sum_{i=1..n+1-k} M[i,i+k] = A173997(n, k) with 1 <= k <= floor((n + 1)/2).
Sum_{i=1..n} Sum_{j=1..n} M[i,j] = 2*A006918(n-1).
Sum_{i=1..n} Sum_{j=1..n} M[i,j]^2 = A350050(n+1).

A353453 a(n) is the permanent of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 4, 64, 576, 7844, 63524, 882772, 11713408, 252996564, 5879980400, 184839020672, 5698866739200, 229815005974352, 9350598794677712, 480306381374466176, 23741710999960266176, 1446802666239931811472, 86153125248221968292928, 6197781268948296566634304

Offset: 0

Stefano Spezia, Apr 19 2022

nonn

			a(8) = 7844:
    0,  1,  0,  0,  0,  0,  0,  0;
    1,  0,  1,  2,  0,  0,  0,  0;
    0,  1,  0,  1,  2,  3,  0,  0;
    0,  2,  1,  0,  1,  2,  3,  4;
    0,  0,  2,  1,  0,  1,  2,  3;
    0,  0,  3,  2,  1,  0,  1,  2;
    0,  0,  0,  3,  2,  1,  0,  1;
    0,  0,  0,  4,  3,  2,  1,  0.

Cf. A085750, A085807.

Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353452 (determinant).

Join[{1},Table[Permanent[Table[If[Min[i,j]

a(n) = matpermanent(matrix(n, n, i, j, if ((min(i,j) < max(i,j)) && (max(i,j) <= 2*min(i,j)), abs(i-j)))); \\ Michel Marcus, Apr 20 2022

from sympy import Matrix
def A353453(n): return Matrix(n, n, lambda i, j: abs(i-j) if min(i,j)Chai Wah Wu, Aug 29 2023

Sum_{i=1..n+1-k} M[i,i+k] = A173997(n, k) with 1 <= k <= floor((n + 1)/2).
Sum_{i=1..n} Sum_{j=1..n} M[i,j] = 2*A006918(n-1).
Sum_{i=1..n} Sum_{j=1..n} M[i,j]^2 = A350050(n+1).

A330237 Square array T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros); read by antidiagonals; n, k >= 1.

Original entry on oeis.org

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, 11, 7, 5, 3, 1, 1, 3, 5, 7, 11, 10, 12, 6, 4, 2, 0, 2, 4, 6, 12, 10, 11, 11, 13, 5, 3, 1, 1, 3, 5, 13, 11, 11, 12, 10, 12, 14, 4, 2, 0, 2, 4, 14, 12, 10, 12, 13, 11, 11, 13, 15, 3, 1, 1, 3, 15, 13, 11, 11, 13, 14, 12, 10, 12, 14, 16, 2, 0, 2, 16

Offset: 1

M. F. Hasler, Dec 06 2019

A digit-wise analog of A049581.

The binary operator T: N x N -> N is commutative, therefore this table is symmetric and it does not matter in which direction the antidiagonals are read. It would also be sufficient to specify only the lower half of the square table: see A330238 for this variant. The operator is also defined for either argument equal to 0, which is the neutral element: T(x,0) = 0 for all x. Therefore we omit row & column 0 here, see A330240 for the table including these. Every element is its opposite or inverse, as shown by the zero diagonal T(x,x) = 0.

			The square array starts as follows:
   n | k=1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
  ---+-----------------------------------------------------------
   1 |   0  1  2  3  4  5  6  7  8 11 10 11 12 13 14 15 16 17 ...
   2 |   1  0  1  2  3  4  5  6  7 12 11 10 11 12 13 14 15 16 ...
   3 |   2  1  0  1  2  3  4  5  6 13 12 11 10 11 12 13 14 15 ...
   4 |   3  2  1  0  1  2  3  4  5 14 13 12 11 10 11 12 13 14 ...
   5 |   4  3  2  1  0  1  2  3  4 15 14 13 12 11 10 11 12 13 ...
   6 |   5  4  3  2  1  0  1  2  3 16 15 14 13 12 11 10 11 12 ...
   7 |   6  5  4  3  2  1  0  1  2 17 16 15 14 13 12 11 10 11 ...
   8 |   7  6  5  4  3  2  1  0  1 18 17 16 15 14 13 12 11 10 ...
   9 |   8  7  6  5  4  3  2  1  0 19 18 17 16 15 14 13 12 11 ...
  10 |  11 12 13 14 15 16 17 18 19  0  1  2  3  4  5  6  7  8 ...
  11 |  10 11 12 13 14 15 16 17 18  1  0  1  2  3  4  5  6  7 ...
  12 |  11 10 11 12 13 14 15 16 17  2  1  0  1  2  3  4  5  6 ...
   (...)
It differs from A049581 only if at least one index is > 10.

Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019.
Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019. [Cached copy]

Cf A330240 (variant including row & column 0), A330237 (lower left triangle), A049581 (T(n,k) = |n-k|).

T(a,b)=fromdigits(abs(Vec(digits(min(a,b)),-logint(a=max(a,b),10)-1)-digits(a)))

A330238 Triangle T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros); n >= k >= 1.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 0, 11, 10, 11, 12, 13, 14, 15, 16, 17, 2, 1, 0, 12, 11, 10, 11, 12, 13, 14, 15, 16, 3, 2, 1, 0, 13, 12, 11, 10, 11, 12, 13, 14, 15, 4, 3, 2, 1, 0

Offset: 1

M. F. Hasler, Dec 06 2019

A digit-wise analog of A049581.

The binary operator T: N x N -> N is commutative, so we need only the lower half of the symmetric square table A330238 or A330240 (including n, k = 0). Also, 0 is the neutral element: T(x,0) = x for all x, therefore we omit row & column 0. The trivial diagonal T(x,x) = 0 could also be omitted but serves as an end-of-row marker and makes indexing simpler and more natural.

			The triangle starts as follows:
    n | k=1  2   3   4   5   6   7   8   9  10  11
   ---+-------------------------------------------
    1 |  0,
    2 |  1,  0,
    3 |  2,  1,  0,
    4 |  3,  2,  1,  0,
    5 |  4,  3,  2,  1,  0,
    6 |  5,  4,  3,  2,  1,  0,
    7 |  6,  5,  4,  3,  2,  1,  0,
    8 |  7,  6,  5,  4,  3,  2,  1,  0,
    9 |  8,  7,  6,  5,  4,  3,  2,  1,  0,
   10 | 11, 12, 13, 14, 15, 16, 17, 18, 19,  0,
   11 | 10, 11, 12, 13, 14, 15, 16, 17, 18,  1,  0,
   12 | 11, 10, 11, 12, 13, 14, 15, 16, 17,  2,  1,  0,
    (...)

Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019.
Eric Angelini, The box ■ operation, personal blog "Cinquante signes", and post to the SeqFan list, Dec 06 2019. [Cached copy]

Cf. A330237 (same as a square array read by antidiagonals), A330240 (idem, including row & column 0), A049581 (T(n,k) = |n-k|).

A330238(n,k)=fromdigits(digits(n)-abs(Vec(digits(k),-logint(n,10)-1))) \\ see A330240 for a more general function not limited to 1 <= k <= n

A352967 Array read by antidiagonals: A(i, j) = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0, with i >= 0 and j >= 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 3, 1, 1, 3, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 0, 2, 4, 0, 0, 0, 0

Offset: 0

Stefano Spezia, Apr 21 2022

			The array begins:
    0, 0, 0, 0, 0, 0, 0, 0, ...
    0, 0, 1, 0, 0, 0, 0, 0, ...
    0, 1, 0, 1, 2, 0, 0, 0, ...
    0, 0, 1, 0, 1, 2, 3, 0, ...
    0, 0, 2, 1, 0, 1, 2, 3, ...
    0, 0, 0, 2, 1, 0, 1, 2, ...
    0, 0, 0, 3, 2, 1, 0, 1, ...
    0, 0, 0, 0, 3, 2, 1, 0, ...
    ...

Cf. A003983, A049581, A051125, A307018 (antidiagonal half-sums), A353452, A353453.

Mathematica
```
A[i_,j_]:=If[Min[i, j]
				
```

A265129 Triangle read by rows, formed as the sum of the two versions of the natural numbers filling an equilateral triangle.

Original entry on oeis.org

2, 5, 5, 10, 10, 10, 17, 17, 17, 17, 26, 26, 26, 26, 26, 37, 37, 37, 37, 37, 37, 50, 50, 50, 50, 50, 50, 50, 65, 65, 65, 65, 65, 65, 65, 65, 82, 82, 82, 82, 82, 82, 82, 82, 82, 101, 101, 101, 101, 101, 101, 101, 101, 101, 101

Offset: 1

Craig Knecht, Dec 02 2015

The natural numbers can sequentially fill a right- or left-handed equilateral triangle. Componentwise addition of the values of these two triangles produces the present triangle.
The row sums for this triangle give A034262.
The difference between the right- and left-handed triangles produces A049581.

			Displayed as a triangle:
   2;
   5  5;
  10 10 10;
  17 17 17 17;
  26 26 26 26 26;
  37 37 37 37 37 37;
  ...

Craig Knecht, Sum of right and left triangles.
Craig Knecht, Difference between right and left triangles A049581.

Column k=1 gives A002522.
Cf. A049581 (difference of triangles), A034262 (row sum of triangle), A069894 (center column).
Cf. A071237.

seq(seq(n^2+1,k=1..n),n=1..10); # Georg Fischer, Oct 01 2021

T(n,k) = n^2 + 1 for k = 1..n and n >= 1. - Georg Fischer, Oct 01 2021

Sum_{k=1..n} k * T(n,k) = A071237(n). - Alois P. Heinz, Oct 01 2021

Row 6 with T(6,k)=37 inserted by Georg Fischer, Oct 01 2021

A351154 a(n) is the permanent of the n X n matrix M(n) that is defined as M[i,j,n] = A351153(n, min(i, j)) + abs(i - j).

Original entry on oeis.org

1, 1, 7, 169, 10388, 1324344, 305668180, 116145817656, 67770421715800, 57594670663866124, 68393751368082128320, 109765035421144948709232, 231657098706747226470685920, 628412716450312334529486247152, 2149132484027947970192241804640128, 9113755489596517688997731211571700256

Offset: 0

Stefano Spezia, Feb 02 2022

nonn

Conjectures: (Start)
det(M(0)) = det(M(1)) = 1 and det(M(n)) = -(n - 2)! for n > 1.
abs(det(M(n))) = abs(A159333(n-2)). (End)

			a(3) = 169:
    1    2    3
    2    4    5
    3    5    6
a(4) = 10388:
    1    2    3    4
    2    5    6    7
    3    6    8    9
    4    7    9   10

Cf. A000142, A003983, A049581, A159333, A351153.

A351153[n_,k_]:=n(k-1)-k(k-3)/2; M[i_,j_,n_]:=A351153[n,Min[i,j]]+Abs[i-j]; a[n_]:=Permanent[Table[M[i,j,n],{i,n},{j,n}]]; Join[{1},Array[a,15]]

t(n, k) = n*(k-1) - k*(k-3)/2; \\ A351153
a(n) = matpermanent(matrix(n, n, i, j, t(n, min(i, j)) + abs(i - j))); \\ Michel Marcus, Feb 03 2022

A068344 Square array read by antidiagonals of T(n,k) = sign(n-k).

Original entry on oeis.org

0, -1, 1, -1, 0, 1, -1, -1, 1, 1, -1, -1, 0, 1, 1, -1, -1, -1, 1, 1, 1, -1, -1, -1, 0, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 0, 1, 1, 1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, 0, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 0, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1

Offset: 0

Henry Bottomley, Mar 06 2002

			The start of the array is:
   0;
  -1,   1;
  -1,   0,   1;
  -1,  -1,   1,   1;
  -1,  -1,   0,   1,   1;
  ...
- _Boris Putievskiy_, Dec 24 2012

Cf. A049581, A057427, A057428, A002260, A004736. As a straight sequence, a(n)=0 when n is in A046092. A023532 seen as a triangle is half this square.

a(n-1) = sign(A002260(n) - A004736(n)) or a(n-1) = sign((n-t*(t+1)/2) - ((t*t+3*t+4)/2-n)) where t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012

cp's OEIS Frontend

A344839 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = abs(n * 2^max(0, w(k)-w(n)) - k * 2^max(0, w(n)-w(k))) (where w = A070939).

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A330240 Square array T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros), read by antidiagonals, n, k >= 0.

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A353452 a(n) is the determinant of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

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A353453 a(n) is the permanent of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

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A330237 Square array T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros); read by antidiagonals; n, k >= 1.

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A330238 Triangle T(n,k): concatenate the absolute differences of the digits of n and k (the smaller one padded with leading zeros); n >= k >= 1.

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A352967 Array read by antidiagonals: A(i, j) = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0, with i >= 0 and j >= 0.

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A265129 Triangle read by rows, formed as the sum of the two versions of the natural numbers filling an equilateral triangle.

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