A342354 M(n,k) = 2*n^2 + 2*k + 1 for 0 <= k <= n and M(n,k) = 2*k^2 + 4*k - 2*n + 1 for 0 <= n <= k; square array M(n,k) read by ascending antidiagonals (n, k >= 0).

1, 3, 7, 9, 5, 17, 19, 11, 15, 31, 33, 21, 13, 29, 49, 51, 35, 23, 27, 47, 71, 73, 53, 37, 25, 45, 69, 97, 99, 75, 55, 39, 43, 67, 95, 127, 129, 101, 77, 57, 41, 65, 93, 125, 161, 163, 131, 103, 79, 59, 63, 91, 123, 159, 199, 201, 165, 133, 105, 81, 61, 89, 121, 157, 197, 241, 243, 203, 167, 135, 107, 83, 87, 119, 155, 195, 239, 287

Offset

0,2

Comments

This is a square array defined by J. M. Bergot in A005917 (originally by mistake in A047926). Here is the edited description of the array by this contributor.

Construct an array M with M(0,n) = 2*n^2 + 4*n + 1 = A056220(n+1), M(n,0) = 2*n^2 + 1 = A058331(n) and M(n,n) = 2*n*(n+1) + 1 = A001844(n). Row(n) begins with all the increasing odd numbers from A058331(n) to A001844(n) and column(n) begins with all the decreasing odd numbers from A056220(n+1) to A001844(n). The sum of the terms in row(n) plus those in column(n) minus M(n,n) equals A005917(n+1).

Links

Table of n, a(n) for n=0..77.

Formula

O.g.f. for rectangular M: (x^4*y^4 + 4*x^3*y^4 + 3*x^4*y^2 - 18*x^3*y^3 - x^2*y^4 + 8*x^3*y^2 + 4*x^2*y^3 - 10*x^3*y + 10*x^2*y^2 - 2*x*y^3 + 8*x^2*y + 4*x*y^2 + 3*x^2 - 18*x*y - y^2 + 4*y + 1)/((1 - x)^3*(1 - y)^3*(1 - x*y)^2).

O.g.f. for triangular T: (x^8*y^4 + 4*x^7*y^4 - x^6*y^4 - 18*x^6*y^3 + 3*x^6*y^2 + 4*x^5*y^3 + 8*x^5*y^2 - 2*x^4*y^3 + 10*x^4*y^2 - 10*x^4*y + 4*x^3*y^2 + 8*x^3*y - x^2*y^2 - 18*x^2*y + 3*x^2 + 4*x*y + 1)/((1 - x)^3*(1 - x*y)^3*(1 - x^2*y)^2).

Example

Square array M(n,k) (n, k >= 0) begins:
   1,  7, 17, 31, 49, 71, 97, 127, ...
   3,  5, 15, 29, 47, 69, 95, 125, ...
   9, 11, 13, 27, 45, 67, 93, 123, ...
  19, 21, 23, 25, 43, 65, 91, 121, ...
  33, 35, 37, 39, 41, 63, 89, 119, ...
  51, 53, 55, 57, 59, 61, 87, 117, ...
  73, 75, 77, 79, 81, 83, 85, 115, ...
  ...
The triangular array T(n,k) = M(n-k,k) (with rows n >= 0 and columns k = 0..n) is obtained by reading array M by ascending antidiagonals:
   1;
   3,  7;
   9,  5, 17;
  19, 11, 15, 31;
  33, 21, 13, 29, 49;
  51, 35, 23, 27, 47, 71;
  73, 53, 37, 25, 45, 69, 97;
  99, 75, 55, 39, 43, 67, 95, 127;
  ...

Prog

(PARI) tabl(nn) = {my(M=matrix(nn+1, nn+1)); for(n=1, nn+1, for(k=1, nn+1, M[n, k]=if(k == n, 2*n^2-2*n+1, if(k < n, 2*n^2-4*n+2*k+1, 2*k^2-2*n+1)))); M}

Crossrefs

Cf. A001844, A005917, A047926, A056220, A058331.

Antidiagonal sums are in A342362.

Sequence in context: A038135 A087655 A016621 * A011271 A005534 A113014

Adjacent sequences: A342351 A342352 A342353 * A342355 A342356 A342357

Keyword

nonn,tabl

Author

Petros Hadjicostas, Mar 08 2021

Status

approved