A317617 Triangle T read by rows: T(n, k) = (n^3 + n)/2 + (k - (n + 1)/2)*(n mod 2).

1, 5, 5, 14, 15, 16, 34, 34, 34, 34, 63, 64, 65, 66, 67, 111, 111, 111, 111, 111, 111, 172, 173, 174, 175, 176, 177, 178, 260, 260, 260, 260, 260, 260, 260, 260, 365, 366, 367, 368, 369, 370, 371, 372, 373, 505, 505, 505, 505, 505, 505, 505, 505, 505, 505, 666

Offset: 1

Author

Stefano Spezia, Aug 01 2018

Keywords

• nonn
• tabl

Comments

T(n, k) is the sum of the terms of the k-th column of an n X n square matrix M formed by writing the numbers 1, ..., n^2 successively forward and backward along the rows in zig-zag pattern (proved). The n X n square matrix M is defined as M[i, j, n] = j + n*(i - 1) if i is odd and M[i, j, n] = n*i - j + 1 if i is even (see the examples below).

The rows of even indices of the triangle T are made of all the same repeating number.

Examples

			n\k|   1   2   3   4   5   6
---+------------------------
1  |   1
2  |   5   5
3  |  14  15  16
4  |  34  34  34  34
5  |  63  64  65  66  67
6  | 111 111 111 111 111 111
...
For n = 1 the matrix M is
  1
with column sum 1.
For n = 2 the matrix M is
  1, 2
  4, 3
with column sums 5, 5.
For n = 3 the matrix M is
  1, 2, 3
  6, 5, 4
  7, 8, 9
with column sums 14, 15, 16.
		

Links

• Stefano Spezia, First 150 rows of the triangle, flattened

Crossrefs

Cf. A006003, A000027, A000035, A037270 (row sums).

A317614(n): the trace of the n X n square matrix M.

A074147(n): the elements of the antidiagonal of the n X n square matrix M.

A241016(n): the triangle of the row sums of the n X n square matrix M.

A246697(n): the right diagonal of the triangle T.

Programs

GAP

A317617 := function(n)
local i, j, t;
for i in [1 .. n] do
   for j in [1 .. i] do
      t := (i^3 + i)/2 + (j - (i + 1)/2)*(i mod 2);
      Print(t, "\t");
   od;
   Print("\n");
od;
end;
A317617(11); # yields sequence in triangular form

GAP

Flat(List([1..11],n->List([1..n],k->(n^3+n)/2+(k-(n+1)/2)*(n mod 2)))); # Muniru A Asiru, Aug 24 2018

Magma

[[(n^3 + n)/2 + (k - (n + 1)/2)*(n mod 2): k in [1..n]]: n in [1..11]];

Maple

a:=(n,k)->(n^3+n)/2+(k-(n+1)/2)*modp(n,2): seq(seq(a(n,k),k=1..n),n=1..11); # Muniru A Asiru, Aug 24 2018

Mathematica

f[n_] := Table[SeriesCoefficient[(x*(x*(5 - 7*y) + x^4*(1 - 2*y) - x^3*(-3 + y) - 3*x^2*(-1 + y) + y))/((-1 + x)^4*(1 + x)^2*(-1 + y)^2), {x, 0, i}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f, 11]]
T[i_, j_, n_] := If[OddQ@ i, j + n*(i - 1), n*i - j + 1]; f[n_] := Plus @@@ Transpose[ Table[T[i, j, n], {i, n}, {j, n}]]; Array[f, 11] // Flatten  (* Robert G. Wilson v, Aug 01 2018 *)
f[n_] := Table[SeriesCoefficient[1/4 E^(-x + y) (1 - x - 2 y + E^(2 x) (-1 + 3 x + 6 x^2 + 2 x^3 + 2 y)), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 1, n}]; Flatten[Array[f, 11]] (* Stefano Spezia, Jan 10 2019 *)

Maxima

sjoin(v, j) := apply(sconcat, rest(join(makelist(j, length(v)), v)))$ display_triangle(n) := for i from 1 thru n do disp(sjoin(makelist((i^3+i)/2+(j-(i+1)/2)*mod(i, 2), j, 1, i), " ")); display_triangle(10);

PARI

M(i,j,n) = if (i % 2, j + n*(i-1), n*i - j + 1);
T(n, k) = sum(i=1, n, M(i,k,n));
tabl(nn) = for(n=1, nn, for(k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Aug 09 2018

R

# by formula
for (n in 1:11){
   t <- c(n, "")
   for(j in 1:n){
      t <- c(t, (n^3+n)/2+(j-(n+1)/2)*(n%%2), "")
   }
   cat(t, "\n")
} # yields sequence in triangular form
(MATLAB and FreeMat)
for(i=1:11);
   for(j=1:i);
      t=(i^3 + i)/2 + (j - (i + 1)/2)*mod(i,2);
      fprintf('%0.f\t', t);
   end
   fprintf('\n');
end % yields sequence in triangular form

Formula

T(n, k) = A006003(n) + (k - (A000027(n) + 1)/2)*A000035(n).

G.f.: x*(x*(5 - 7*y) + x^4*(1 - 2*y) - x^3*(- 3 + y) - 3*x^2*(- 1 + y) + y)/((-1 + x)^4*(1 + x)^2*(-1 + y)^2).

E.g.f.: (1/4)*exp(-x + y)*(1 - x - 2*y + exp(2*x)*(-1 + 3*x + 6*x^2 + 2*x^3 + 2*y)). - Stefano Spezia, Jan 10 2019