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

1,2

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.

Links

Table of n, a(n) for n=1..56.

Stefano Spezia, First 150 rows of the triangle, flattened

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

Example

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.

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 *)

Prog

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

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.

Sequence in context: A286257 A168418 A266440 * A294750 A304300 A321653

Adjacent sequences: A317614 A317615 A317616 * A317618 A317619 A317620

Keyword

nonn,tabl

Author

Stefano Spezia, Aug 01 2018

Status

approved