A360145 Triangle read by rows where row n is the largest (or middle or n-th) column of the reverse pyramid summation of order n described in A359087.

1, 2, 4, 3, 7, 19, 4, 10, 28, 78, 5, 13, 37, 105, 301, 6, 16, 46, 132, 382, 1108, 7, 19, 55, 159, 463, 1351, 3951, 8, 22, 64, 186, 544, 1594, 4680, 13758, 9, 25, 73, 213, 625, 1837, 5409, 15945, 47049, 10, 28, 82, 240, 706, 2080, 6138, 18132, 53610, 158616, 11, 31, 91, 267, 787, 2323, 6867, 20319, 60171, 178299, 528619

Offset: 1

Author

Bernard Schott, Jan 27 2023

Keywords

• nonn
• tabl

Comments

The integer that is at the k-th row of the middle column of this pyramid of order n will be noted T(n,k).

Each row has n terms.

Examples

			Triangle begins:
   n=1:  1;
   n=2:  2,  4;
   n=3:  3,  7, 19;
   n=4:  4, 10, 28,  78;
   n=5:  5, 13, 37, 105, 301;
   n=6:  6, 16, 46, 132, 382, 1108;
   ...
For n=5, the reverse pyramid summation is as follows and row 5 here is the middle column 5,13,37,...
  1    2    3    4    5    4    3    2    1
       6    9   12   13   12    9    6
           27   34   37   34   27
                98  105   98
                    301
		

Crossrefs

Cf. A132894, A359087 (right diagonal).

Columns k=1..3: A000027, A016777, A017173.

Programs

PARI

f(v) = if (#v == 1, v, vector(#v-2, i, v[i]+v[i+1]+v[i+2]));
row(n) = my(u = concat([1..n], Vecrev([1..n-1])), v=u, w = vector(n)); for (i=1, n, w[i] = v[#v\2+1]; v = f(v);); w; \\ Michel Marcus, Jan 30 2023

Formula

T(n,1) = n.

T(n,2) = 3n - 2.

T(n,3) = 9n - 8.

T(n,4) = 27n - 30.

T(n,5) = 81n - 104.

T(n,n) = A359087(n).

T(n,k) = 3^(k-1)*n - 2*A132894(k-1) for 1 <= k <= n (conjectured).