A359087 a(n) is equal to the last point of a reverse pyramid summation with base 1, 2, 3, ..., n-2, n-1, n, n-1, n-2, ..., 3, 2, 1.

1, 4, 19, 78, 301, 1108, 3951, 13758, 47049, 158616, 528619, 1745098, 5715429, 18593032, 60136183, 193525002, 620046513, 1978886448, 6293809971, 19955385762, 63094947981, 198990438408, 626141673375, 1966085927898, 6161660863929, 19276374528468, 60206635741131

Offset: 1

Author

Moosa Nasir, Dec 15 2022

Keywords

• nonn

Comments

Each element in the pyramid below the base is equal to the sum of the top left, top, and top right elements.

Each row has 2*n-(1+2*r) elements where r is the row number starting from 0.

The sum of elements in the first row is n^2.

The total number of elements in the pyramid is n^2.

Examples

			For n = 3:
  1  2  3  2  1
     6  7  6
       19
so a(3) = 19.
For n = 4:
  1   2   3   4   3   2   1
      6   9  10   9   6
         25  28  25
             78
so a(4) = 78.
		

Crossrefs

Cf. A004737, A027907, A132894.

Programs

C

unsigned long tri(int n, int k)
{
    if (n == 0 && k == 0) return 1;
    if(k < -n || k > n) return 0;
    return tri(n - 1, k - 1) + tri(n - 1, k) + tri(n - 1, k + 1);
}
unsigned long a(int n)
{
    unsigned long sum = 0;
    sum += tri(n - 1,0) * n;
    for (int i = 1; i < n; i++)
    {
        sum += 2 * tri(n - 1,n - i) * i;
    }
    return sum;
}

Maple

f:= proc(n) local L,i;
  L:= [seq(i,i=1..n),seq(n-i,i=1..n-1)];
  for i from 1 to n-1 do
    L:= L[1..-3] + L[2..-2] + L[3..-1]
  od;
  op(L)
end proc:
map(f, [$1..30]); # Robert Israel, Dec 17 2022

Mathematica

f[n_] := Module[{L, i}, L = Range[n]~Join~Table[n-i, {i, 1, n-1}]; For[i = 1, i <= n-1, i++, L = L[[1;;-3]] + L[[2;;-2]] + L[[3;;-1]]]; L[[1]]];
f /@ Range[30] (* Jean-François Alcover, Jan 25 2023, after Robert Israel *)

Formula

a(n) = Sum_{k=1..2*n-1} A004737(k + (n-1)^2) * A027907(k + (n-1)^2 - 1).

Empirical g.f.: x/(1-3*x)^2 - 2*x^2/((1+x)^(1/2)*(1-3*x)^(3/2)). - Robert Israel, Dec 17 2022

a(n) = n*3^(n-1) - 2*A132894(n-1) (conjectured). - Bernard Schott, Dec 20 2022