A357042 The sum of the numbers of the central diamond of the multiplication table [1..k] X [1..k] for k=2*n-1.

1, 20, 117, 400, 1025, 2196, 4165, 7232, 11745, 18100, 26741, 38160, 52897, 71540, 94725, 123136, 157505, 198612, 247285, 304400, 370881, 447700, 535877, 636480, 750625, 879476, 1024245, 1186192, 1366625, 1566900, 1788421, 2032640, 2301057, 2595220, 2916725, 3267216, 3648385

Offset: 1

Author

Nicolay Avilov, Sep 18 2022

Keywords

• nonn
• easy

Comments

a(n) is the sum of the elements of the multiplication table, forming the maximum diamond in its center.

Examples

			In the multiplication table [1..3] X [1..3]: a(2) = 2+2+4+6+6 = 20;
In the multiplication table [1..5] X [1..5]: a(3) = 3+4+3+6+6+8+9+8+12+12+15+16+15 = 117.
For n=3, the multiplication table [1..5] X [1..5] and the terms summed are
  *   1  2  3  4  5
   -----------------
  1|        3
  2|     4  6  8
  3|  3  6  9 12 15
  4|     8 12 16
  5|       15
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Nicolay Avilov, Drawing for a(1)-a(5)
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000290, A001844, A003991.

Cf. A000583, A015237.

Programs

Mathematica

A357042[n_] := n^2*(2*(n-1)*n + 1); Array[A357042, 50] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 20, 117, 400, 1025}, 50] (* Paolo Xausa, Oct 03 2024 *)

Formula

a(n) = n^2*(2*n^2 - 2*n + 1).

a(n) = 2*A000583(n) - A015237(n).

From Stefano Spezia, Sep 19 2022: (Start)

G.f.: x*(1 + 15*x + 27*x^2 + 5*x^3)/(1 - x)^5.

a(n) = A000290(n)*A001844(n-1). (End)