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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
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
1,2
COMMENTS
a(n) is the sum of the elements of the multiplication table, forming the maximum diamond in its center.
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)
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Nicolay Avilov, Sep 18 2022
STATUS
approved