A357042 - OEIS

%I #77 Oct 04 2024 00:27:08

%S 1,20,117,400,1025,2196,4165,7232,11745,18100,26741,38160,52897,71540,

%T 94725,123136,157505,198612,247285,304400,370881,447700,535877,636480,

%U 750625,879476,1024245,1186192,1366625,1566900,1788421,2032640,2301057,2595220,2916725,3267216,3648385

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

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

%H Paolo Xausa, <a href="/A357042/b357042.txt">Table of n, a(n) for n = 1..10000</a>

%H Nicolay Avilov, <a href="/A357042/a357042.jpg">Drawing for a(1)-a(5)</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

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

%F From _Stefano Spezia_, Sep 19 2022: (Start)

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

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

%e In the multiplication table [1..3] X [1..3]: a(2) = 2+2+4+6+6 = 20;

%e 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.

%e For n=3, the multiplication table [1..5] X [1..5] and the terms summed are

%e   *   1  2  3  4  5

%e    -----------------

%e   1|        3

%e   2|     4  6  8

%e   3|  3  6  9 12 15

%e   4|     8 12 16

%e   5|       15

%t A357042[n_] := n^2*(2*(n-1)*n + 1); Array[A357042, 50] (* or *)

%t LinearRecurrence[{5, -10, 10, -5, 1}, {1, 20, 117, 400, 1025}, 50] (* _Paolo Xausa_, Oct 03 2024 *)

%Y Cf. A000290, A001844, A003991.

%Y Cf. A000583, A015237.

%K nonn,easy

%O 1,2

%A _Nicolay Avilov_, Sep 18 2022