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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.

%I A357042 #77 Oct 04 2024 00:27:08
%S A357042 1,20,117,400,1025,2196,4165,7232,11745,18100,26741,38160,52897,71540,
%T A357042 94725,123136,157505,198612,247285,304400,370881,447700,535877,636480,
%U A357042 750625,879476,1024245,1186192,1366625,1566900,1788421,2032640,2301057,2595220,2916725,3267216,3648385
%N A357042 The sum of the numbers of the central diamond of the multiplication table [1..k] X [1..k] for k=2*n-1.
%C A357042 a(n) is the sum of the elements of the multiplication table, forming the maximum diamond in its center.
%H A357042 Paolo Xausa, <a href="/A357042/b357042.txt">Table of n, a(n) for n = 1..10000</a>
%H A357042 Nicolay Avilov, <a href="/A357042/a357042.jpg">Drawing for a(1)-a(5)</a>
%H A357042 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A357042 a(n) = n^2*(2*n^2 - 2*n + 1).
%F A357042 a(n) = 2*A000583(n) - A015237(n).
%F A357042 From _Stefano Spezia_, Sep 19 2022: (Start)
%F A357042 G.f.: x*(1 + 15*x + 27*x^2 + 5*x^3)/(1 - x)^5.
%F A357042 a(n) = A000290(n)*A001844(n-1). (End)
%e A357042 In the multiplication table [1..3] X [1..3]: a(2) = 2+2+4+6+6 = 20;
%e A357042 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 A357042 For n=3, the multiplication table [1..5] X [1..5] and the terms summed are
%e A357042   *   1  2  3  4  5
%e A357042    -----------------
%e A357042   1|        3
%e A357042   2|     4  6  8
%e A357042   3|  3  6  9 12 15
%e A357042   4|     8 12 16
%e A357042   5|       15
%t A357042 A357042[n_] := n^2*(2*(n-1)*n + 1); Array[A357042, 50] (* or *)
%t A357042 LinearRecurrence[{5, -10, 10, -5, 1}, {1, 20, 117, 400, 1025}, 50] (* _Paolo Xausa_, Oct 03 2024 *)
%Y A357042 Cf. A000290, A001844, A003991.
%Y A357042 Cf. A000583, A015237.
%K A357042 nonn,easy
%O A357042 1,2
%A A357042 _Nicolay Avilov_, Sep 18 2022