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a(n) = n^4/2 - n^3 + 3*n^2/2 - n + 1 = (n^2 + 1)*(n^2 - 2*n + 2)/2.
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%I #46 Feb 09 2024 08:40:50

%S 1,1,5,25,85,221,481,925,1625,2665,4141,6161,8845,12325,16745,22261,

%T 29041,37265,47125,58825,72581,88621,107185,128525,152905,180601,

%U 211901,247105,286525,330485,379321,433381,493025,558625,630565,709241

%N a(n) = n^4/2 - n^3 + 3*n^2/2 - n + 1 = (n^2 + 1)*(n^2 - 2*n + 2)/2.

%C On an n X (n - 1)(n - 2)/2 X n(n - 1)/2 cuboid with n >= 5, the two points at greatest surface distance from a corner are the opposite corner and the point 1 in from each of the two edges on a smallest face which meet at the opposite corner; this greatest surface distance is sqrt(a(n)).

%C Subsequence of A080827, see formula. - _David James Sycamore_, Jul 31 2018

%H Harry J. Smith, <a href="/A058919/b058919.txt">Table of n, a(n) for n = 0..500</a>

%H Henry Bottomley, <a href="http://se16.info/js/cuboid.htm#Numerical">Source</a>

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

%F G.f.: (1 - 4*x + 10*x^2 + 5*x^4)/(1 - x)^5. - _Colin Barker_, Jan 01 2012

%F a(n) = A002522(n)*A002522(n-1)/2, with A002522(-1)=2. - _Bruno Berselli_, Nov 11 2014

%F a(n) = A080827(n^2-n+1). - _David James Sycamore_, Jul 31 2018

%F E.g.f.: exp(x)*(2 + 4*x^2 + 4*x^3 + x^4)/2. - _Stefano Spezia_, Oct 08 2022

%F For n>1, a(n) = A000217(n-1)^2 + (A000217(n-1)+1)^2 = (A000217(n)-1)^2 + (A000217(n-2)-1)^2. - _Charlie Marion_, Feb 08 2024

%p A058919:=n->n^4/2 - n^3 + 3*n^2/2 - n + 1; seq(A058919(n), n=0..30); # _Wesley Ivan Hurt_, May 10 2014

%t Table[n^4/2 - n^3 + 3 n^2/2 - n + 1, {n, 0, 30}] (* _Wesley Ivan Hurt_, May 10 2014 *)

%o (PARI) { for (n = 0, 500, write("b058919.txt", n, " ", (n^4 + 3*n^2)/2 - n^3 - n + 1); ) } \\ _Harry J. Smith_, Jun 23 2009

%o (Magma) [n^4/2 - n^3 + 3*n^2/2 - n + 1: n in [0..30]]; // _Wesley Ivan Hurt_, May 10 2014

%Y For n >= 4 the sequence is a subsequence of A007692.

%Y Cf. A002522, A080827.

%K nonn,easy

%O 0,3

%A _Henry Bottomley_, Jan 11 2001