A071233 - OEIS

%I #33 Aug 05 2024 08:43:57

%S 0,10,40,102,208,370,600,910,1312,1818,2440,3190,4080,5122,6328,7710,

%T 9280,11050,13032,15238,17680,20370,23320,26542,30048,33850,37960,

%U 42390,47152,52258,57720,63550,69760,76362,83368,90790,98640,106930,115672,124878,134560

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

%C For n > 1, a(n) is the sum of the numbers appearing along the outside border of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows. - _Wesley Ivan Hurt_, May 13 2021

%D T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.

%H Vincenzo Librandi, <a href="/A071233/b071233.txt">Table of n, a(n) for n = 1..2000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = 2*A062158(n).

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

%F G.f.: 2*x*(5+x^2)/(1 - x)^4 - _Harvey P. Dale_, Jun 27 2021

%F E.g.f.: 2*exp(x)*x*(5 + 5*x + x^2). - _Stefano Spezia_, Apr 22 2023

%F a(n) = (n-1)*A005893(n). - _G. C. Greubel_, Aug 05 2024

%e From _Wesley Ivan Hurt_, May 13 2021: (Start)

%e Given the 4 X 4 square array below,

%e   [  1   2   3   4 ]

%e   [  5   6   7   8 ]

%e   [  9  10  11  12 ]

%e   [ 13  14  15  16 ]

%e the sum of the elements along the outside border is 1+2+3+4+8+12+16+15+14+13+9+5 = 102. Thus a(4) = 102. (End)

%t Table[2(n-1)(n^2+1),{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,10,40,102},50] (* _Harvey P. Dale_, Jun 27 2021 *)

%o (Magma) [2*(n-1)*(n^2+1): n in [1..50]]; // _Vincenzo Librandi_, Jun 14 2011

%o (SageMath)

%o def A071233(n): return 2*(n-1)*(n^2+1)

%o [A071233(n) for n in range(1,51)] # _G. C. Greubel_, Aug 05 2024

%Y Cf. A005893, A062158.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_, Jun 11 2002