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%I #29 Feb 22 2024 17:45:37
%S 13,17,53,73,125,161,229,281,365,433,533,617,733,833,965,1081,1229,
%T 1361,1525,1673,1853,2017,2213,2393,2605,2801,3029,3241,3485,3713,
%U 3973,4217,4493,4753,5045,5321,5629,5921,6245,6553,6893,7217,7573,7913,8285,8641
%N Numbers of the form (4k+3)^2+4 or (4k+5)^2-8.
%C For every n=2k the triple (a(2k-1)^2, a(2k)^2 , a(2k+1)^2) is an arithmetic progression, i.e., 2*a(2k)^2 = a(2k-1)^2 + a(2k+1)^2.
%C In general a triple((x-y)^2,z^2,(x+y)^2) is an arithmetic progression if and only if x^2+y^2=z^2, e.g., (17^2, 53^2, 73^2).
%C The first differences of this sequence is the interleaved sequence 4,36,20,52,36,68,52,....
%H Paolo Xausa, <a href="/A214393/b214393.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
%F G.f.: (13-9*x+19*x^2-7*x^3)/((1+x)*(1-x)^3).
%F a(n) = 4*n*(n+3)+6*(-1)^n+7.
%F 2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.
%e a(5) = 2*a(4) - 2*a(2) + a(1) = 2*125 - 2*53 + 17 = 161.
%t A214393[n_] := 4*n*(n+3) + 6*(-1)^n + 7; Array[A214393, 50, 0] (* or *)
%t LinearRecurrence[{2, 0, -2, 1}, {13, 17, 53, 73}, 50] (* _Paolo Xausa_, Feb 22 2024 *)
%o (Magma) I:=[13, 17, 53, 73]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];
%o (Maxima) A214393(n):=4*n*(n+3)+6*(-1)^n+7$
%o makelist(A214393(n),n,0,30); /* _Martin Ettl_, Nov 01 2012 */
%Y Cf. A178218, A214345.
%K nonn,easy
%O 0,1
%A _Yasir Karamelghani Gasmallah_, Jul 15 2012
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