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%I #15 Jun 17 2017 04:00:50
%S 17,23,65,89,149,191,269,329,425,503,617,713,845,959,1109,1241,1409,
%T 1559,1745,1913,2117,2303,2525,2729,2969,3191,3449,3689,3965,4223,
%U 4517,4793,5105,5399,5729,6041,6389,6719,7085,7433,7817,8183,8585,8969,9389,9791,10229,10649,11105,11543,12017,12473,12965,13439,13949
%N Numbers of the form ((6k+5)^2+9)/2 or 2(3k+4)^2-9.
%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.
%C The first differences of this sequence is the interleaved sequence 6,42,24,60,42,78.... = 9*n*(39-27*(-1)^n)/2.
%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.: (17-11*x+19*x^2-7*x^3)/((1+x)*(1-x)^3).
%F a(n) = (6*n*(3*n+10)+27*(-1)^n+41)/4.
%F 2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.
%e For n = 7, a(7)=2*a(6)-2*a(4)+a(3)=2*269-2*149+89=329.
%t LinearRecurrence[{2,0,-2,1},{17,23,65,89},60] (* _Harvey P. Dale_, Aug 07 2015 *)
%o I:=[17, 23, 65, 89]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];
%Y Cf. A178218, A214345.
%K nonn,easy
%O 0,1
%A _Yasir Karamelghani Gasmallah_, Jul 19 2012
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