A214405 - OEIS

%I #10 Sep 08 2022 08:46:02

%S 1,29,41,85,113,173,217,293,353,445,521,629,721,845,953,1093,1217,

%T 1373,1513,1685,1841,2029,2201,2405,2593,2813,3017,3253,3473,3725,

%U 3961,4229,4481,4765,5033,5333,5617,5933,6233,6565,6881,7229,7561,7925,8273,8653

%N Numbers of the form (4k+3)^2-8 or (4k+5)^2+4.

%C For every odd n the triple (a(n-1)^2, a(n)^2 , a(n+1)^2) is an arithmetic progression, i.e., 2*a(n)^2 = a(n-1)^2 + a(n+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 28,12,44,28,60,44....

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

%F O.G.f.: (1+27*x-17*x^2+5*x^3)/((1+x)*(1-x)^3).

%F a(n) = 4*n*(n+3)-6*(-1)^n+7.

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

%e a(4) = 2*a(3) - 2*a(1) + a(0) = 2*85 - 2*29 + 1 = 113.

%o (Magma) I:=[1, 29, 41, 85]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];

%o (Maxima) A214405(n):=4*n*(n+3)-6*(-1)^n+7$

%o makelist(A214405(n),n,0,30); /* _Martin Ettl_, Nov 01 2012 */

%Y Cf. A178218, A214345.

%K nonn,easy

%O 1,2

%A _Yasir Karamelghani Gasmallah_, Jul 15 2012