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A214345 Interleaved reading of A073577 and A053755.

%I A214345 #36 Feb 22 2024 20:24:16
%S A214345 5,7,17,23,37,47,65,79,101,119,145,167,197,223,257,287,325,359,401,
%T A214345 439,485,527,577,623,677,727,785,839,901,959,1025,1087,1157,1223,1297,
%U A214345 1367,1445,1519,1601,1679,1765,1847,1937,2023,2117,2207,2305,2399,2501
%N A214345 Interleaved reading of A073577 and A053755.
%C A214345 The elements of this sequence satisfy the property that 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. 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 : in the case of this sequence 7^2, 17^2, and 23^2 is such a triple (i.e. 15-8 =7, 17, 8+15=23, and 8^2+15^2=17^2) .
%C A214345 The first differences of such a sequence is always an interleaved sequence; in this case the interleaved sequence is 2,10,6,14,10,... (A142954).
%H A214345 Guenther Schrack, <a href="/A214345/b214345.txt">Table of n, a(n) for n = 0..10001</a>
%H A214345 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A214345 a(2n+1) = A073577(n+1); a(2n) = A053755(n+1).
%F A214345 a(n+1)-a(n) = A142954(n+1).
%F A214345 a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
%F A214345 G.f.: (x^3-3*x^2+3*x-5)/((x-1)^3*(x+1)).
%F A214345 a(n) = (2*n*(n+4)+3*(-1)^n+7)/2.
%F A214345 2*a(2n)^2 = a(2n-1)^2 + a(2n+1)^2.
%F A214345 a(n) = 4*(n+1) + a(n-2) for n > 1; a(-n) = a(n-4). - _Guenther Schrack_, Oct 24 2018
%F A214345 E.g.f.: (5 + 5*x + x^2)*cosh(x) + (2 + 5*x + x^2)*sinh(x). - _Stefano Spezia_, Feb 22 2024
%e A214345 For n = 7, a(7)=2*a(6)-2*a(4)+a(3)=2*65-2*37+23=79
%p A214345 seq(coeff(series((x^3-3*x^2+3*x-5)/((x-1)^3*(x+1)),x,n+1), x, n), n = 0 .. 50); # _Muniru A Asiru_, Oct 26 2018
%t A214345 LinearRecurrence[{2,0,-2,1},{5,7,17,23},50] (* _Harvey P. Dale_, Apr 02 2018 *)
%o A214345 (Magma) I:=[5, 7, 17, 23];[n le 4 select I[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..75]];
%o A214345 (Maxima) A214345(n):=(2*n*(n+4)+3*(-1)^n+7)/2$
%o A214345 makelist(A214345(n),n,0,30); /* _Martin Ettl_, Nov 01 2012 */
%o A214345 (GAP) a:=[7,17];; for n in [3..50] do a[n]:=4*(n+1)+a[n-2]; od; Concatenation([5],a); # _Muniru A Asiru_, Oct 26 2018
%Y A214345 Cf. A053755, A073577, A178218.
%Y A214345 First differences: A142954; 2-element moving average (a(n-1) + a(n))/2: A002378. - _Guenther Schrack_, Oct 25 2018
%K A214345 nonn,easy
%O A214345 0,1
%A A214345 _Yasir Karamelghani Gasmallah_, Jul 13 2012