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A160372 The 4-tuple (2, ((2*n+1)^2-1)/2, ((2*n+3)^2-1)/2, a(n)), where a(n) = 4*(3 + 20n + 42n^2 + 32n^3 + 8n^4), has Diophantus's property that the product of any two distinct terms plus one is a square.

%I A160372 #43 Jan 22 2024 01:19:30
%S A160372 420,2380,7812,19404,40612,75660,129540,208012,317604,465612,660100,
%T A160372 909900,1224612,1614604,2091012,2665740,3351460,4161612,5110404,
%U A160372 6212812,7484580,8942220,10603012,12485004,14607012,16988620,19650180,22612812,25898404,29529612,33529860
%N A160372 The 4-tuple (2, ((2*n+1)^2-1)/2, ((2*n+3)^2-1)/2, a(n)), where a(n) = 4*(3 + 20n + 42n^2 + 32n^3 + 8n^4), has Diophantus's property that the product of any two distinct terms plus one is a square.
%H A160372 Paolo Xausa, <a href="/A160372/b160372.txt">Table of n, a(n) for n = 1..10000</a>
%H A160372 Lenny Jones, <a href="https://www.jstor.org/stable/2691315">A polynomial Approach to a Diophantine Problem</a>, Math. Mag. 72 (1999), 52-55.
%H A160372 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantusProperty.html">Diophantus Property</a>.
%H A160372 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A160372 G.f.: 4*x*(3*x^4-14*x^3+28*x^2+70*x+105)/(1-x)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 [corrected by _R. J. Mathar_, Sep 16 2009]
%F A160372 a(n) = 4 * (2*n + 1) * (2*n + 3) * (2*n^2 + 4*n + 1). - _Paolo Xausa_, Jan 16 2024
%F A160372 From _Amiram Eldar_, Jan 22 2024: (Start)
%F A160372 Sum_{n>=1} 1/a(n) = -cot(Pi/sqrt(2))*Pi/(8*sqrt(2)) - 5/24.
%F A160372 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 - cosec(Pi/sqrt(2))*Pi/(8*sqrt(2)) - 1/24. (End)
%e A160372 For n=2, we get (2,12,24,2380), and 2*12+1 = 25 = 5^2, 2*24+1 = 49 = 7^2, 2*2380+1 = 4761 = 69^2, 12*24+1 = 289 = 17^2, 12*2380+1 = 28561 = 169^2, 4*2380+1 = 57121 = 239^2.
%t A160372 A160372[n_] := 4 (2 n + 1) (2 n + 3) (2 n^2 + 4 n + 1);
%t A160372 Array[A160372, 50] (* _Paolo Xausa_, Jan 16 2024 *)
%Y A160372 Cf. A086302.
%K A160372 nonn,easy
%O A160372 1,1
%A A160372 _John W. Layman_, May 11 2009