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A157914 a(n) = 8*n^2 - 1.

%I A157914 #43 Jul 16 2025 16:24:30
%S A157914 7,31,71,127,199,287,391,511,647,799,967,1151,1351,1567,1799,2047,
%T A157914 2311,2591,2887,3199,3527,3871,4231,4607,4999,5407,5831,6271,6727,
%U A157914 7199,7687,8191,8711,9247,9799,10367,10951,11551,12167,12799,13447,14111,14791
%N A157914 a(n) = 8*n^2 - 1.
%C A157914 The identity (8*n^2 - 1)^2 - (16*n^2 - 4)*(2*n)^2 = 1 can be written as a(n)^2 - A158443(n)*A005843(n)^2 = 1.
%C A157914 Sequence found by reading the line from 7, in the direction 7, 31, ..., in the square spiral whose vertices are the triangular numbers A000217. - _Omar E. Pol_, Sep 03 2011
%C A157914 Bisection of A195605 (odd part). - _Bruno Berselli_, Sep 21 2011
%C A157914 The identity (8*n^2 - 1)^2 - (64*n^2 - 16)*(n)^2 = 1 can be written as a(n)^2 - A157913(n)*(n)^2 = 1. - _Vincenzo Librandi_, Feb 09 2012
%H A157914 Vincenzo Librandi, <a href="/A157914/b157914.txt">Table of n, a(n) for n = 1..10000</a>
%H A157914 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A157914 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A157914 G.f: x*(7+10*x-x^2)/(1-x)^3.
%F A157914 a(n) = A139098(n) - 1. - _Omar E. Pol_, Sep 03 2011
%F A157914 E.g.f.: (8*x^2 + 8*x - 1)*exp(x) + 1. - _G. C. Greubel_, Jul 15 2017
%F A157914 From _Amiram Eldar_, Feb 04 2021: (Start)
%F A157914 Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(8))*cot(Pi/sqrt(8)))/2.
%F A157914 Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(8))*csc(Pi/sqrt(8)) - 1)/2.
%F A157914 Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(8))*csc(Pi/sqrt(8)).
%F A157914 Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(8))/sqrt(2). (End)
%t A157914 Table[8n^2-1,{n,50}]
%t A157914 LinearRecurrence[{3,-3,1},{7,31,71},50] (* _Harvey P. Dale_, Jul 16 2025 *)
%o A157914 (Magma) I:=[7, 31, 71]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];
%o A157914 (PARI) a(n)=8*n^2-1 \\ _Charles R Greathouse IV_, Sep 03 2011
%Y A157914 Cf. A157913, A158443, A005843, A139098, A195605.
%K A157914 nonn,easy
%O A157914 1,1
%A A157914 _Vincenzo Librandi_, Mar 09 2009