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%I #92 Nov 22 2023 12:12:38
%S 3,11,27,51,83,123,171,227,291,363,443,531,627,731,843,963,1091,1227,
%T 1371,1523,1683,1851,2027,2211,2403,2603,2811,3027,3251,3483,3723,
%U 3971,4227,4491,4763,5043,5331,5627,5931,6243,6563,6891,7227,7571,7923,8283,8651,9027,9411
%N a(n) = 4*n*(n+1) + 3.
%C One-fourth the sum of the three terms produced by the division of complex numbers (2*n-3+(2*n-1)*i)/(2*n+1+(2*n+3)*i). For (b+c*i)/(d+e*i) the three terms in parentheses are ((b*d+c*e)+(c*d-b*e)*i/(d^2+e^2). By substituting b=2*n-3, c=2*n-1, d=2*n+1, and e=2*n+3 one gets a(n). - _J. M. Bergot_, Sep 10 2015
%C The continued fraction expansion of sqrt(a(n)) is [2n+1; {2n+1, 4n+2}]. - _Magus K. Chu_, Sep 08 2022
%H Vincenzo Librandi, <a href="/A164897/b164897.txt">Table of n, a(n) for n = 0..900</a>
%H R. M. Green and Tianyuan Xu, <a href="https://arxiv.org/abs/2204.09765">2-roots for simply laced Weyl groups</a>, arXiv:2204.09765 [math.RT], 2022.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A000124(2*n) + A000124(2*n+1) = A069894(n)+1.
%F a(n+1) - a(n) = 8n+8 = A008590(n+1) (first differences).
%F a(n+1) - 2*a(n) + a(n-1) = 8 = A010731(n) (second differences).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n>2.
%F G.f.: (3+2*x+3*x^2) / (1-x)^3.
%F Sum_{k=n+1..2*n+1} a(k) - Sum_{k=0..n} a(k) = (2*n+2)^3. - _Bruno Berselli_, Jan 24 2011
%F E.g.f.: (4x^2 + 8x + 1)*exp(x). - _G. C. Greubel_, Sep 22 2015
%F a(n)^2 = A222465(n)*A222465(n+1) - 12. - _Ezhilarasu Velayutham_, Mar 18 2020
%F Sum_{n>=0} 1/a(n) = tanh(Pi/sqrt(2))*Pi/(4*sqrt(2)). - _Amiram Eldar_, Aug 21 2022
%F a(n) = A059100(2*n+1). - _Dimitri Papadopoulos_, Nov 21 2023
%p A164897:=n->4*n*(n+1)+3: seq(A164897(n), n=0..100); # _Wesley Ivan Hurt_, Sep 10 2015
%t Table[4 n (n + 1) + 3, {n, 0, 50}] (* _Harvey P. Dale_, Jan 23 2011 *)
%o (Magma) [4*n*(n+1)+3: n in [0..50]]; // _Vincenzo Librandi_, Apr 24 2011
%o (PARI) a(n)=4*n*(n+1)+3 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A000124, A008590, A010731, A016743, A069894.
%Y Odd-indexed terms of A059100.
%K nonn,easy
%O 0,1
%A _Paul Curtz_, Aug 30 2009
%E Definition simplified by _R. J. Mathar_, Sep 16 2009
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