cp's OEIS Frontend

A084159 Pell oblongs.

%I A084159 #51 Aug 03 2024 11:18:01
%S A084159 1,3,21,119,697,4059,23661,137903,803761,4684659,27304197,159140519,
%T A084159 927538921,5406093003,31509019101,183648021599,1070379110497,
%U A084159 6238626641379,36361380737781,211929657785303,1235216565974041,7199369738058939,41961001862379597,244566641436218639
%N A084159 Pell oblongs.
%C A084159 Essentially the same as A046727.
%D A084159 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 60 at p. 123.
%H A084159 Vincenzo Librandi, <a href="/A084159/b084159.txt">Table of n, a(n) for n = 0..1000</a>
%H A084159 P. E. Trier, <a href="https://archim.org.uk/eureka/archive/Eureka-4.pdf">"Almost Isosceles" Right-Angled Triangles</a>, Eureka, No. 4, May 1940, pp. 9 - 11.
%H A084159 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-1).
%F A084159 a(n) = ((sqrt(2)+1)^(2*n+1) - (sqrt(2)-1)^(2*n+1) + 2*(-1)^n)/4.
%F A084159 a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3). - _Paul Curtz_, May 17 2008
%F A084159 G.f.: (1-x)^2/((1+x)*(1-6*x+x^2)). - _R. J. Mathar_, Sep 17 2008
%F A084159 a(n) = A078057(n)*A001333(n). - _R. J. Mathar_, Jul 08 2009
%F A084159 a(n) = A001333(n)*A001333(n+1).
%F A084159 From _Peter Bala_, May 01 2012: (Start)
%F A084159 a(n) = (-1)^n*R(n,-4), where R(n,x) is the n-th row polynomial of A211955.
%F A084159 a(n) = (-1)^n*1/u*T(n,u)*T(n+1,u) with u = sqrt(-1) and T(n,x) the Chebyshev polynomial of the first kind.
%F A084159 a(n) = (-1)^n + 4*Sum_{k = 1..n} (-1)^(n-k)*8^(k-1)*binomial(n+k,2*k).
%F A084159 Recurrence equations: a(n) = 6*a(n-1) - a(n-2) + 4*(-1)^n, with a(0) = 1 and a(1) = 3; a(n)*a(n-2) = a(n-1)*(a(n-1)+4*(-1)^n).
%F A084159 Sum_{k >= 0} (-1)^k/a(k) = 1/sqrt(2).
%F A084159 1 - 2*(Sum_{k = 0..n} (-1)^k/a(k))^2 = (-1)^(n+1)/A090390(n+1). (End)
%F A084159 a(n) = (A001333(2*n+1) + (-1)^n)/2. - _G. C. Greubel_, Oct 11 2022
%F A084159 E.g.f.: exp(-x)*(1 + exp(4*x)*(cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x)))/2. - _Stefano Spezia_, Aug 03 2024
%t A084159 b[n_]:= Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[2], n]]];
%t A084159 Join[{1}, Table[b[n+1], {n,50}]*Table[b[n], {n,50}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 15 2011 *)
%t A084159 LinearRecurrence[{5,5,-1},{1,3,21},30] (* _Harvey P. Dale_, Aug 04 2019 *)
%o A084159 (Magma) [Floor(((Sqrt(2)+1)^(2*n+1)-(Sqrt(2)-1)^(2*n+1)+2*(-1)^n)/4): n in [0..30]]; // _Vincenzo Librandi_, Aug 13 2011
%o A084159 (SageMath) [(lucas_number2(2*n+1, 2, -1) + 2*(-1)^n)/4 for n in range(31)] # _G. C. Greubel_, Oct 11 2022
%Y A084159 Cf. A046727 (same sequence except for first term).
%Y A084159 Cf. A001333, A001654, A078057, A084158, A084175, A090390, A182432, A211955.
%K A084159 easy,nonn
%O A084159 0,2
%A A084159 _Paul Barry_, May 18 2003