Search: id:A046176
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%I A046176
%S A046176 1,35,1189,40391,1372105,46611179,1583407981,53789260175,
%T A046176 1827251437969,62072759630771,2108646576008245,71631910824649559,
%U A046176 2433376321462076761,82663163018885960315,2808114166320660573949
%N A046176 Indices of square numbers which are also hexagonal.
%C A046176 Bisection (even part) of Chebyshev sequence with Diophantine property.
%C A046176 (3*b(n))^2 - 2*(2*a(n+1))^2 = 1 with companion sequence b(n)=A077420(n),
               n>=0.
%C A046176 Sequence also refers to inradius of primitive Pythagorean triangles with 
               consecutive legs, odd followed by even. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Apr 23 2003
%H A046176 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A046176 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HexagonalSquareNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A046176 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A046176 a(n)=34*a(n-1)-a(n-2), a(0)=-1, a(1)=1.
%F A046176 a(n+1)= S(2*n, 6)= S(n, 34) + S(n-1, 34), n>=1, with S(n, x) := U(n, 
               x/2) Chebyshev's polynomials of the second kind. See A049310. S(n, 
               34)=A029547(n).
%F A046176 G.f.: x*(1+x)/(1-34*x+x^2).
%F A046176 a(n+1)= sum((-1)^k*binomial(2*n-k, k)*6^(2*(n-k)), k=0..n), n>=0.
%F A046176 a(n)=A001109(2n+1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 23 
               2003
%F A046176 Define f[x,s] = s x + Sqrt[(s^2-1)x^2+1]; f[0,s]=0. a(n) = f[f[a(n-1),
               3],3]. - Marcos Carreira, Dec 27 2006
%F A046176 (1) a(n) = (sqrt(2)/8)(3+2*sqrt(2))*(17+12*sqrt(2))^(n-1) -(sqrt(2)/8)(3-2*sqrt(2))*(17-12*sqrt(2))^(n-1) 
               - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 22 2008
%F A046176 (2) a(n) = (sqrt(2)/8)*(17+12*sqrt(2))^(n-1/2) -(sqrt(2)/8)*(17-12*sqrt(2))^(n-1/
               2) - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 22 2008
%F A046176 (3) a(n) = (sqrt(2)/8)*(3+2*sqrt(2))^(2n-1) -(sqrt(2)/8)*(3-2*sqrt(2))^(2n-1) 
               - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 22 2008
%F A046176 (4) a(n) = (sqrt(2)/8)*(1+sqrt(2))^(4n-2) -(sqrt(2)/8)*(1-sqrt(2))^(4n-2) 
               - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 22 2008
%F A046176 (5) a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3) - Antonio A. Olivares (olivares14031(AT)yahoo.com), 
               Mar 22 2008
%F A046176 a(n+1)=17*a(n)+6*(8*a(n)^2+1)^0.5 for n>=0 [From Richard Choulet (richardchoulet(AT)yahoo.fr), 
               May 01 2009]
%t A046176 q=9;s=0;lst={};Do[s+=n;If[Sqrt[q*s+1]==Floor[Sqrt[q*s+1]],AppendTo[lst,
               Sqrt[q*s+1]]],{n,0,9!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Apr 02 2009]
%Y A046176 Cf. A008844, A046177.
%Y A046176 Cf. A001109.
%Y A046176 Sequence in context: A002453 A049395 A115492 this_sequence A029546 A126158 
               A095153
%Y A046176 Adjacent sequences: A046173 A046174 A046175 this_sequence A046177 A046178 
               A046179
%K A046176 nonn
%O A046176 1,2
%A A046176 Eric Weisstein (eric(AT)weisstein.com)
%E A046176 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Nov 29 2002

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