Search: id:A001541
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%I A001541 M3037 N1231
%S A001541 1,3,17,99,577,3363,19601,114243,665857,3880899,22619537,131836323,
%T A001541 768398401,4478554083,26102926097,152139002499,886731088897,
%U A001541 5168247530883,30122754096401,175568277047523,1023286908188737
%N A001541 a(0) = 1, a(1) = 3; for n > 1, a(n) = 6a(n-1) - a(n-2).
%C A001541 Chebyshev polynomials of the first kind evaluated at 3.
%C A001541 a(n) solves for x in x^2 - 8*y^2 = 1, the corresponding y being A001109(n). 
               For n>0, the ratios a(n)/A001090(n) may be obtained as convergents 
               to sqrt(8): either successive convergents of [3; -6] or odd convergents 
               of [2; 1, 4]. - Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 09 2003
%C A001541 Formula: ((-1+sqrt(2))^n+(1+sqrt(2))^n+(1-sqrt(2))^n+(-1-sqrt(2))^n)/
               4 (with interpolated zeros) E.g.f. cosh(x)cosh(sqrt(2)x) (with interpolated 
               zeros). - Paul Barry (pbarry(AT)wit.ie), Sep 18 2003
%C A001541 Also gives solutions to the equation x^2-1=floor(x*r*floor(x/r)) where 
               r=sqrt(8) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 14 2004
%C A001541 Appears to give all solutions >1 to the equation : x^2=ceiling(x*r*floor(x/
               r)) where r=sqrt(2). - Benoit Cloitre, Feb 24, 2004
%C A001541 a(n+1) - A001542(n+1) = A090390(n+1) - A046729(n) = A001653(n); a(n+1) 
               - 4*A079291(n+1) = (-1)^(n+1). Formula generated by the floretion 
               - .5'i + .5'j - .5i' + .5j' - 'ii' + 'jj' - 2'kk' + 'ij' + .5'ik' 
               + 'ji' + .5'jk' + .5'ki' + .5'kj' + e - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), 
               Nov 16 2004
%C A001541 This sequence give numbers n such that (n-1)*(n+1)/2 = j^2 = a square. 
               Remark : (i-1)*(i+1)/2 = (i^2-1)/2 = -1 = i^2 with i=sqrt(-1) so 
               i is also in the sequence. - Pierre CAMI (pierrecami(AT)tele2.fr), 
               Apr 20 2005
%C A001541 a(n) is prime for n = {1, 2, 4, 8}. Prime a(n) are {3, 17, 577, 665857}, 
               which belong to A001601(n). a(2k-1) is divisible by a(1) = 3. a(4k-2) 
               is dovisible by a(2) = 17. a(8k-4) is divisible by a(4) = 577. a(16k-8) 
               is divisible by a(8) = 665857. - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Nov 24 2006
%C A001541 A001541(n)=A001333(2*n) [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), 
               Aug 13 2008]
%C A001541 The upper principal convergents to 2^(1/2), beginning with 3/2, 17/12, 
               99/70, 577/408, comprise a strictly decreasing sequence; essentially, 
               numerators=A001541 and denominators=A001542. - Clark Kimberling (ck6(AT)evansville.edu), 
               Aug 26 2008
%C A001541 Odd Pell numbers [A001541^2-2*A001542^2=1] [From Vincenzo Librandi (vincenzo.librandi(AT)ti.it), 
               Aug 01 2009]
%D A001541 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001541 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001541 I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), 
               181-193.
%D A001541 H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles 
               Correspondance Math\'{e}matique, 4 (1878), 161-169.
%D A001541 J. W. L. Glaisher, On Eulerian numbers (formulae, residues, end-figures), 
               with the values of the first twenty-seven, Quarterly Journal of Mathematics, 
               vol. 45, 1914, pp. 1-51.
%D A001541 D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. 
               J., 4 (1935), 323-340.
%D A001541 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli 
               and Euler, Annals Math., 36 (1935), 637-649.
%D A001541 Problem E1976, Amer. Math. Monthly, 75 (1968), 683-684.
%H A001541 T. D. Noe, <a href="b001541.txt">Table of n, a(n) for n=0..200</a>
%H A001541 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A001541 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A001541 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A001541 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001541 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001541 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A001541 a(n) = ((3+2*sqrt(2))^n + (3-2*sqrt(2))^n)/2.
%F A001541 a(n) = 3*A001109(n)-A001109(n-1), n >= 1. - Barry Williams and Wolfdieter 
               Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), May 05 2000.
%F A001541 G.f.: (1-3*x)/(1-6*x+x^2) - Barry Williams and Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               May 05 2000.
%F A001541 a(n) = sqrt{8*[(A001109(n))^2] + 1} = T(n, 3), with Chebyshev's T-polynomials 
               A053120.
%F A001541 a(n) ~ (1/2)*(sqrt(2) + 1)^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 
               2002
%F A001541 For all elements x of the sequence, 2*x^2 - 2 is a square. Lim. as n 
               -> inf. of a(n)/a(n-1) = 3 + 2*sqrt(2). - Gregory V. Richardson (omomom(AT)hotmail.com), 
               Oct 10 2002. [Corrected by Peter Pein, Mar 09 2009]
%F A001541 E.g.f.: exp(3x)cosh(2sqrt(2)x). Binomial transform of A084128. - Paul 
               Barry (pbarry(AT)wit.ie), May 16 2003
%F A001541 For n>=1, a(n) = A001652(n) - A001652(n-1) - Charlie Marion (charliem(AT)bestweb.net), 
               Jul 01 2003
%F A001541 For n>0, a(n)^2 +1=2*A001653(n-1)*A001653(n); e.g. 17^2+1=290=2*5*29 
               - Charlie Marion (charliemath(AT)verizon.net), Dec 21 2003
%F A001541 a(n) = Sum_{k>=0} binomial(2*n, 2*k)*2^k = Sum_{k>=0} A086645(n, k)*2^k 
               . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 29 2004
%F A001541 a(n)*A002315(n+k)=A001652(2n+k)+A001652(k)+1; e.g. 3*1393=4069+119+1; 
               for k>0, a(n+k)*A002315(n)=A001652(2n+k)-A001652(k-1); e.g. 577*7=4059-20 
               - Charlie Marion (charliemath(AT)verizon.net), Mar 17 2003
%F A001541 a(n)^2+a(n+1)^2=2*(A001653(2n+1)-A001652(2n)); e.g., 1^2+3^2=2(5-0); 
               17^2+99^2=2(5741-696) - Charlie Marion (charliemath(AT)verizon.net), 
               Mar 17 2003
%F A001541 A053141(n+1) + A055997(n+1) = a(n+1) + A001109(n+1). - Creighton Dement 
               (creighton.k.dement(AT)uni-oldenburg.de), Sep 16 2004
%F A001541 For n>k, a(n)*A001653(k)=A011900(n+k)+A053141(n-k-1); e.g. 99*5=495=493+2. 
               For n<=k, a(n)*A001653(k)=A011900(n+k)+A053141(k-n); e.g. 3*29=87=85+2 
               - Charlie Marion (charliemath(AT)optonline.net), Oct 18 2004
%F A001541 a(n) = Sqrt[ A055997(2n) ]. - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Nov 24 2006
%F A001541 a(2n) = A056771(n). a(2n+1) = 3*A077420(n). - Alexander Adamchuk (alex(AT)kolmogorov.com), 
               Feb 01 2007
%F A001541 (A000129(n)^2)*4+(-1)^n - Vim Wenders (vim(AT)gmx.li), Mar 28 2007
%F A001541 2*a(k)*A001653(n)*A001653(n+k)=A001653(n)^2+A0001653(n+k)^2+A001542(k)^2; 
               e.g., 2*3*5*29=5^2+29^2+2^2; 2*99*29*5741=2*99*29*5741=29^2+5741^2+70^2 
               - Charlie Marion (charliemath(AT)optonline.net), Oct 12 2007
%F A001541 a(n) = Cosh[2n*ArcSinh[1]] - Herbert Kociemba (kociemba(AT)t-online.de), 
               Apr 24 2008
%p A001541 a[0]:=1: a[1]:=3: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], 
               n=0..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 26 
               2006
%p A001541 A001541:=-(-1+3*z)/(1-6*z+z**2); [S. Plouffe in his 1992 dissertation.]
%o A001541 (PARI) a(n)=real((3+quadgen(32))^n)
%o A001541 (PARI) a(n)=subst(poltchebi(abs(n)),x,3)
%o A001541 (PARI) a(n)=if(n<0,a(-n),polsym(1-6*x+x^2,n)[n+1]/2)
%o A001541 (PARI) a(n)=([1,2,2;2,1,2;2,2,3]^n)[3,3] - Vim Wenders (vim(AT)gmx.li), 
               Mar 28 2007
%Y A001541 Bisection of A001333. A003499(n)=2a(n). Cf. A046090, A001109, A053142.
%Y A001541 Cf. A084130.
%Y A001541 Cf. A001109.
%Y A001541 Cf. A055997 = numbers n such that n(n-1)/2 is a square. Cf. A001601.
%Y A001541 Cf. A056771, A077420.
%Y A001541 Cf. A005319.
%Y A001541 Sequence in context: A142988 A056660 A155610 this_sequence A161940 A074565 
               A054365
%Y A001541 Adjacent sequences: A001538 A001539 A001540 this_sequence A001542 A001543 
               A001544
%K A001541 nonn,easy,nice
%O A001541 0,2
%A A001541 N. J. A. Sloane (njas(AT)research.att.com).
%E A001541 More terms from Clark Kimberling (ck6(AT)evansville.edu), Aug 26 2008

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