Search: id:A001075 Results 1-1 of 1 results found. %I A001075 M1769 N0700 %S A001075 1,2,7,26,97,362,1351,5042,18817,70226,262087,978122,3650401,13623482, %T A001075 50843527,189750626,708158977,2642885282,9863382151,36810643322, %U A001075 137379191137,512706121226,1913445293767,7141075053842,26650854921601 %N A001075 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - a(n-2). %C A001075 Chebyshev's T(n,x) polynomials evaluated at x=2. %C A001075 x = 2^n - 1 is prime if and only if x divides a(2^(n-2)). %C A001075 Any k in the sequence is succeeded by 2*k + sqrt{3*(k^2 - 1)} - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 28 2002 %C A001075 a(n) solves for x in x^2 - 3*y^2 = 1, the corresponding y being given by A001353(n). The solution ratios a(n)/A001353(n) are obtained as convergents of the continued fraction expansion of sqrt(3): either as successive convergents of [2;-4] or as odd convergents of [1;1, 2]. - Lekraj Beedassy (blekraj(AT)yahoo.com), Sep 19 2003 %C A001075 a(n) is half the central value in a list of three consecutive integers, the lengths of the sides of a triangle with integer sides and area. - Eugene McDonnell (eemcd(AT)mac.com), Oct 19 2003 %C A001075 a(3+6k)-1 and a(3+6k)+1 are consecutive odd powerful numbers. See A076445. - T. D. Noe (noe(AT)sspectra.com), May 04 2006 %C A001075 a(n)=2*a(n-1)+3*A001353(n-1). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 21 2006 %C A001075 The intermediate convergents to 3^(1/2), beginning with 3/2, 12/7, 45/ 26, 168/97, comprise a strictly increasing sequence; essentially, numerators=A005320, denominators=A001075. - Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 %C A001075 The upper principal convergents to 3^(1/2), beginning with 2/1, 7/4, 26/15, 97/56, comprise a strictly decreasing sequence; numerators=A001075, denominators=A001353. - Clark Kimberling (ck6(AT)evansville.edu), Aug 27 2008 %C A001075 a(n+1) is the Hankel transform of A000108(n)+A000984(n)=(n+2)*Catalan(n). [From Paul Barry (pbarry(AT)wit.ie), Aug 11 2009] %D A001075 H. Brocard, Notes e'le'mentaires sur le proble`me de Peel, Nouvelles Correspondance Math\'{e}matique, 4 (1878), 161-169. %D A001075 E. I. Emerson, Recurrent sequences in the equation DQ^2 = R^2 + N, Fib. Quart., 7 (1969), 231-242. %D A001075 Clark Kimberling, "Best lower and upper approximates to irrational numbers, " Elemente der Mathematik, 52 (1997) 122-126. %D A001075 Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966. %D A001075 Mcdonnell, Eugene, "Heron's Rule and Integer-Area Triangles", Vector 12.3 (January 1996) pp. 133-142 %D A001075 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). %D A001075 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). %D A001075 P.-F. Teilhet, Reply to Query 2094, L'Interm\'{e}diaire des Math\'{e}maticiens, 10 (1903), 235-238. %D A001075 F. V. Waugh and M. W. Maxfield, Side-and-diagonal numbers, Math. Mag., 40 (1967), 74-83. %H A001075 T. D. Noe, Table of n, a(n) for n=0..200 %H A001075 C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002. %H A001075 Chris Caldwell, Primality Proving, Arndt's theorem. %H A001075 Tanya Khovanova, Recursive Sequences %H A001075 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001075 S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992. %H A001075 Index entries for sequences related to Chebyshev polynomials. %H A001075 Index entries for two-way infinite sequences %H A001075 Index entries for sequences related to linear recurrences with constant coefficients %F A001075 For all elements x of the sequence, 12*x^2 -12 is a square. Lim. as n-> Inf. a(n)/a(n-1) = 2 + sqrt(3) = (4 + sqrt(12))/2 which preserves the kinship with the equation "12*x^2 - 12 is a square" where the initial "12" ends up appearing as a square root. - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 10 2002 %F A001075 a(n) = (S(n, 4) - S(n-2, 4))/2 = T(n, 2), with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. U, resp. T, are Chebyshev's polynomials of the second, resp. first, kind. S(n-1, 4) = A001353(n), n>=0. See A049310 and A053120. %F A001075 a(n) = 2^(-n)*Sum_{k>=0} binomial(2n, 2k)*3^k = 2^(-n)*Sum_{k>=0} A086645(n, k)*3^k. - Philippe DELEHAM, Mar 01, 2004 %F A001075 a(n) = ((2+sqrt(3))^n + (2-sqrt(3))^n)/2; a(n) = ceiling((1/2)*(2+sqrt(3))^(n)). %F A001075 a(n) = cosh( n * ln( 2 + sqrt(3))). %F A001075 a(n)=sum{k=0..floor(n/2); C(n, 2k)2^(n-2k)3^k } - Paul Barry (pbarry(AT)wit.ie), May 08 2003 %F A001075 G.f.: (1-2x)/(1-4x+x^2). E.g.f.: exp(2x)cosh(sqrt(3)x). a(n)=4a(n-1)-a(n-2)=a(-n). %F A001075 a(n+2) = 2*a(n+1) + 3*Sum_{k>=0} a(n-k)*2^k. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Mar 03 2004 %F A001075 a(n) = left term of M^n * [1,0] where M = the 2 X 2 matrix [2,3; 1,2]. Right term = A001353(n). Example: a(4) = 97 since M^4 * [1,0] = [A001075(4), A001353(4)] = [97, 56]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2006 %F A001075 Binomial transform of A026150: (1, 1, 4, 10, 28, 76,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007 %F A001075 First differences of A001571. - njas, Nov 03 2009 %F A001075 Sequence satisfies -3 = f(a(n), a(n+1)) where f(u, v) = u^2 + v^2 - 4*u*v. - Michael Somos Sep 19 2008 %e A001075 2^6 -1 = 63 does not divide a(2^4) = 708158977, therefore 63 is composite. 2^5 -1 = 31 divides a(2^3) = 18817, therefore 31 is prime. %p A001075 A001075:=-(-1+2*z)/(1-4*z+z**2); [S. Plouffe in his 1992 dissertation.] %t A001075 Table[ Ceiling[(1/2)*(2 + Sqrt[3])^n], {n, 0, 24}] %o A001075 (PARI) a(n)=subst(poltchebi(abs(n)),x,2) %o A001075 (PARI) a(n)=real((2+quadgen(12))^abs(n)) %o A001075 (PARI) a(n)=polsym(1-4*x+x^2,abs(n))[1+abs(n)]/2 %o A001075 (Other) sage: [lucas_number2(n,4,1)/2 for n in xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2009] %Y A001075 Cf. A065918, A071954. a(n) = sqrt(1+3*A001353(n)) (cf. Richardson comment). %Y A001075 Cf. A001353, A001571, A001834, A003500, A016064, A082840. %Y A001075 Bisections are A011943 and A094347. %Y A001075 Cf. A001353. %Y A001075 Cf. A026150. %Y A001075 Sequence in context: A087448 A129273 A055988 this_sequence A113436 A126223 A114121 %Y A001075 Adjacent sequences: A001072 A001073 A001074 this_sequence A001076 A001077 A001078 %K A001075 nonn,easy,nice,new %O A001075 0,2 %A A001075 N. J. A. Sloane (njas(AT)research.att.com). %E A001075 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 10 2000 %E A001075 Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 31 2002 Search completed in 0.002 seconds