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 A001541 a(0) = 1, a(1) = 3; for n > 1, a(n) = 6*a(n-1) - a(n-2). (Formerly M3037 N1231) 111
 1, 3, 17, 99, 577, 3363, 19601, 114243, 665857, 3880899, 22619537, 131836323, 768398401, 4478554083, 26102926097, 152139002499, 886731088897, 5168247530883, 30122754096401, 175568277047523, 1023286908188737, 5964153172084899, 34761632124320657 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Chebyshev polynomials of the first kind evaluated at 3. This sequence gives the values of x in solutions of the Diophantine equation x^2 - 8*y^2 = 1, the corresponding values of y are in A001109. 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, Sep 09 2003 [edited by Jon E. Schoenfield, May 04 2014] Also gives solutions to the equation x^2 - 1 = floor(x*r*floor(x/r)) where r = sqrt(8). - Benoit Cloitre, Feb 14 2004 Appears to give all solutions greater than 1 to the equation: x^2 = ceiling(x*r*floor(x/r)) where r = sqrt(2). - Benoit Cloitre, Feb 24 2004 This sequence give numbers n such that (n-1)*(n+1)/2 is a perfect 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, Apr 20 2005 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 divisible by a(2) = 17. a(8k-4) is divisible by a(4) = 577. a(16k-8) is divisible by a(8) = 665857. - Alexander Adamchuk, Nov 24 2006 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, Aug 26 2008 Also index of sequence A082532 for which A082532(n) = 1. - Carmine Suriano, Sep 07 2010 Numbers n such that sigma(n-1) and sigma(n+1) are both odd numbers. - Juri-Stepan Gerasimov, Mar 28 2011 Also, numbers such that floor(a(n)^2/2) is a square: base 2 analog of A031149, A204502, A204514, A204516, A204518, A204520, A004275, A001075. - M. F. Hasler, Jan 15 2012 Numbers such that 2n^2 - 2 is a square. Also integer square roots of the expression 2*n^2 + 1, at values of n given by A001542. Also see A228405 regarding 2n^2 -+ 2^k generally for k >= 0. - Richard R. Forberg, Aug 20 2013 Values of x (or y) in the solutions to x^2 - 6xy + y^2 + 8 = 0. - Colin Barker, Feb 04 2014 Panda and Ray call the numbers in this sequence the Lucas-balancing numbers C_n (see references and links). Partial sums of X or X+1 of Pythagorean triples (X,X+1,Z). - Peter M. Chema, Feb 03 2017 a(n)/A001542(n) is the closest rational approximation to sqrt(2) with a numerator not larger than a(n), and 2*A001542(n)/a(n) is the closest rational approximation to sqrt(2) with a denominator not larger than a(n). These rational approximations together with those obtained from the sequences A001653 and A002315 give a complete set of closest rational approximations to sqrt(2) with restricted numerator or denominator. a(n)/A001542(n) > sqrt(2) > 2*A001542(n)/a(n). - A.H.M. Smeets, May 28 2017 x = a(n), y = A001542(n) are solutions of the Diophantine equation x^2 - 2y^2 = 1 (Pell equation). x = 2*A001542(n), y = a(n) are solutions of the Diophantine equation x^2 - 2y^2 = -2. Both together give the set of fractional approximations for sqrt(2) obtained from limited fractions obtained from continued fraction representation to sqrt(2). - A.H.M. Smeets, Jun 22 2017 a(n) is the radius of the n-th circle among the sequence of circles generated as follows: Starting with a unit circle centered at the origin, every subsequent circle touches the previous circle as well as the two limbs of hyperbola x^2 - y^2 = 1, and lies in the region y > 0. - Kaushal Agrawal, Nov 10 2018 All of the positive integer solutions of a*b+1=x^2, a*c+1=y^2, b*c+1=z^2, x+z=2*y, 0 infinity} a(n)/a(n-1) = 3 + 2*sqrt(2). - Gregory V. Richardson, Oct 10 2002 [corrected by Peter Pein, Mar 09 2009] a(n) = 3*A001109(n) - A001109(n-1), n >= 1. - Barry E. Williams and Wolfdieter Lang, May 05 2000 For n >= 1, a(n) = A001652(n) - A001652(n-1). - Charlie Marion, Jul 01 2003 From Paul Barry, Sep 18 2003: (Start) a(n) = ((-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). (End) For n > 0, a(n)^2 + 1 = 2*A001653(n-1)*A001653(n). - Charlie Marion, Dec 21 2003 a(n)^2 + a(n+1)^2 = 2*(A001653(2*n+1) - A001652(2*n)). - Charlie Marion, Mar 17 2003 a(n) = Sum_{k >= 0} binomial(2*n, 2*k)*2^k = Sum_{k >= 0} A086645(n, k)*2^k. - Philippe Deléham, Feb 29 2004 a(n)*A002315(n+k) = A001652(2*n+k) + A001652(k) + 1; for k > 0, a(n+k)*A002315(n) = A001652(2*n+k) - A001652(k-1). - Charlie Marion, Mar 17 2003 For n > k, a(n)*A001653(k) = A011900(n+k) + A053141(n-k-1). For n <= k, a(n)*A001653(k) = A011900(n+k) + A053141(k-n). - Charlie Marion, Oct 18 2004 A053141(n+1) + A055997(n+1) = a(n+1) + A001109(n+1). - Creighton Dement, Sep 16 2004 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, Nov 16 2004 a(n) = sqrt( A055997(2*n) ). - Alexander Adamchuk, Nov 24 2006 a(2n) = A056771(n). a(2*n+1) = 3*A077420(n). - Alexander Adamchuk, Feb 01 2007 a(n) = (A000129(n)^2)*4 + (-1)^n. - Vim Wenders, Mar 28 2007 2*a(k)*A001653(n)*A001653(n+k) = A001653(n)^2 + A001653(n+k)^2 + A001542(k)^2. - Charlie Marion, Oct 12 2007 a(n) = A001333(2*n). - Ctibor O. Zizka, Aug 13 2008 A028982(a(n)-1) + 2 = A028982(a(n)+1). - Juri-Stepan Gerasimov, Mar 28 2011 a(n) = 2*A001108(n) + 1. - Paul Weisenhorn, Dec 17 2011 a(n) = sqrt(2*x^2 + 1) with x being A001542(n). - Zak Seidov, Jan 30 2013 a(2n) = 2*a(n)^2 - 1 = a(n)^2 + 2*A001542(n)^2. a(2*n+1) = 1 + 2*A002315(n)^2. - Steven J. Haker, Dec 04 2013 a(n) = 3*a(n-1) + 4*A001542(n-1); e.g., a(4) = 99 = 3*17 + 4*12. - Zak Seidov, Dec 19 2013 a(n) = cos(n * arccos(3)) = cosh(n * log(3 + 2*sqrt(2))). - Daniel Suteu, Jul 28 2016 From Ilya Gutkovskiy, Jul 28 2016: (Start) Inverse binomial transform of A084130. Exponential convolution of A000079 and A084058. Sum_{n>=0} (-1)^n*a(n)/n! = cosh(2*sqrt(2))/exp(3) = 0.4226407909842764637... (End) a(2*n+1) = 2*a(n)*a(n+1) - 3. - Timothy L. Tiffin, Oct 12 2016 a(n) = a(-n) for all n in Z. - Michael Somos, Jan 20 2017 a(2^n) = A001601(n+1). - A.H.M. Smeets, May 28 2017 a(A298210(n)) = A002350(2*n^2). - A.H.M. Smeets, Jan 25 2018 a(n) = S(n, 6) - 3*S(n-1, 6), for n >= 0, with S(n, 6) = A001109(n+1), (Chebyshev S of A049310). See the first comment and the formula a(n) = T(n, 3). - Wolfdieter Lang, Nov 22 2020 From Peter Bala, Dec 31 2021: (Start) a(n) = [x^n] (3*x + sqrt(1 + 8*x^2))^n. The Gauss congruences a(n*p^k) == a(n*p^(k-1)) hold for all prime p and positive integers n and k. O.g.f. A(x) = 1 + x*d/dx(log(B(x))), where B(x) = 1/sqrt(1 - 6*x + x^2) is the o.g.f. of A001850. (End) From Peter Bala, Aug 17 2022: (Start) Sum_{n >= 1} 1/(a(n) - 2/a(n)) = 1/2. Sum_{n >= 1} (-1)^(n+1)/(a(n) + 1/a(n)) = 1/4. Sum_{n >= 1} 1/(a(n)^2 - 2) = 1/2 - 1/sqrt(8). (End) EXAMPLE 99^2 + 99^2 = 140^2 + 2. - Carmine Suriano, Jan 05 2015 G.f. = 1 + 3*x + 17*x^2 + 99*x^3 + 577*x^4 + 3363*x^5 + 19601*x^6 + 114243*x^7 + ... MAPLE a:=1: a:=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, Jul 26 2006 A001541:=-(-1+3*z)/(1-6*z+z**2); # Simon Plouffe in his 1992 dissertation MATHEMATICA Table[Simplify[(1/2) (3 + 2 Sqrt)^n + (1/2) (3 - 2 Sqrt)^n], {n, 0, 20}] (* Artur Jasinski, Feb 10 2010 *) a[ n_] := If[n == 0, 1, With[{m = Abs @ n}, m Sum[4^i Binomial[m + i, 2 i]/(m + i), {i, 0, m}]]]; (* Michael Somos, Jul 11 2011 *) a[ n_] := ChebyshevT[ n, 3]; (* Michael Somos, Jul 11 2011 *) LinearRecurrence[{6, -1}, {1, 3}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2012 *) PROG (PARI) {a(n) = real((3 + quadgen(32))^n)}; /* Michael Somos, Apr 07 2003 */ (PARI) {a(n) = subst( poltchebi( abs(n)), x, 3)}; /* Michael Somos, Apr 07 2003 */ (PARI) {a(n) = if( n<0, a(-n), polsym(1 - 6*x + x^2, n) [n+1] / 2)}; /* Michael Somos, Apr 07 2003 */ (PARI) {a(n) = polchebyshev( n, 1, 3)}; /* Michael Somos, Jul 11 2011 */ (PARI) a(n)=([1, 2, 2; 2, 1, 2; 2, 2, 3]^n)[3, 3] \\ Vim Wenders, Mar 28 2007 (Magma)[n: n in [1..10000000] |IsSquare(8*(n^2-1))] // Vincenzo Librandi, Nov 18 2010] (Haskell) a001541 n = a001541_list !! (n-1) a001541_list = 1 : 3 : zipWith (-) (map (* 6) \$ tail a001541_list) a001541_list -- Reinhard Zumkeller, Oct 06 2011 (Scheme, with memoization-macro definec) (definec (A001541 n) (cond ((zero? n) 1) ((= 1 n) 3) (else (- (* 6 (A001541 (- n 1))) (A001541 (- n 2)))))) ;; Antti Karttunen, Oct 04 2016 CROSSREFS Bisection of A001333. A003499(n) = 2a(n). Cf. A055997 = numbers n such that n(n-1)/2 is a square. Row 1 of array A188645. Cf. A046090, A001109, A053142, A084130, A001601, A056771, A077420, A005319, A082532, A001542. Cf. A055792 (terms squared), A132592. Sequence in context: A356267 A056660 A155610 * A322242 A356392 A330626 Adjacent sequences:  A001538 A001539 A001540 * A001542 A001543 A001544 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified October 1 15:55 EDT 2022. Contains 357149 sequences. (Running on oeis4.)