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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)
94
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]

From Paul Barry, Sep 18 2003: (Start)

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). (End)

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

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

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 xˆ2 - 2yˆ2 = 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

REFERENCES

Bastida, Julio R. Quadratic properties of a linearly recurrent sequence. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 163--166, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561042 (81e:10009)

H. Brocard, Notes élémentaires sur le problème de Peel, Nouvelle Correspondance Mathématique, 4 (1878), 161-169.

J. W. L. Glaisher, On Eulerian numbers (formulas, residues, end-figures), with the values of the first twenty-seven, Quarterly Journal of Mathematics, vol. 45, 1914, pp. 1-51.

G. K. Panda, Some fascinating properties of balancing numbers, In Proc. of Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium 194 (2009), 185-189.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..200

I. Adler, Three Diophantine equations - Part II, Fib. Quart., 7 (1969), 181-193.

John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.

P. Catarino, H. Campos, P. Vasco, On some identities for balancing and cobalancing numbers, Annales Mathematicae et Informaticae, 45 (2015) pp. 11-24.

S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234.

O. Khadir, K. Liptai, L. Szalay, On the On the Shifted Product of Binary Recurrences, J. Int. Seq. 13 (2010), 10.6.1

J. M. Katri and D. R. Byrkit, Problem E1976, Amer. Math. Monthly, 75 (1968), 683-684.

Tanya Khovanova, Recursive Sequences

D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340.

D. H. Lehmer, A cotangent analogue of continued fractions, Duke Math. J., 4 (1935), 323-340. [Annotated scanned copy]

D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.

Dino Lorenzini, Z. Xiang, Integral points on variable separated curves, Preprint 2016.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Prasanta K. Ray, Curious congruences for balancing numbers, Int. J. Contemp. Math. Sci. 7 (2012), 881-889.

N. J. Wildberger, Pell's equation without irrational numbers, J. Int. Seq. 13 (2010), 10.4.3, Section 4.

Index entries for two-way infinite sequences

Index entries for linear recurrences with constant coefficients, signature (6,-1)

Index entries for sequences related to Chebyshev polynomials.

FORMULA

G.f.: (1-3*x)/(1-6*x+x^2). - Barry Williams and Wolfdieter Lang, May 05 2000

E.g.f.: exp(3x)cosh(2sqrt(2)x). Binomial transform of A084128. - Paul Barry, May 16 2003

From N. J. A. Sloane, May 16 2003: (Start)

a(n) = sqrt(8*((A001109(n))^2) + 1).

a(n) = T(n, 3), with Chebyshev's T-polynomials A053120. (End)

a(n) = ((3+2*sqrt(2))^n + (3-2*sqrt(2))^n)/2.

a(n) = cosh(2n*arcsinh(1)). - Herbert Kociemba, Apr 24 2008

a(n) ~ (1/2)*(sqrt(2) + 1)^(2*n). - Joe Keane (jgk(AT)jgk.org), May 15 2002

For all elements x of the sequence, 2*x^2 - 2 is a square. Lim_{n -> inf.} 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 Williams and Wolfdieter Lang, May 05 2000

For n >= 1, a(n) = A001652(n) - A001652(n-1). - Charlie Marion, Jul 01 2003

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(2n+1) - A001652(2n)). - 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(2n+k) + A001652(k) + 1; for k > 0, a(n+k)*A002315(n) = A001652(2n+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) = sqrt( A055997(2n) ). - Alexander Adamchuk, Nov 24 2006

a(2n) = A056771(n). a(2n+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(2n+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 сonvolution of A000079 and A084058.

Sum_{n>=0} (-1)^n*a(n)/n! = cosh(2*sqrt(2))/exp(3) = 0.4226407909842764637... (End)

a(2n+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

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[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, 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[2])^n + (1/2) (3 - 2 Sqrt[2])^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: A142988 A056660 A155610 * A161940 A074565 A241768

Adjacent sequences:  A001538 A001539 A001540 * A001542 A001543 A001544

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 26 03:15 EDT 2017. Contains 292502 sequences.