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A001542 a(n) = 6*a(n-1) - a(n-2) for n>1, a(0)=0 and a(1)=2.
(Formerly M2030 N0802)
56
0, 2, 12, 70, 408, 2378, 13860, 80782, 470832, 2744210, 15994428, 93222358, 543339720, 3166815962, 18457556052, 107578520350, 627013566048, 3654502875938, 21300003689580, 124145519261542, 723573111879672 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Consider the equation core(x)=core(2x+1) where core(x) is the smallest number such that x*core(x) is a square: solutions are given by a(n)^2, n>0. - Benoit Cloitre, Apr 06 2002

Terms >0 give numbers k which are solutions to the inequality |round(sqrt(2)*k)/k-sqrt(2)|<1/2/sqrt(2)/k^2. - Benoit Cloitre, Feb 06 2006

Also numbers n such that A125650[ 6*n^2 ] is an even perfect square, where A124650(n) is a numerator of n(n+3)/(4(n+1)(n+2)) = Sum[ 1/(k(k+1)(k+2)), {k,1,n} ]. Sequence of numbers 6*n^2 is a bisection of A125651(n). - Alexander Adamchuk, Nov 30 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

Even Pell numbers. - Omar E. Pol, Dec 10 2008

Numbers n such that 2*n^2+1 is a square. - Vladimir Joseph Stephan Orlovsky, Feb 19 2010

These are the integer square roots of the Half-Squares, A007590(n), which occur at values of n given by A001541. Also the numbers produced by adding m + sqrt(floor(m^2/2) + 1) when m = A002315. See array in A227972. - Richard R. Forberg, Aug 31 2013

A001541(n)/a(n) is the closest rational approximation of sqrt(2) with a denominator not larger than a(n), and 2*a(n)/A001541(n) is the closest rational approximation of sqrt(2) with a nominator not larger than 2*a(n). These rational approximations together with those obtained from the sequences  A001653 and A002315 give a complete set of closest rational approximations of sqrt(2) with restricted nominator c.q. denominator. - A.H.M. Smeets, May 28 2017

REFERENCES

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

Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002; p. 480-481.

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 2001, Wiley, p. 77-79.

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..100

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

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

R. J. Hetherington, Letter to N. J. A. Sloane, Oct 26 1974

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

Tanya Khovanova, Recursive Sequences

D. H. Lehmer, On the multiple solutions of the Pell equation, Annals Math., 30 (1928), 66-72.

D. H. Lehmer, On the multiple solutions of the Pell equation (annotated scanned copy)

Mathematical Reflections, Solution to Problem O271, Issue 5, 2013, p 22.

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.

B. Polster, M. Ross, Marching in squares, arXiv preprint arXiv:1503.04658, 2015

Mark A. Shattuck, Tiling proofs of some formulas for the Pell numbers of odd index, Integers, 9 (2009), 53-64.

R. A. Sulanke, Moments, Narayana numbers and the cut and paste for lattice paths

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

FORMULA

a(n) = 2*A001109(n).

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

G.f.: 2*x/(1-6*x+x^2).

a(n) = sqrt{2*(A001541(n))^2-2}/2. - Barry E. Williams, May 07 2000

a(n) = (C^(2n) - C^(-2n))/sqrt(8) where C = sqrt(2) + 1. - Gary W. Adamson, May 11 2003

For all terms x of the sequence, 2*x^2 + 1 is a square. Lim. as n -> Inf. a(n)/a(n-1) = 3 + 2*Sqrt(2). - Gregory V. Richardson, Oct 10 2002

For n > 0: a(n) = A001652(n) + A046090(n) - A001653(n); e.g. 70 = 119 + 120 - 169. Also a(n) = A001652(n - 1) + A046090(n - 1) + A001653(n - 1); e.g. 70 = 20 + 21 + 29. Also a(n)^2 + 1 = A001653(n - 1)*A001653(n); e.g. 12^2 + 1 = 145 = 5*29. Also a(n + 1)^2 = A084703(n + 1) = A001652(n)*A001652(n + 1) + A046090(n)*A046090(n + 1). - Charlie Marion, Jul 01 2003

a(n) = ((1+sqrt(2))^(2*n)-(1-sqrt(2))^(2*n))/(2*sqrt(2)). - Antonio Alberto Olivares, Dec 24 2003

n such that Mod(sigma(2*n^2+1), 2 ) = 1. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 26 2004

2*A001541(k)*A001653(n)*A001653(n+k) = A001653(n)^2 + A001653(n+k)^2 + a2(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, Oct 12 2007

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

For n > 0, a(n) = A001653(n) + A002315(n-1). - Richard R. Forberg, Aug 31 2013

a(n) = 3*a(n-1) + 2*A001541(n-1); e.g., a(4) = 70 = 3*12+2*17. - Zak Seidov, Dec 19 2013

a(n)^2 + 1^2 = A115598(n)^2 + (A115598(n)+1)^2. - Hermann Stamm-Wilbrandt, Jul 27 2014

Sum _{n >= 1} 1/( a(n) + 1/a(n) ) = 1/2. - Peter Bala, Mar 25 2015

E.g.f.: exp(3*x)*sinh(2*sqrt(2)*x)/sqrt(2). - Ilya Gutkovskiy, Dec 07 2016

A007814(a(n)) = A001511(n). See Mathematical Reflections link. - Michel Marcus, Jan 06 2017

a(n) = -a(-n) for all n in Z. - Michael Somos, Jan 20 2017

From A.H.M. Smeets, May 28 2017: (Start)

A051009(n) = a(2^(n-2)).

a(2n) =2*a(2)*A001541(n).

A001541(n)/a(n) > sqrt(2) > 2*a(n)/A001541(n). (End)

EXAMPLE

G.f. = 2*x + 12*x^2 + 70*x^3 + 408*x^4 + 2378*x^5 + 13860*x^6 + ...

MAPLE

A001542:=2*z/(1-6*z+z**2); # conjectured by Simon Plouffe in his 1992 dissertation

MATHEMATICA

f[n_]:=IntegerQ[Sqrt[2*n^2+1]]; Select[Range[0, 2*9! ], f[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

LinearRecurrence[{6, -1}, {0, 2}, 30] (* Harvey P. Dale, Jun 11 2011 *)

PROG

(PARI) for (i=0, 10000, if(Mod(sigma(2*i^2+1), 2)==1, print1(i, ", ")))

(Haskell)

a001542 n = a001542_list !! n

a001542_list =

   0 : 2 : zipWith (-) (map (6 *) $ tail a001542_list) a001542_list

-- Reinhard Zumkeller, Aug 14 2011

(Maxima)

a[0]:0$

a[1]:2$

a[n]:=6*a[n-1]-a[n-2]$

A001542(n):=a[n]$

makelist(A001542(x), x, 0, 30); /* Martin Ettl, Nov 03 2012 */

(MAGMA) [n: n in [0..2*10^7] | IsSquare(2*n^2+1)]; // Vincenzo Librandi, Dec 07 2016

(PARI)  {a(n) = imag( (3 + 2*quadgen(8))^n )}; /* Michael Somos, Jan 20 2017 */

(Python)

l=[0, 2]

for n in xrange(2, 51): l+=[6*l[n - 1] - l[n - 2], ]

print l # Indranil Ghosh, Jun 06 2017

CROSSREFS

Bisection of A000129.

Cf. A001541, A007913, A003499, A125650, A125651, A125652, A001653, A007805, A001353, A001835, A115598, A001108.

Sequence in context: A243771 A026306 A116398 * A059229 A001251 A143357

Adjacent sequences:  A001539 A001540 A001541 * A001543 A001544 A001545

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified June 29 01:52 EDT 2017. Contains 288857 sequences.