A227972 -id:A227972

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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)

+10
70

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 m such that A125650(6*m^2) is an even perfect square, where A124650(m) is a numerator of m*(m+3)/(4*(m+1)*(m+2)) = Sum_{k=1..m} 1/(k*(k+1)*(k+2)). Sequence A033581 is a bisection of A125651. - 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 = {a(n)}. - Clark Kimberling, Aug 26 2008

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

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

These are the integer square roots of the Half-Squares, A007590(k), which occur at values of k given by A001541. Also the numbers produced by adding m + sqrt(floor(m^2/2) + 1) when m is in 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 numerator 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 numerator as well as denominator. - A.H.M. Smeets, May 28 2017

Conjecture: Numbers k such that c/m < k for all natural a^2 + b^2 = c^2 (Pythagorean triples), a < b < c and a+b+c = m. Numbers which correspondingly minimize c/m are A002939. - Lorraine Lee, Jan 31 2020

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 < a < b < c are given by a=a(n), b=A005319(n), c=a(n+1), x=A001541(n), y=A001653(n+1), z=A002315(n) with 0 < n. - Michael Somos, Jun 26 2022

REFERENCES

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

Thomas Koshy, Fibonacci and Lucas Numbers with Applications, 2001, Wiley, pp. 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).

P.-F. Teilhet, Query 2376, L'Intermédiaire des Mathématiciens, 11 (1904), 138-139. - N. J. A. Sloane, Mar 08 2022

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.

Christian Aebi and Grant Cairns, Lattice Equable Parallelograms, arXiv:2006.07566 [math.NT], 2020.

Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

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

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, page xxxv.

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.

E. Kilic, Y. T. Ulutas, and N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, Table 3, k=1.

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; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

B. Polster and M. Ross, Marching in squares, arXiv preprint arXiv:1503.04658 [math.HO], 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.

Soumeya M. Tebtoub, Hacène Belbachir, and László Németh, Integer sequences and ellipse chains inside a hyperbola, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 17-18.

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. Limit_{n->oo} 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

2*A001541(k)*A001653(n)*A001653(n+k) = A001653(n)^2 + A001653(n+k)^2 + a(k)^2; e.g., 2*3*5*29 = 5^2 + 29^2 + 2^2; 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)

a(A298210(n)) = A002349(2*n^2). - A.H.M. Smeets, Jan 25 2018

a(n) = A000129(n)*A002203(n). - Adam Mohamed, Jul 20 2024

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

seq(combinat:-fibonacci(2*n, 2), n = 0..20); # Peter Luschny, Jun 28 2018

MATHEMATICA

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

Fibonacci[2*Range[0, 20], 2] (* G. C. Greubel, Dec 23 2019 *)

Table[2 ChebyshevU[-1 + n, 3], {n, 0, 20}] (* Herbert Kociemba, Jun 05 2022 *)

PROG

(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 */

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

(PARI) vector(21, n, 2*polchebyshev(n-1, 2, 33) ) \\ G. C. Greubel, Dec 23 2019

(Python)

l=[0, 2]

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

print(l) # Indranil Ghosh, Jun 06 2017

(Magma) I:=[0, 2]; [n le 2 select I[n] else 6*Self(n-1) -Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 23 2019

(Sage) [2*chebyshev_U(n-1, 3) for n in (0..20)] # G. C. Greubel, Dec 23 2019

(GAP) a:=[0, 2];; for n in [3..20] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 23 2019

CROSSREFS

Bisection of Pell numbers A000129: {a(n)} and A001653(n+1), n >= 0.

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

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

A228405

Pellian Array, A(n, k) with numbers m such that 2*m^2 +- 2^k is a square, and their corresponding square roots, read downward by diagonals.

+10
13

0, 1, 1, 0, 1, 2, 2, 2, 3, 5, 0, 2, 4, 7, 12, 4, 4, 6, 10, 17, 29, 0, 4, 8, 14, 24, 41, 70, 8, 8, 12, 20, 34, 58, 99, 169, 0, 8, 16, 28, 48, 82, 140, 239, 408, 16, 16, 24, 40, 68, 116, 198, 338, 577, 985, 0, 16, 32, 56, 96, 164, 280, 478, 816, 1393, 2378

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

The left column, A(n,0), is A000129(n), Pell Numbers.

The top row, A(0,k), is A077957(k) plus an initial 0, which is the inverse binomial transform of A000129.

These may be considered initializing values, or results, depending the perspective taken, since there are several ways to generate the array. See Formula section for details.

The columns of the array hold all values, in sequential order, of numbers m such that 2m^2 + 2^k or 2m^2 - 2^k are squares, and their corresponding square roots in the next column, which then form the "next round" of m values for k+1.

For example A(n,0) are numbers such that 2m^2 +- 1 are squares, the integer square roots of each are in A(n,1), which are then numbers m such that 2m^2 +- 2 are squares, with those square roots in A(n,2), etc.

A(n, k)/A(n,k-2) = 2; A(n,k)/A(n,k-1) converges to sqrt(2) for large n.

A(n,k)/A(n-1,k) converges to 1 + sqrt(2) for large n.

The other columns of this array hold current OEIS sequences as follows:

A(n,1) = A001333(n); A(n,2) = A163271(n); A(n,3) = A002203(n);

Bisections of these column-oriented sequences also appear in the OEIS, corresponding to the even and odd rows of the array, which in turn correspond to the two different recursive square root equations in the formula section below.

Farey fraction denominators interleave columns 0 and 1, and the corresponding numerators interleave columns 1 and 2, for approximating sqrt(2). See A002965 and A119016, respectively.

The other rows of this array hold current OEIS sequences as follows:

A(1,k) = A016116(k); A(2,k) = A029744(k) less the initial 1;

A(3,k) = A070875(k); A(4,k) = A091523(k) less the initial 8.

The Pell Numbers (A000219) are the only initializing set of numbers where the two recursive square root equations (see below) produce exclusively integer values, for all iterations of k. For any other initial values only even iterations (at k = 2, 4, ...) produce integers.

The numbers in this array, especially the first three columns, are also integer square roots of these expressions: floor(m^2/2), floor(m^2/2 + 1), floor (m^2/2 - 1). See A227972 for specific arrangements and relationships. Also: ceiling(m^2/2), ceiling(m^2/2 + 1), ceiling (m^2/2 -1), m^2+1, m^2-1, m^2*(m^2-1)/2, m^2*(m^2-1)/2, in various different arrangements. Many of these involve: A000129(2n)/2 = A001109(n).

A001109 also appears when multiplying adjacent columns: A(n,k) * A(n,k+1) = (k+1) * A001109(n), for all k.

LINKS

Seiichi Manyama, Antidiagonals n = 0..139, flattened

MacTutor, D'Arcy Thompson on Greek irrationals

D'Arcy Thompson, Excess and Defect: Or the Little More and the Little Less, Mind, New Series, Vol. 38, No. 149 (Jan., 1929), pp. 43-55 (13 pages). See page 50.

FORMULA

If using the left column and top row to initialize: A(n,k) = A(n,k-1) + A(n-1,k-1).

If using only the top row to initialize, then each column for k = i is the binomial transform of A(0,k) restricted to k=> i as input to the transform with an appropriate down shift of index. The inverse binomial transform with a similar condition can produce each row from A000129.

If using only the first two rows to initialize then the Pell equation produces each column, as: A(n,k) = 2*A(n-1, k) + A(n-2, k).

If using only the left column (A000219(n) = Pell Numbers) to initialize then the following two equations will produce each row:

A(n,k) = sqrt(2*A(n,k-1) + (-2)^(k-1)) for even rows

A(n,k) = sqrt(2*A(n,k-1) - (-2)^(k-1)) for odd rows.

Interestingly, any portion of the array can also be filled "backwards" given the top row and any column k, using only: A(n,k-1) = A(n-1,k-1) + A(n-1, k), or if given any column and its column number by rearranging the sqrt recursions above.

EXAMPLE

With row # as n. and column # as k, and n, k =>0, the array begins:

0, 1, 0, 2, 0, 4, 0, 8, ...

1, 1, 2, 2, 4, 4, 8, 8, ...

2, 3, 4, 6, 8, 12, 16, 24, ...

5, 7, 10, 14, 20, 28, 40, 56, ...

12, 17, 24, 34, 48, 68, 96, 136, ...

29, 41, 58, 82, 116, 164, 232, 328, ...

70, 99, 140, 198, 280, 396, 560, 792, ...

169, 239, 338, 478, 676, 956, 1352, 1912, ...

408, 577, 816, 1154, 1632, 2308, 3264, 4616, ...

CROSSREFS

Cf. A000219, A077957, A001333, A163271, A002203, A001109, A227972.

Main diagonal is A007070.

KEYWORD

nonn,tabl,look

AUTHOR

Richard R. Forberg, Aug 21 2013

STATUS

approved

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