A119016 -id:A119016 - cp's OEIS frontend

A002965 Interleave denominators (A000129) and numerators (A001333) of convergents to sqrt(2).

0, 1, 1, 1, 2, 3, 5, 7, 12, 17, 29, 41, 70, 99, 169, 239, 408, 577, 985, 1393, 2378, 3363, 5741, 8119, 13860, 19601, 33461, 47321, 80782, 114243, 195025, 275807, 470832, 665857, 1136689, 1607521, 2744210, 3880899, 6625109, 9369319, 15994428, 22619537

Offset: 0

N. J. A. Sloane

Denominators of Farey fraction approximations to sqrt(2). The fractions are 1/0, 0/1, 1/1, 2/1, 3/2, 4/3, 7/5, 10/7, 17/12, .... See A082766(n+2) or A119016 for the numerators. "Add" (meaning here to add the numerators and add the denominators, not to add the fractions) 1/0 to 1/1 to make the fraction bigger: 2/1. Now 2/1 is too big, so add 1/1 to make the fraction smaller: 3/2, 4/3. Now 4/3 is too small, so add 3/2 to make the fraction bigger: 7/5, 10/7, ... Because the continued fraction for sqrt(2) is all 2's, it will always take exactly two terms here to switch from a number that's bigger than sqrt(2) to one that's less. A097545/A097546 gives the similar sequence for Pi. A119014/A119015 gives the similar sequence for e. - Joshua Zucker, May 09 2006
The principal and intermediate convergents to 2^(1/2) begin with 1/1, 3/2 4/3, 7/5, 10/7; essentially, numerators=A143607, denominators=A002965. - Clark Kimberling, Aug 27 2008
(a(2n)*a(2n+1))^2 is a triangular square. - Hugh Darwen, Feb 23 2012
a(2n) are the interleaved values of m such that 2*m^2+1 and 2*m^2-1 are squares, respectively; a(2n+1) are the interleaved values of their corresponding integer square roots. - Richard R. Forberg, Aug 19 2013
Coefficients of (sqrt(2)+1)^n are a(2n)*sqrt(2)+a(2n+1). - John Molokach, Nov 29 2015
Apart from the first two terms, this is the sequence of denominators of the convergents of the continued fraction expansion sqrt(2) = 1/(1 - 1/(2 + 1/(1 - 1/(2 + 1/(1 - ....))))). - Peter Bala, Feb 02 2017
Limit_{n->infinity} a(2n+1)/a(2n) = sqrt(2); lim_{n->infinity} a(2n)/a(2n-1) = (2+sqrt(2))/2. - Ctibor O. Zizka, Oct 28 2018

			The convergents are rational numbers given by the recurrence relation p/q -> (p + 2*q)/(p + q). Starting with 1/1, the next three convergents are (1 + 2*1)/(1 + 1) = 3/2, (3 + 2*2)/(3 + 2) = 7/5, and (7 + 2*5)/(7 + 5) = 17/12. The sequence puts the denominator first, so a(2) through a(9) are 1, 1, 2, 3, 5, 7, 12, 17. - _Michael B. Porter_, Jul 18 2016

C. Brezinski, History of Continued Fractions and Padé Approximants. Springer-Verlag, Berlin, 1991, p. 24.
Jay Kappraff, Musical Proportions at the Basis of Systems of Architectural Proportion both Ancient and Modern, in Volume I of K. Williams and M.J. Ostwald (eds.), Architecture and Mathematics from Antiquity to the Future, DOI 10.1007/978-3-319-00143-2_27, Springer International Publishing Switzerland 2015. See Eq. 32.7.
Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Guelena Strehler, Chess Fractal, April 2016, p. 24.

T. D. Noe, Table of n, a(n) for n = 0..500
Damanvir Singh Binner, Proofs of Chappelon and Alfonsín Conjectures On Square Frobenius Numbers and its Relationship to Simultaneous Pell's Equations, arXiv:2112.15474 [math.NT], 2021.
Jonathan Chappelon and Jorge Luis Ramírez Alfonsín, The Square Frobenius Number, arXiv:2006.14219 [math.NT], 2020.
H. S. M. Coxeter, The role of intermediate convergents in Tait's explanation for phyllotaxis, J. Algebra 20 (1972), 167-175.
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
Pierre Lamothe, En marge du problème des cercles tangents
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
Dave Rusin, Farey fractions on sci.math [Broken link]
Dave Rusin, Farey fractions on sci.math [Cached copy]
K. Williams, The sacred cult revisited: the pavement of the baptistery of San Giovanni, Florence, Math. Intellig., 16 (No. 2, 1994), 18-24.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Cf. A000129(n) = a(2n), A001333(n) = a(2n+1).

Cf. A001109, A155046.

a:=[0,1];; for n in [3..45] do a[n]:=a[n-1]+a[n-2-((n-1) mod 2)]; od; a; # Muniru A Asiru, Oct 28 2018

import Data.List (transpose)
a002965 n = a002965_list !! n
a002965_list = concat $ transpose [a000129_list, a001333_list]
-- Reinhard Zumkeller, Jan 01 2014

a=new Array(); a[0]=0; a[1]=1;
for (i=2;i<50;i+=2) {a[i]=a[i-1]+a[i-2];a[i+1]=a[i]+a[i-2];}
document.write(a); // Jon Perry, Sep 12 2012

I:=[0,1,1,1]; [n le 4 select I[n] else 2*Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi, Nov 30 2015

A002965 := proc(n) option remember; if n <= 0 then 0; elif n <= 3 then 1; else 2*A002965(n-2)+A002965(n-4); fi; end;
A002965:=-(1+2*z+z**2+z**3)/(-1+2*z**2+z**4); # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for two leading terms

LinearRecurrence[{0, 2, 0, 1}, {0, 1, 1, 1}, 42] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
With[{c=Convergents[Sqrt[2],20]},Join[{0,1},Riffle[Denominator[c], Numerator[c]]]] (* Harvey P. Dale, Oct 03 2012 *)

PARI
```
a(n)=if(n<4,n>0,2*a(n-2)+a(n-4))
```

x='x+O('x^100); concat(0, Vec((x+x^2-x^3)/(1-2*x^2-x^4))) \\ Altug Alkan, Dec 04 2015

a(n) = 2*a(n-2) + a(n-4) if n>3; a(0)=0, a(1)=a(2)=a(3)=1.
a(2*n) = a(2*n-1) + a(2*n-2) and a(2*n+1) = 2*a(2*n) - a(2*n-1).
G.f.: (x+x^2-x^3)/(1-2*x^2-x^4).
a(0)=0, a(1)=1, a(n) = a(n-1) + a(2*[(n-2)/2]). - Franklin T. Adams-Watters, Jan 31 2006
For n > 0, a(2*n) = a(2*n-1) + a(2*n-2) and a(2*n+1) = a(2*n) + a(2*n-2). - Jon Perry, Sep 12 2012
a(n) = (((sqrt(2) - 2)*(-1)^n + 2 + sqrt(2))*(1 + sqrt(2))^(floor(n/2)) - ((2 + sqrt(2))*(-1)^n -2 + sqrt(2))*(1 - sqrt(2))^(floor(n/2)))/8. - Ilya Gutkovskiy, Jul 18 2016
a(n) = a(n-1) + a(n-2-(n mod 2)); a(0)=0, a(1)=1. - Ctibor O. Zizka, Oct 28 2018

Thanks to Michael Somos for several comments which improved this entry.

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.

Original entry on oeis.org

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

Offset: 0

Richard R. Forberg, Aug 21 2013

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.

			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, ...

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.

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

Main diagonal is A007070.

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.

A143607 Numerators of principal and intermediate convergents to 2^(1/2).

Original entry on oeis.org

1, 3, 4, 7, 10, 17, 24, 41, 58, 99, 140, 239, 338, 577, 816, 1393, 1970, 3363, 4756, 8119, 11482, 19601, 27720, 47321, 66922, 114243, 161564, 275807, 390050, 665857, 941664, 1607521, 2273378, 3880899, 5488420, 9369319, 13250218, 22619537, 31988856, 54608393

Offset: 1

Clark Kimberling, Aug 27 2008

Sequence is essentially A082766 (by omitting two terms A082766(0) and A082766(2)). - L. Edson Jeffery, Apr 04 2011

a(n) = A119016(n+2) for n>=2. - Georg Fischer, Oct 07 2018

			The principal and intermediate convergents to 2^(1/2) begin with 1/1, 3/2 4/3, 7/5, 10/7, ...

Serge Lang, Introduction to Diophantine Approximations, Addison-Wesley, New York, 1966.

Colin Barker, Table of n, a(n) for n = 1..1000
Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Cf. A002965 (denominators), A082766, A119016.

a:=[1,3,4,7,10];; for n in [6..40] do a[n]:=2*a[n-2]+a[n-4]; od; a; # Muniru A Asiru, Oct 07 2018

seq(coeff(series(x*(1+x)*(1+2*x+x^3)/(1-2*x^2-x^4),x,n+1), x, n), n = 1 .. 40); # Muniru A Asiru, Oct 07 2018

CoefficientList[Series[(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4), {x, 0, 50}], x] (* or *)
LinearRecurrence[{0, 2, 0, 1}, {1, 3, 4, 7, 10}, 40] (* Stefano Spezia, Oct 08 2018; signature amended by Georg Fischer, Apr 02 2019 *)

Vec(x*(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4) + O(x^60)) \\ Colin Barker, Jul 28 2017

From Colin Barker, Jul 28 2017: (Start)
G.f.: x*(1 + x)*(1 + 2*x + x^3) / (1 - 2*x^2 - x^4).
a(n) = 2*a(n-2) + a(n-4) for n>5.
(End)

A082766 Series ratios converge alternately to sqrt(2) and 1+sqrt(1/2).

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 17, 24, 41, 58, 99, 140, 239, 338, 577, 816, 1393, 1970, 3363, 4756, 8119, 11482, 19601, 27720, 47321, 66922, 114243, 161564, 275807, 390050, 665857, 941664, 1607521, 2273378, 3880899, 5488420, 9369319, 13250218, 22619537

Offset: 1

Gary W. Adamson, May 24 2003

nonn

a(2n+2)/a(2n+1) converges to sqrt(2).
a(2n+1)/a(2n) converges to 1+sqrt(1/2).
a(n+2)/a(n) converges to 1+sqrt(2).
a(2n) is A001333, a(2n+1) is A052542.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Haocong Song and Wen Wu, Hankel determinants of a Sturmian sequence, arXiv:2007.09940 [math.CO], 2020. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Cf. A001333, A052542. See A119016 for another version.

import Data.List (transpose)
a082766 n = a082766_list !! (n-1)
a082766_list = concat $ transpose [a052542_list, tail a001333_list]
-- Reinhard Zumkeller, Feb 24 2015

Rest[CoefficientList[Series[x (1 - x^2 + x) (x^2 + 1)/(1 - 2 x^2 - x^4), {x, 0, 50}], x]] (* G. C. Greubel, Nov 28 2017 *)
LinearRecurrence[{0,2,0,1},{1,1,2,3,4},50] (* Harvey P. Dale, Dec 15 2022 *)

x='x+O('x^30); Vec(x*(1+x-x^2)*(x^2+1)/(1-2*x^2-x^4)) \\ G. C. Greubel, Nov 28 2017

a(2n) = a(2n-1) + a(2n-2); a(2n+1) = a(2n) + a(2n-2)

O.g.f.: x*(1+x-x^2)*(x^2+1)/(1-2*x^2-x^4). - R. J. Mathar, Aug 08 2008

Edited by Don Reble, Nov 04 2005

A142879 a(n) = 5*a(n-3) - a(n-6) with terms 1..6 as 0, 1, 2, 5, 7, 9.

Original entry on oeis.org

0, 1, 2, 5, 7, 9, 25, 34, 43, 120, 163, 206, 575, 781, 987, 2755, 3742, 4729, 13200, 17929, 22658, 63245, 85903, 108561, 303025, 411586, 520147, 1451880, 1972027, 2492174, 6956375, 9448549, 11940723, 33329995, 45270718, 57211441, 159693600

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 28 2008

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,5,0,0,-1).

Cf. A119016, A002965, A002530, A048788.

Clear[a, n]; a[0] = 0; a[1] = 1; a[n_] := a[n] = If[Mod[n, 3] == 0, 2*a[n - 1] + a[n - 2], If[Mod[n, 3] == 1, a[n - 1] + a[n - 2], 2*a[n - 1] - a[n - 2]]]; b = Table[a[n], {n, 0, 50}]
LinearRecurrence[{0,0,5,0,0,-1},{0,1,2,5,7,9},40] (* Harvey P. Dale, Apr 06 2016 *)

a=vector(20); a[1]=1; a[2]=2; for(n=3, #a, if(n%3==0, a[n]=2*a[n-1]+a[n-2], if(n%3==1, a[n]=a[n-1]+a[n-2], a[n]=2*a[n-1]-a[n-2]))); concat(0, a) \\ Colin Barker, Jan 30 2016

concat(0, Vec(x^2*(1+2*x+5*x^2+2*x^3-x^4)/(1-5*x^3+x^6) + O(x^50))) \\ Colin Barker, Jan 30 2016

a(n) = 2*a(n - 1) + a(n - 2) if 3 | n, a(n) = a(n - 1) + a(n - 2) if n = 1 mod 3, and a(n) = 2*a(n - 1) - a(n - 2) if n = 2 mod 3.

G.f.: x^2*(1+2*x+5*x^2+2*x^3-x^4) / (1-5*x^3+x^6). - Colin Barker, Jan 08 2013

New name from Colin Barker and Charles R Greathouse IV, Jan 08 2013

A142881 a(0) = 0, a(1) = 1, after which, if n=3k: a(n) = 2a(n-1) - a(n-2), if n=3k+1: a(n) = a(n-1) + a(n-2), if n=3k+2: a(n) = 2a(n-1) + a(n-2).

Original entry on oeis.org

0, 1, 2, 3, 5, 13, 21, 34, 89, 144, 233, 610, 987, 1597, 4181, 6765, 10946, 28657, 46368, 75025, 196418, 317811, 514229, 1346269, 2178309, 3524578, 9227465, 14930352, 24157817, 63245986, 102334155, 165580141, 433494437, 701408733, 1134903170

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 28 2008

The original name of the sequence was: A modulo three switched recursion (third kind): a(n)=If[Mod[n, 3] ==2, 2*a(n - 1) + a(n - 2), If[Mod[n, 3] == 1, a(n - 1) + a(n - 2), 2*a(n - 1) - a(n - 2)]].

How is this related to A000045 ? - Antti Karttunen, Jan 29 2016

Antti Karttunen, Table of n, a(n) for n = 0..120
Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,-1).

Cf. A000045, A119016, A002965, A002530, A048788.

Clear[a, n]; a[0] = 0; a[1] = 1; a[n_] := a[n] = If[Mod[n, 3] == 2, 2*a[n - 1] + a[n - 2], If[Mod[n, 3] == 1, a[n - 1] + a[n - 2], 2*a[n - 1] - a[n - 2]]]; b = Table[a[n], {n, 0, 50}]

a=vector(100); a[1]=1; a[2]=2; for(n=3, #a, if(n%3==0, a[n]=2*a[n-1]-a[n-2], if(n%3==1, a[n]=a[n-1]+a[n-2], a[n]=2*a[n-1]+a[n-2]))); concat(0, a) \\ Colin Barker, Jan 30 2016

a(n) = If[Mod[n, 3] == 2, 2*a(n - 1) + a(n - 2), If[Mod[n, 3] == 1, a(n - 1) + a(n - 2), 2*a(n - 1) - a(n - 2)]].
a(n) = 7*a(n-3)-a(n-6). G.f.: -x^2*(x^4+2*x^3-3*x^2-2*x-1) / (x^6-7*x^3+1). [Colin Barker, Jan 08 2013]
a(0) = 0, a(1) = 1, after which, if n is a multiple of 3, a(n) = 2*a(n-1) - a(n-2), else, if n is of the form 3k+1, a(n) = a(n-1) + a(n-2), and otherwise [when n is of the form 3k+2], a(n) = 2*a(n-1) + a(n-2). - Antti Karttunen, Jan 29 2016, after the original name of the sequence.

Offset corrected and sequence edited by Antti Karttunen, Jan 29 2016

cp's OEIS Frontend

A002965 Interleave denominators (A000129) and numerators (A001333) of convergents to sqrt(2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Haskell

JavaScript

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A143607 Numerators of principal and intermediate convergents to 2^(1/2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

A082766 Series ratios converge alternately to sqrt(2) and 1+sqrt(1/2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A142879 a(n) = 5*a(n-3) - a(n-6) with terms 1..6 as 0, 1, 2, 5, 7, 9.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A142881 a(0) = 0, a(1) = 1, after which, if n=3k: a(n) = 2*a(n-1) - a(n-2), if n=3k+1: a(n) = a(n-1) + a(n-2), if n=3k+2: a(n) = 2*a(n-1) + a(n-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A142881 a(0) = 0, a(1) = 1, after which, if n=3k: a(n) = 2a(n-1) - a(n-2), if n=3k+1: a(n) = a(n-1) + a(n-2), if n=3k+2: a(n) = 2a(n-1) + a(n-2).