A002965 -id:A002965 - cp's OEIS frontend

A278928 Decimal expansion of sqrt(sqrt(2) + 1).

1, 5, 5, 3, 7, 7, 3, 9, 7, 4, 0, 3, 0, 0, 3, 7, 3, 0, 7, 3, 4, 4, 1, 5, 8, 9, 5, 3, 0, 6, 3, 1, 4, 6, 9, 4, 8, 1, 6, 4, 5, 8, 3, 4, 9, 9, 4, 1, 0, 3, 0, 7, 8, 3, 6, 3, 3, 2, 6, 7, 1, 1, 4, 8, 3, 3, 3, 6, 7, 5, 2, 5, 6, 7, 8, 8, 7, 3, 3, 1, 0, 2, 7, 2, 7, 9

Offset: 1

Bobby Jacobs, Dec 01 2016

A quartic integer with minimal polynomial x^4 - 2*x^2 - 1. - Charles R Greathouse IV, Dec 01 2016
Suppose f(n) has the recurrence f(2*n) = f(2*n - 1) + f(2*n - 2) and f(2*n + 1) = f(2*n) + f(2*n - 2), where f(0) and f(1) are not both 0. Then, lim_{n -> oo} f(n)^(1/n) is this constant.
Apart from the first digit, the same as A190283. - R. J. Mathar, Dec 09 2016
Imaginary part of sqrt(1 + i)^3, where i is the imaginary unit such that i^2 = -1. See A154747 for real part. - Alonso del Arte, Sep 09 2019

			1.553773974030037307344158953063146948164583499410307836332671...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 7.4, p. 466.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 4.

Cf. A002965, A014176, A154747, A190283, A245592.

Cf. A309948 and A309949 for real and imaginary parts of sqrt(1 + i).

Sqrt(1+Sqrt(2)); // G. C. Greubel, Apr 14 2018

Digits:=100: evalf(sqrt(sqrt(2)+1)); # Wesley Ivan Hurt, Dec 01 2016

RealDigits[Sqrt[Sqrt[2] + 1], 10, 100][[1]] (* Wesley Ivan Hurt, Dec 01 2016 *)

sqrt(sqrt(2)+1) \\ Charles R Greathouse IV, Dec 01 2016

polrootsreal(x^4 - 2*x^2 - 1)[2] \\ Charles R Greathouse IV, Dec 01 2016

Equals 1/A154747.
Limit_{n -> oo} A002965(n)^(1/n).
From Peter Bala, Jul 01 2024: (Start)
This constant occurs in the evaluation of Integral_{x = 0..Pi/2} 1/(1 + sin^4(x)) dx = Pi/4 * sqrt(sqrt(2) + 1).
Equals 2*Sum_{n >= 0} (-1/16)^n * binomial(4*n, 2*n) (a slowly converging series). (End)
Equals 2^(3/4)*cos(Pi/8). - Vaclav Kotesovec, Jul 01 2024
Equals Product_{k>=0} coth(Pi/4 + k*Pi/2). - Antonio Graciá Llorente, Dec 19 2024
Equals sqrt(A014176) = 1/A154747 = exp(A245592). - Hugo Pfoertner, Dec 19 2024

A331101 Denominators of the best approximations for sqrt(2).

Original entry on oeis.org

1, 2, 3, 5, 12, 17, 29, 70, 99, 169, 408, 577, 985, 2378, 3363, 5741, 13860, 19601, 33461, 80782, 114243, 195025, 470832, 665857, 1136689, 2744210, 3880899, 6625109, 15994428, 22619537, 38613965, 93222358, 131836323, 225058681, 543339720, 768398401, 1311738121, 3166815962

Offset: 1

Gerhard Kirchner, Jan 09 2020

For numerators, see A331115.
Let w = sqrt(2). Each of the principal convergents 1/1, 3/2, 7/5, 17/12, ..., see A002965, represents a best approximation for w because no other fraction with a smaller denominator is closer to w.
However, with abs(3/2 - w) > abs(4/3 - w)> abs(7/5 - w), 4/3 is another best approximation which has to be inserted. Generally, after each principal convergent p/q, we must insert the correspondent intermediate convergent 2q/p = (2/1), 4/3, (10/7), 24/17, ..., if it is closer to w than p/q (terms without brackets).
It is a well-known fact that the geometric mean sqrt(a*b) of two factors a and b is closer to the smaller one. As w is the geometric mean of p/q and 2q/p, the second term is inserted if it is smaller than p/q. This applies to every second intermediate convergent because the principal convergents alternately undershoot and overshoot w.

			The fractions are 1/1, 3/2, 4/3, 7/5, 17/12, 24/17, ...
Let w = sqrt(2) again. The first four principal convergents are, see comments, 1/1 (which is less than w), 3/2 (greater than w), 7/5 (less than w), 17/12 (greater than w). After 3/2, the fraction 2 * 2/3 = 4/3 is inserted because 4/3 < 3/2 and therefore w - 4/3 < 3/2 - w (0.081... < 0.085...). After 7/5, the fraction 2 * 5/7 = 10/7 is not inserted, because 10/7 > 7/5 etc.

Gerhard Kirchner, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A002965, A331115, A001333, A143607, A000129.

Vec(x*(1 + 2*x + 3*x^2 - x^3 - x^5) / (1 - 6*x^3 + x^6) + O(x^40)) \\ Colin Barker, Jan 09 2020

If n mod 3 = 2: a(n) = 3*a(n - 1) - a(n - 2), otherwise: a(n) = a(n - 1) + a(n - 2), with a(1) = 1, a(2) = 2.
a(3n - 2) = w/4*D(2n - 1), a(3n - 1) = w/4*D(2n), a(3n) = 1/2*S(2n), for n>0 with w = sqrt(2) and S(n) = (1 + w)^n + (1 - w)^n and D(n) = (1 + w)^n - (1 - w)^n.
From Colin Barker, Jan 09 2020: (Start)
G.f.: x*(1 + 2*x + 3*x^2 - x^3 - x^5) / (1 - 6*x^3 + x^6).
a(n) = 6*a(n - 3) - a(n - 6) for n > 6.
(End)

A331115 Numerators of the best approximations for sqrt(2).

Original entry on oeis.org

1, 3, 4, 7, 17, 24, 41, 99, 140, 239, 577, 816, 1393, 3363, 4756, 8119, 19601, 27720, 47321, 114243, 161564, 275807, 665857, 941664, 1607521, 3880899, 5488420, 9369319, 22619537, 31988856, 54608393, 131836323, 186444716, 318281039, 768398401, 1086679440, 1855077841, 4478554083, 6333631924

Offset: 1

Gerhard Kirchner, Jan 10 2020

Every principal convergent, see A002965, and every second intermediate convergent is a best approximation for sqrt(2). The numerators of these convergents are the terms of the current sequence. For denominators and more details, see A331101.

			The principal convergents are 1/1, 3/2, 7/5, 17/12, ... and 1,3,7,17,... the corresponding numerators, see A001333. Intermediate convergents: (2/1), 4/3, (10/7), 24/17, ... (best approximations without brackets). Numerators: 4,24,... (subsequence of A143607). All these numerators sorted: 1,3,4,7,17,24,...

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A002965, A331101, A001333, A143607, A000129.

LinearRecurrence[{0,0,6,0,0,-1},{1,3,4,7,17,24},50] (* Harvey P. Dale, Nov 26 2024 *)

Vec(x*(1 + x)*(1 + 2*x + 2*x^2 - x^3) / (1 - 6*x^3 + x^6) + O(x^40)) \\ Colin Barker, Jan 10 2020

If n mod 3 = 2: a(n) = 3*a(n-1) - a(n-2), otherwise: a(n) = a(n-1) + a(n-2), for n>2 with a(1)=1, a(2)=3.
a(3n-2) = 1/2*S(2n-1), a(3n-1) = 1/2*S(2n), a(3n) = w/2*D(2n), for n>0 with w = sqrt(2) and S(n) = (1+w)^n + (1-w)^n and D(n) = (1+w)^n - (1-w)^n.
From Colin Barker, Jan 10 2020: (Start)
G.f.: x*(1 + x)*(1 + 2*x + 2*x^2 - x^3) / (1 - 6*x^3 + x^6).
a(n) = 6*a(n-3) - a(n-6) for n>6.
(End)

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

A142880 a(n) = 7*a(n-3) - a(n-6).

Original entry on oeis.org

0, 1, 2, 3, 8, 13, 21, 55, 89, 144, 377, 610, 987, 2584, 4181, 6765, 17711, 28657, 46368, 121393, 196418, 317811, 832040, 1346269, 2178309, 5702887, 9227465, 14930352, 39088169, 63245986, 102334155, 267914296, 433494437, 701408733, 1836311903

Offset: 0

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

Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,-1).

Cf. A119016, A002965, A002530, A048788.

a[0] = 0; a[1] = 1;
a[n_] := a[n] = If[Mod[n, 3] == 1, 2*a[n - 1] + a[n - 2], If[Mod[n, 3] == 0, a[n - 1] + a[n - 2], 2*a[n - 1] - a[n - 2]]];
Table[a[n], {n, 0, 50}]
LinearRecurrence[{0,0,7,0,0,-1},{0,1,2,3,8,13},40] (* Harvey P. Dale, Jul 17 2021 *)

G.f.: -x*(1+x)*(x^3 - 2*x^2 - x - 1) / ( 1 - 7*x^3 + x^6 ).
a(3n) = A033888(n).
a(3n+1) = A033890(n).
a(3n+2)= A033891(n).
a(n) = 2*a(n-1) + a(n-2) if n == 1 (mod 3).
a(n) = a(n-1) + a(n-2) if n == 0 (mod 3).
a(n) = 2*a(n-1) - a(n-2) if n == 2 (mod 3).

A160572 Elements of A160444, pairs of consecutive entries swapped.

Original entry on oeis.org

1, 0, 1, 1, 4, 2, 10, 6, 28, 16, 76, 44, 208, 120, 568, 328, 1552, 896, 4240, 2448, 11584, 6688, 31648, 18272, 86464, 49920, 236224, 136384, 645376, 372608, 1763200, 1017984, 4817152, 2781184, 13160704, 7598336, 35955712, 20759040, 98232832

Offset: 1

Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 20 2009

The case k=3 of a family of sequences defined by a(1)=1, a(2)=0, a(2n+1)=a(2n-1)+k*a(2n), a(2n+2)=a(2n)+a(2n-1), each congruent to one of the sequences mentioned in A160444 by pairwise interchanges. The case k=2 is covered by swapping pairs in A002965.
Each of the two subsequences b(n) obtained by bisection has a limiting ratio b(n+1)/b(n)=1+sqrt(k) by Binet's Formula. In a logarithmic plot of the sequence a(n) one therefore sees a staircase, the two edges at each step alternately marked by one of the two subsequences.
Matrix M = [[1 3] [1 1]] is iterated with starting vector [1 0]^T. Since M has eigenvectors [+-sqrt(3) 1]^T with eigenvalues 1 +- sqrt(3), we have lim xn/yn = 1+sqrt(3) for all nonzero integer starting vectors. - Hagen von Eitzen, May 22 2009

			k=2: 1,0,1,1,3,2,7,5,17,12,41,29,99,70,239,169,577,408,1393,985
k=3: 1,0,1,1,4,2,10,6,28,16,76,44,208,120,568,328,1552... (here)
k=4: 1,0,1,1,5,2,13,7,41,20,121,61,365,182,1093,547,3281,..
k=5: 1,0,1,1,6,2,16,8,56,24,176,80,576,256,1856,832,6016,2688,..
k=6: 1,0,1,1,7,2,19,9,73,28,241,101,847,342,2899,1189,..
k=7: 1,0,1,1,8,2,22,10,92,32,316,124,1184,440,4264,1624,..
k=8: 1,0,1,1,9,2,25,11,113,36,401,149,1593,550,5993,2143,..
k=9: 1,0,1,1,10,2,28,12,136,40,496,176,2080,672,8128,2752,..
k=10: 1,0,1,1,11,2,31,13,161,44,601,205,2651,806,10711,3457,..

W. Limbrunner, Das Quadrat, ein Wunder der Geometrie (in German)
Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, 2).

Cf. A160444, A002605 (bisection), A026150 (bisection).

a(2*n+1)=A160444(2*n+2). a(2*n+2)=A160444(2*n+1).
G.f.: -x*(1-x^2+x^3)/(-1+2*x^2+(k-1)*x^4). a(n)=2*a(n-2)+(k-1)*a(n-4) at k=3. - R. J. Mathar, May 22 2009
a(1)=1, a(2)=0, and for n>=1: a(2*n+1) = a(2*n-1)+3*a(2*n), a(2*n+2) = a(2*n+1)+a(2*n). Or: Let c1 = 1+sqrt(3), c2 = 1-sqrt(3). Then a(2*n+1) = (c1^n + c2^n)/2, a(2*n+2) = (c1^n - c2^n)/(2*sqrt(3)) for n >= 0. - Hagen von Eitzen, May 22 2009

Edited by R. J. Mathar, May 22 2009

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A278928 Decimal expansion of sqrt(sqrt(2) + 1).

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A331101 Denominators of the best approximations for sqrt(2).

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A331115 Numerators of the best approximations for sqrt(2).

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

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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).

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A142880 a(n) = 7*a(n-3) - a(n-6).

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A160572 Elements of A160444, pairs of consecutive entries swapped.

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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).