A083332 -id:A083332 - cp's OEIS frontend

A002531 a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 2a(2n) + a(2*n-1); a(0) = a(1) = 1.

1, 1, 2, 5, 7, 19, 26, 71, 97, 265, 362, 989, 1351, 3691, 5042, 13775, 18817, 51409, 70226, 191861, 262087, 716035, 978122, 2672279, 3650401, 9973081, 13623482, 37220045, 50843527, 138907099, 189750626, 518408351, 708158977, 1934726305

Offset: 0

N. J. A. Sloane

Numerators of continued fraction convergents to sqrt(3), for n >= 1.
For the denominators see A002530.
Consider the mapping f(a/b) = (a + 3*b)/(a + b). Taking a = b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the convergents 1/1, 2/1, 5/3, 7/4, 19/11, ... converging to sqrt(3). Sequence contains the numerators. - Amarnath Murthy, Mar 22 2003
In the Murthy comment if we take a = 0, b = 1 then the denominator of the reduced fraction is a(n+1). A083336(n)/a(n+1) converges to sqrt(3). - Mario Catalani (mario.catalani(AT)unito.it), Apr 26 2003
If signs are disregarded, all terms of A002316 appear to be elements of this sequence. - Creighton Dement, Jun 11 2007
2^(-floor(n/2))*(1 + sqrt(3))^n = a(n) + A002530(n)*sqrt(3); integers in the real quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 10 2018
Let T(n) = A000034(n), U(n) = A002530(n), V(n) = a(n), x(n) = U(n)/V(n). Then T(n*m) * U(n+m) = U(n)*V(m) + U(m)*V(n), T(n*m) * V(n+m) = 3*U(n)*U(m) + V(m)*V(n), x(n+m) = (x(n) + x(m))/(1 + 3*x(n)*x(m)). - Michael Somos, Nov 29 2022

			1 + 1/(1 + 1/(2 + 1/(1 + 1/2))) = 19/11 so a(5) = 19.
Convergents are 1, 2, 5/3, 7/4, 19/11, 26/15, 71/41, 97/56, 265/153, 362/209, 989/571, 1351/780, 3691/2131, ... = A002531/A002530.
G.f. = 1 + x + 2*x^2 + 5*x^3 + 7*x^4 + 19*x^5 + 26*x^6 + 71*x^7 + ... - _Michael Somos_, Mar 22 2022

I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 181.
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).
A. Tarn, Approximations to certain square roots and the series of numbers connected therewith, Mathematical Questions and Solutions from the Educational Times, 1 (1916), 8-12.

Harry J. Smith, Table of n, a(n) for n = 0..2000
MacTutor, D'Arcy Thompson on Greek irrationals
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
Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]
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 48.
Hein van Winkel, Q-quadrangles inscribed in a circle, 2014. See Table 1. [Reference from Antreas Hatzipolakis, Jul 14 2014]
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).
Index entries for sequences related to Chebyshev polynomials.

Bisections are A001075 and A001834.
Cf. A002530 (denominators), A048788.
Cf. A002316.
Cf. A083332, A199710, A026150, A053120.

a:=[1,1,2,5];; for n in [5..40] do a[n]:=4*a[n-2]-a[n-4]; od; a; # G. C. Greubel, Nov 16 2018

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1 +x-2*x^2+x^3)/(1-4*x^2+x^4))); // G. C. Greubel, Nov 16 2018

A002531 := proc(n) option remember; if n=0 then 0 elif n=1 then 1 elif n=2 then 1 elif type(n,odd) then A002531(n-1)+A002531(n-2) else 2*A002531(n-1)+A002531(n-2) fi; end; [ seq(A002531(n), n=0..50) ];
with(numtheory): tp := cfrac (tan(Pi/3),100): seq(nthnumer(tp,i), i=-1..32 ); # Zerinvary Lajos, Feb 07 2007
A002531:=(1+z-2*z**2+z**3)/(1-4*z**2+z**4); # Simon Plouffe; see his 1992 dissertation

Insert[Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[3], n]]], {n, 1, 40}], 1, 1] (* Stefan Steinerberger, Apr 01 2006 *)
Join[{1},Numerator[Convergents[Sqrt[3],40]]] (* Harvey P. Dale, Jan 23 2012 *)
CoefficientList[Series[(1 + x - 2 x^2 + x^3)/(1 - 4 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 01 2014 *)
LinearRecurrence[{0, 4, 0, -1}, {1, 1, 2, 5}, 35] (* Robert G. Wilson v, Feb 11 2018 *)
a[ n_] := ChebyshevT[n, Sqrt[-1/2]]*Sqrt[2]^Mod[n,2]/I^n //Simplify; (* Michael Somos, Mar 22 2022 *)
a[ n_] := If[n<0, (-1)^n*a[-n], SeriesCoefficient[ (1 + x - 2*x^2 + x^3) / (1 - 4*x^2 + x^4), {x, 0, n}]]; (* Michael Somos, Sep 23 2024 *)

a(n)=contfracpnqn(vector(n,i,1+(i>1)*(i%2)))[1,1]

apply( {A002531(n,w=quadgen(12))=real((2+w)^(n\/2)*if(bittest(n, 0), w-1, 1))}, [0..30]) \\ M. F. Hasler, Nov 04 2019

{a(n) = if(n<0, (-1)^n*a(-n), polcoeff( (1 + x - 2*x^2 + x^3) / (1 - 4*x^2 + x^4) + x*O(x^n), n))}; /* Michael Somos, Sep 23 2024 */

s=((1+x-2*x^2+x^3)/(1-4*x^2+x^4)).series(x,40); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 16 2018

G.f.: (1 + x - 2*x^2 + x^3)/(1 - 4*x^2 + x^4).
a(2*n) = a(2*n-1) + a(2*n-2), a(2*n+1) = 2*a(2*n) + a(2*n-1), n > 0.
a(2*n) = (1/2)*((2 + sqrt(3))^n+(2 - sqrt(3))^n); a(2*n) = A003500(n)/2; a(2*n+1) = round(1/(1 + sqrt(3))*(2 + sqrt(3))^n). - Benoit Cloitre, Dec 15 2002
a(n) = ((1 + sqrt(3))^n + (1 - sqrt(3))^n)/(2*2^floor(n/2)). - Bruno Berselli, Nov 10 2011
a(n) = A080040(n)/(2*2^floor(n/2)). - Ralf Stephan, Sep 08 2013
a(2*n) = (-1)^n*T(2*n,u) and a(2*n+1) = (-1)^n*1/u*T(2*n+1,u), where u = sqrt(-1/2) and T(n,x) denotes the Chebyshev polynomial of the first kind. - Peter Bala, May 01 2012
a(n) = (-sqrt(2)*i)^n*T(n, sqrt(2)*i/2)*2^(-floor(n/2)) = A026150(n)*2^(-floor(n/2)), n >= 0, with i = sqrt(-1) and the Chebyshev T polynomials (A053120). - Wolfdieter Lang, Feb 10 2018
From Franck Maminirina Ramaharo, Nov 14 2018: (Start)
a(n) = ((1 - sqrt(2))*(-1)^n + 1 + sqrt(2))*(((sqrt(2) - sqrt(6))/2)^n + ((sqrt(6) + sqrt(2))/2)^n)/4.
E.g.f.: cosh(sqrt(3/2)*x)*(sqrt(2)*sinh(x/sqrt(2)) + cosh(x/sqrt(2))). (End)
a(n) = (-1)^n*a(-n) for all n in Z. - Michael Somos, Mar 22 2022
a(n) = 4*a(n-2) - a(n-4). - Boštjan Gec, Sep 21 2023

Name edited (as by discussion in A002530) by M. F. Hasler, Nov 04 2019

A147590 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n-1 digits 0.

Original entry on oeis.org

1, 14, 124, 1016, 8176, 65504, 524224, 4194176, 33554176, 268434944, 2147482624, 17179867136, 137438949376, 1099511619584, 8796093005824, 70368744144896, 562949953355776, 4503599627239424, 36028797018701824, 288230376151187456

Offset: 1

Omar E. Pol, Nov 08 2008

a(n) is the number whose binary representation is A147589(n).

			     1_10 is 1_2;
    14_10 is 1110_2;
   124_10 is 1111100_2;
  1016_10 is 1111111000_2.

Nathaniel Johnston, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A020540, A138118, A147537, A147589.

List([1..25], n-> 2^(n-2)*(4^n-2)); # G. C. Greubel, Jul 27 2019

[8^n/4-2^(n-1): n in [1..25]]; // Vincenzo Librandi, Jul 27 2019

seq(8^n/4-2^(n-1),n=1..25); # Nathaniel Johnston, Apr 30 2011

LinearRecurrence[{10,-16},{1,14},30] (* Harvey P. Dale, Oct 10 2014 *)
Table[8^n / 4 - 2^(n - 1), {n, 25}] (* Vincenzo Librandi, Jul 27 2019 *)

vector(25, n, 2^(n-2)*(4^n-2)) \\ G. C. Greubel, Jul 27 2019

[2^(n-2)*(4^n-2) for n in (1..25)] # G. C. Greubel, Jul 27 2019

a(n) = A147537(n)/2.
From R. J. Mathar, Jul 13 2009: (Start)
a(n) = 8^n/4 - 2^(n-1) = A083332(2n-2).
a(n) = 10*a(n-1) - 16*a(n-2).
G.f.: x*(1+4*x)/((1-2*x)*(1-8*x)). (End)
From César Aguilera, Jul 26 2019: (Start)
Lim_{n->infinity} a(n)/a(n-1) = 8;
a(n)/a(n-1) = 8 + 6/A083420(n). (End)
E.g.f.: (1/4)*(exp(2*x)*(-2 + exp(6*x)) + 1). - Stefano Spezia, Aug 05 2019
a(n) = A020540(n - 1)/4. - Jon Maiga, Aug 05 2019

More terms from R. J. Mathar, Jul 13 2009

Typo in a(12) corrected by Omar E. Pol, Jul 20 2009

A199710 Expansion of (1+x-14x^2+13x^3)/(1-28x^2+169x^4).

Original entry on oeis.org

1, 1, 14, 41, 223, 979, 3878, 20483, 70897, 408073, 1329734, 7964417, 25250959, 154039339, 482301806, 2967115019, 9237038497, 57046572241, 177128072702, 1095861584537, 3398526529663, 21043253658307, 65224098543926, 404010494645843, 1251923775716881

Offset: 0

Bruno Berselli, Nov 10 2011

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,28,0,-169).

Cf. A002531, A083332.

I:=[1,1,14,41]; [n le 4 select I[n] else 28*Self(n-2)-169*Self(n-4): n in [1..25]];

LinearRecurrence[{0, 28, 0, -169}, {1, 1, 14, 41}, 25]
CoefficientList[Series[(1+x-14x^2+13x^3)/(1-28x^2+169x^4),{x,0,30}],x] (* Harvey P. Dale, Nov 08 2017 *)

makelist(expand(((1+3*sqrt(3))^n+(1-3*sqrt(3))^n)/(2*2^floor(n/2))),n,0,24);

Vec((1+x-14*x^2+13*x^3)/(1-28*x^2+169*x^4)+O(x^25))

G.f.: (1+x-14*x^2+13*x^3)/(1-28*x^2+169*x^4).
a(n) = ((1+3*sqrt(3))^n+(1-3*sqrt(3))^n)/(2*2^floor(n/2)).
a(n) = 28*a(n-2)-169*a(n-4).

A083333 a(n) = 10a(n-2) - 16a(n-4) for n>=4, with a(0)=a(1)=1, a(2)=6, a(3)=10.

Original entry on oeis.org

1, 1, 6, 10, 44, 84, 344, 680, 2736, 5456, 21856, 43680, 174784, 349504, 1398144, 2796160, 11184896, 22369536, 89478656, 178956800, 715828224, 1431655424, 5726623744, 11453245440, 45812985856, 91625967616, 366503878656

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-16).

Cf. A016131, A082412 (bisections).

Cf. A001045, A016116, A083332.

I:=[1,1,6,10]; [n le 4 select I[n] else 10*Self(n-2) -16*Self(n-4): n in [1..41]]; // G. C. Greubel, Dec 27 2024

CoefficientList[Series[(1+x-4x^2)/(1-10x^2+16x^4), {x, 0, 30}], x]
LinearRecurrence[{0,10,0,-16},{1,1,6,10},30] (* Harvey P. Dale, Aug 04 2024 *)

def A083333(n): return 2^((n-1)/2)*( (n%2)*(2^(n+1) -1) + ((n+1)%2)*sqrt(2)*(2^(n+1) +1))/3
print([A083333(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

G.f.: (1+x-4*x^2)/(1-10*x^2+16*x^4).
Limit_{n -> oo} A083332(n)/a(n) = 3.
a(n) = A001045(n+1)*A016116(n). - R. J. Mathar, Jul 08 2009
From G. C. Greubel, Dec 27 2024: (Start)
a(n) = (1/3)*2^((n-3)/2)*( (1-(-1)^n)*(2^(n+1) - 1) + (1+(-1)^n)*sqrt(2)*(2^(n+1) + 1) ).
E.g.f.: (1/3)*(2*cosh(2*sqrt(2)*x) + cosh(sqrt(2)*x)) + (1/(3*sqrt(2)))*(2*sinh(2*sqrt(2)*x) - sinh(sqrt(2)*x)). (End)

cp's OEIS Frontend

A002531 a(2*n) = a(2*n-1) + a(2*n-2), a(2*n+1) = 2*a(2*n) + a(2*n-1); a(0) = a(1) = 1.

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A147590 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n-1 digits 0.

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A199710 Expansion of (1+x-14*x^2+13*x^3)/(1-28*x^2+169*x^4).

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A083333 a(n) = 10*a(n-2) - 16*a(n-4) for n>=4, with a(0)=a(1)=1, a(2)=6, a(3)=10.

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A002531 a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 2a(2n) + a(2*n-1); a(0) = a(1) = 1.

A199710 Expansion of (1+x-14x^2+13x^3)/(1-28x^2+169x^4).

A083333 a(n) = 10a(n-2) - 16a(n-4) for n>=4, with a(0)=a(1)=1, a(2)=6, a(3)=10.