A010541 -id:A010541 - cp's OEIS frontend

A024036 a(n) = 4^n - 1.

0, 3, 15, 63, 255, 1023, 4095, 16383, 65535, 262143, 1048575, 4194303, 16777215, 67108863, 268435455, 1073741823, 4294967295, 17179869183, 68719476735, 274877906943, 1099511627775, 4398046511103, 17592186044415, 70368744177663, 281474976710655

Offset: 0

N. J. A. Sloane

This sequence is the normalized length per iteration of the space-filling Peano-Hilbert curve. The curve remains in a square, but its length increases without bound. The length of the curve, after n iterations in a unit square, is a(n)*2^(-n) where a(n) = 4*a(n-1)+3. This is the sequence of a(n) values. a(n)*(2^(-n)*2^(-n)) tends to 1, the area of the square where the curve is generated, as n increases. The ratio between the number of segments of the curve at the n-th iteration (A015521) and a(n) tends to 4/5 as n increases. - Giorgio Balzarotti, Mar 16 2006
Numbers whose base-4 representation is 333....3. - Zerinvary Lajos, Feb 03 2007
From Eric Desbiaux, Jun 28 2009: (Start)
It appears that for a given area, a square n^2 can be divided into n^2+1 other squares.
It's a rotation and zoom out of a Cartesian plan, which creates squares with side
= sqrt( (n^2) / (n^2+1) ) --> A010503|A010532|A010541... --> limit 1,
and diagonal sqrt(2*sqrt((n^2)/(n^2+1))) --> A010767|... --> limit A002193.
(End)
Also the total number of line segments after the n-th stage in the H tree, if 4^(n-1) H's are added at the n-th stage to the structure in which every "H" is formed by 3 line segments. A164346 (the first differences of this sequence) gives the number of line segments added at the n-th stage. - Omar E. Pol, Feb 16 2013
a(n) is the cumulative number of segment deletions in a Koch snowflake after (n+1) iterations. - Ivan N. Ianakiev, Nov 22 2013
Inverse binomial transform of A005057. - Wesley Ivan Hurt, Apr 04 2014
For n > 0, a(n) is one-third the partial sums of A002063(n-1). - J. M. Bergot, May 23 2014
Also the cyclomatic number of the n-Sierpinski tetrahedron graph. - Eric W. Weisstein, Sep 18 2017

			G.f. = 3*x + 15*x^2 + 63*x^3 + 255*x^4 + 1023*x^5 + 4095*x^6 + ...

Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Felix Fröhlich, Table of n, a(n) for n = 0..99
Alexander V. Kitaev, Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 15 (2019), 046, 53 pages; arXiv preprint, arXiv:1809.00122 [math.CA], 2018-2019.
Eric Weisstein's World of Mathematics, Cyclomatic Number.
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A000051, A000120, A000225, A000302, A002001, A002063, A002193, A002450, A005057, A010503, A010532, A010541, A010767, A015521, A020988, A027637 (partial products), A078904 (partial sums), A079978, A080674, A164346 (first differences), A178789, A179857, A248721.

a024036 = (subtract 1) . a000302
a024036_list = iterate ((+ 3) . (* 4)) 0
-- Reinhard Zumkeller, Oct 03 2012

A024036:=n->4^n-1; seq(A024036(n), n=0..30); # Wesley Ivan Hurt, Apr 04 2014

Array[4^# - 1 &, 50, 0] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
(* Start from Eric W. Weisstein, Sep 19 2017 *)
Table[4^n - 1, {n, 0, 20}]
4^Range[0, 20] - 1
LinearRecurrence[{5, -4}, {0, 3}, 20]
CoefficientList[Series[3 x/(1 - 5 x + 4 x^2), {x, 0, 20}], x]
(* End *)

for(n=0, 100, print1(4^n-1, ", ")) \\ Felix Fröhlich, Jul 04 2014

[gaussian_binomial(2*n,1, 2) for n in range(21)] # Zerinvary Lajos, May 28 2009

[stirling_number2(2*n+1, 2) for n in range(21)] # Zerinvary Lajos, Nov 26 2009

a(n) = 3*A002450(n). - N. J. A. Sloane, Feb 19 2004
G.f.: 3*x/((-1+x)*(-1+4*x)) = 1/(-1+x) - 1/(-1+4*x). - R. J. Mathar, Nov 23 2007
E.g.f.: exp(4*x) - exp(x). - Mohammad K. Azarian, Jan 14 2009
a(n) = A000051(n)*A000225(n). - Reinhard Zumkeller, Feb 14 2009
A079978(a(n)) = 1. - Reinhard Zumkeller, Nov 22 2009
a(n) = A179857(A000225(n)), for n > 0; a(n) > A179857(m), for m < A000225(n). - Reinhard Zumkeller, Jul 31 2010
a(n) = 4*a(n-1) + 3, with a(0) = 0. - Vincenzo Librandi, Aug 01 2010
A000120(a(n)) = 2*n. - Reinhard Zumkeller, Feb 07 2011
a(n) = (3/2)*A020988(n). - Omar E. Pol, Mar 15 2012
a(n) = (Sum_{i=0..n} A002001(i)) - 1 = A178789(n+1) - 3. - Ivan N. Ianakiev, Nov 22 2013
a(n) = n*E(2*n-1,1)/B(2*n,1), for n > 0, where E(n,x) denotes the Euler polynomials and B(n,x) the Bernoulli polynomials. - Peter Luschny, Apr 04 2014
a(n) = A000302(n) - 1. - Sean A. Irvine, Jun 18 2019
Sum_{n>=1} 1/a(n) = A248721. - Amiram Eldar, Nov 13 2020
a(n) = A080674(n) - A002450(n). - Elmo R. Oliveira, Dec 02 2023

More terms Wesley Ivan Hurt, Apr 04 2014

A040080 Continued fraction for sqrt(90).

Original entry on oeis.org

9, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2, 18, 2

Offset: 0

N. J. A. Sloane

			9.4868329805051379959966806... = 9 + 1/(2 + 1/(18 + 1/(2 + 1/(18 + ...)))). - _Harry J. Smith_, Jun 11 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0, 1).

Cf. A010541 Decimal expansion. - Harry J. Smith, Jun 11 2009

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[90],300] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2011 *)
PadRight[{9},120,{18,2}] (* Harvey P. Dale, Jun 13 2017 *)

{ allocatemem(932245000); default(realprecision, 32000); x=contfrac(sqrt(90)); for (n=0, 20000, write("b040080.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 11 2009

a(n) = if (!n, 9, if (n % 2, 2, 18)); \\ Michel Marcus, Jan 03 2016

A041160 Numerators of continued fraction convergents to sqrt(90).

Original entry on oeis.org

9, 19, 351, 721, 13329, 27379, 506151, 1039681, 19220409, 39480499, 729869391, 1499219281, 27715816449, 56930852179, 1052471155671, 2161873163521, 39966188099049, 82094249361619, 1517662676608191

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

Cf. A010541, A041161.

Numerator[Convergents[Sqrt[90], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
Table[(2 + (-1)^n) ((3 - Sqrt[10])^(n + 1) + (3 + Sqrt[10])^(n + 1))/2, {n, 0, 30}] (* Bruno Berselli, Oct 30 2013 *)
LinearRecurrence[{0,38,0,-1},{9,19,351,721},30] (* Harvey P. Dale, May 12 2018 *)

G.f.: (1 + x)*(9 + 10*x - x^2) / (1 - 38*x^2 + x^4). [Bruno Berselli, Oct 30 2013]

a(n) = (2+(-1)^n)*((3-sqrt(10))^(n+1)+(3+sqrt(10))^(n+1))/2. [Bruno Berselli, Oct 30 2013]

A171542 Decimal expansion of sqrt(27/70).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient = <2 3/2 ; -1 3/2 | 5/2 1/2>.

			sqrt(27/70) = 3*sqrt(210)/70 = 0.621059003408118796016514178164...

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Wikipedia, Table of Clebsch-Gordan coefficients

Cf. A010482, A010522, A010541, A171537.

RealDigits[Sqrt[27/70],10,120][[1]] (* Harvey P. Dale, Apr 26 2011 *)

Equals A010482/A010522 = A010541*A171537/10.

cp's OEIS Frontend

A248313 Egyptian fraction representation of sqrt(90) (A010541) using a greedy function.

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A024036 a(n) = 4^n - 1.

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A040080 Continued fraction for sqrt(90).

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A041160 Numerators of continued fraction convergents to sqrt(90).

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A171542 Decimal expansion of sqrt(27/70).

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