A010767 -id:A010767 - 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

A378142 a(n) = n + floor(ns/r) + floor(nt/r), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

Original entry on oeis.org

3, 6, 10, 13, 17, 21, 24, 28, 31, 35, 39, 42, 46, 49, 53, 57, 61, 64, 67, 71, 74, 79, 82, 85, 89, 92, 97, 100, 104, 107, 110, 115, 118, 122, 125, 128, 133, 136, 140, 143, 146, 150, 154, 158, 161, 165, 168, 172, 176, 179, 183, 186, 190, 194, 197, 201, 204

Offset: 1

Clark Kimberling, Jan 13 2025

nonn

The sequences A378142, A378185, A379510, partition the positive integers (A000027), as proved at A184812:
A378142: 3,6,10,13,17,21,24,28,32,35,...
A378185: 2,5,8,11,14,18,20,23,26,29,,...
A379510: 1,4,7,9,12,15,16,19,22,25,27,...
For each integer k >= 1, write "a" if k=A378142(n) for some n, "b" if k=A378185(n) for some n, and "c" if k=A379510(n) for some n. Concatenating these letters for k = 1,2,3,... spells the following infinite word:
cbacbacbcabcabccabcbacbacbcabcacbcabcbacbacbcacbacbcabcbacbcabcacbacbcabcabcbcacbacbacbcabcabccbacbacb...

Cf. A000027, A010767, A184812, A378185, A379510.

r=2^(1/4); s=2^(1/2); t=2^(3/4);
a[n_]:=n+Floor[n*s/r]+Floor[n*t/r];
b[n_]:=n+Floor[n*r/s]+Floor[n*t/s];
c[n_]:=n+Floor[n*r/t]+Floor[n*s/t];
Table[a[n], {n, 1, 120}]  (* A378142 *)
Table[b[n], {n, 1, 120}]  (* A378185 *)
Table[c[n], {n, 1, 120}]  (* A379510 *)

a(n) = n + [w*n] + [w^2 n], where w = 2^(1/4) and [ ] = floor.

Name corrected by Clark Kimberling, Jan 20 2025

A228497 Decimal expansion of the fourth root of 1/2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 23 2013

Height of a equilateral square antiprism of edge length 1. - R. J. Mathar, Mar 06 2025

			0.8408964152537145430311254762...

E. W. Weisstein, Antiprism, MathWorld
Index entries for algebraic numbers, degree 4.

Cf. A010503, A010767.

Maple
```
evalf(1/root[4](2)) ;
```

RealDigits[Surd[1/2,4],10,120][[1]] (* Harvey P. Dale, Oct 05 2014 *)

sqrtn(2,-4) \\ Charles R Greathouse IV, Apr 15 2014

One divided by A010767.
Square root of A010503.
Equals Product_{k>=1} (1 + (-1)^k/(4*k+1)). - Amiram Eldar, Aug 10 2020

A203146 Decimal expansion of Gamma(7/8).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 29 2011

			1.0896523574228969512523767551028929711478700677675651205137...

Index to sequences related to the Gamma function.

Cf. A010767, A068465, A203143, A231863.

RealDigits[Gamma[7/8], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

gamma(7/8) \\ Charles R Greathouse IV, Mar 25 2014

A068465 = A231863 * A010767 * A203143 * this. - R. J. Mathar, Jan 15 2021

A367842 Decimal expansion of limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Dec 02 2023

Limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(1/n) = sqrt(2*Pi).

			1.23456019539799897381741853007827189474437277093956302475669920823457...

Cf. A010767, A092040, A074962, A367898, A367899.

RealDigits[(2*Pi)^(1/4)/Glaisher, 10, 120][[1]]
Exp[Integrate[x*Log[Gamma[x]], {x, 0, 1}]]

Equals (2*Pi)^(1/4) / A, where A = A074962 is the Glaisher-Kinkelin constant.
Equals A010767 * A092040 / A074962.
Equals exp(Integral_{x=0..1} x*log(Gamma(x)) dx).

A011011 Decimal expansion of 4th root of 14.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			1.934336420267669308917861835...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

RealDigits[14^(1/4), 10, 100][[1]] (* Alonso del Arte, Mar 14 2015 *)

sqrtn(14,4) \\ Charles R Greathouse IV, Apr 15 2014

Equals A010767*A011005. - R. J. Mathar, Mar 10 2025

A179613 Continued fraction for 2^(1/4).

Original entry on oeis.org

1, 5, 3, 1, 1, 40, 5, 1, 1, 25, 2, 3, 1, 6, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 7, 2, 7, 1, 1, 1, 2, 1, 1, 32, 4, 1, 6, 2, 1, 1, 1, 15, 1, 5, 1, 4, 1, 1, 1, 3, 1, 3, 7, 2, 7, 1, 1, 3, 31, 1, 3, 1, 3, 1, 9, 18, 4, 5, 3, 1, 2, 3, 2, 1, 3, 7, 1, 3, 1, 9, 10, 2, 1, 2, 1, 14, 1, 17, 1, 2, 2, 1, 7, 1, 5, 3, 14, 1

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 20 2010

2^(1/4) = 1.1892071150027210667174999705604759152929720...

Cf. A010767 (decimal expansion), A002945.

Mathematica
```
ContinuedFraction[2^(1/4),200]
```

Offset changed by Andrew Howroyd, Jul 07 2024

A248557 Decimal expansion of (Pi/2)^(1/4)/Gamma(3/4).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Dec 15 2014

			0.913579138156116821407242593401222089701963916393469...

MathOverflow, How to calculate the infinite sum of this double series?, 2014.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.

Cf. A010767, A068465, A092040, A175573, A222068, A251992.

RealDigits[(Pi/2)^(1/4)/Gamma[3/4], 10, 103] // First

(Pi/2)^(1/4)/gamma(3/4) \\ Michel Marcus, Dec 15 2014

Also equals theta_2(0,exp(-Pi)), where 'theta' is the elliptic theta function.
Equals A175573 / exp(4*A251992/Pi + Pi/4).
Equals Product_{k>=1} tanh(k*Pi/2). - Amiram Eldar, Jun 12 2021

A309949 Decimal expansion of the imaginary part of the square root of 1 + i.

Original entry on oeis.org

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

Offset: 0

Alonso del Arte, Aug 24 2019

i is the imaginary unit such that i^2 = -1.

Multiplied by -1, this is the imaginary part of the square root of 1 - i. And also the real part of -sqrt(1 + i) - i + sqrt(1 + i)^3, which is a unit in Q(sqrt(1 + i)).

			Im(sqrt(1 + i)) = 0.45508986056222734130435775782247...

Cf. A000108, A010767, A182168, A309948 (real part).

Digits := 120: Re(sqrt(-1 - I))*10^95:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Sep 20 2019

RealDigits[Sqrt[1/Sqrt[2] - 1/2], 10, 100][[1]]

imag(sqrt(1+I)) \\ Michel Marcus, Sep 16 2019

Equals sqrt(1/sqrt(2) - 1/2) = 2^(1/4) * sin(Pi/8).
Equals sqrt((sqrt(2) - 1)/2) = A010767 * A182168. - Bernard Schott, Sep 16 2019
Equals Re(sqrt(-1 - i)). - Peter Luschny, Sep 20 2019
Equals Product_{k>=0} ((8*k - 1)*(8*k + 4))/((8*k - 2)*(8*k + 5)). - Antonio Graciá Llorente, Feb 24 2024

A329219 Decimal expansion of 2^(10/12) = 2^(5/6).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 08 2019

2^(10/12) is the ratio of the frequencies of the pitches in a minor seventh (e.g., D4-C5) in 12-tone equal temperament.

			1.78179743...

Wikipedia, Equal temperament
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Frequency ratios of musical intervals:
Perfect unison:   2^(0/12) = 1.0000000000
Minor second:     2^(1/12) = 1.0594630943...  (A010774)
Major second:     2^(2/12) = 1.1224620483...  (A010768)
Minor third:      2^(3/12) = 1.1892071150...  (A010767)
Major third:      2^(4/12) = 1.2599210498...  (A002580)
Perfect fourth:   2^(5/12) = 1.3348398541...  (A329216)
Aug. fourth/
Dim. fifth:       2^(6/12) = 1.4142135623...  (A002193)
Perfect fifth:    2^(7/12) = 1.4983070768...  (A328229)
Minor sixth:      2^(8/12) = 1.5874010519...  (A005480)
Major sixth:      2^(9/12) = 1.6817928305...  (A011006)
Minor seventh:   2^(10/12) = 1.7817974362...  (this sequence)
Major seventh:   2^(11/12) = 1.8877486253...  (A329220)
Perfect octave:  2^(12/12) = 2.0000000000

First[RealDigits[2^(5/6), 10, 100]] (* Paolo Xausa, Apr 27 2024 *)

PARI
```
default(realprecision, 100); 2^(10/12)
```

Equals 2/A010768.

Equals Product_{k>=0} (1 + (-1)^k/(6*k + 1)). - Amiram Eldar, Jul 25 2020

cp's OEIS Frontend

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

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A378142 a(n) = n + floor(n*s/r) + floor(n*t/r), where r=2^(1/4), s=2^(1/2), t=2^(3/4).

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A228497 Decimal expansion of the fourth root of 1/2.

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A203146 Decimal expansion of Gamma(7/8).

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A367842 Decimal expansion of limit_{n->oo} Product_{k=1..n} Gamma(k/n)^(k/n^2).

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A011011 Decimal expansion of 4th root of 14.

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A179613 Continued fraction for 2^(1/4).

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A378142 a(n) = n + floor(ns/r) + floor(nt/r), where r=2^(1/4), s=2^(1/2), t=2^(3/4).