A147590 -id:A147590 - cp's OEIS frontend

A083420 a(n) = 2*4^n - 1.

1, 7, 31, 127, 511, 2047, 8191, 32767, 131071, 524287, 2097151, 8388607, 33554431, 134217727, 536870911, 2147483647, 8589934591, 34359738367, 137438953471, 549755813887, 2199023255551, 8796093022207, 35184372088831, 140737488355327, 562949953421311

Offset: 0

Paul Barry, Apr 29 2003

Sum of divisors of 4^n. - Paul Barry, Oct 13 2005
Subsequence of A000069; A132680(a(n)) = A005408(n). - Reinhard Zumkeller, Aug 26 2007
If x = a(n), y = A000079(n+1) and z = A087289(n), then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014
It seems that a(n) divides A001676(3+4n). Several other entries apparently have this sequence embedded in them, e.g., A014551, A168604, A213243, A213246-8, and A279872. - Tom Copeland, Dec 27 2016
To elaborate on Librandi's comment from 2014: all these numbers, even if prime in Z, are sure not to be prime in Z[sqrt(2)], since a(n) can at least be factored as ((2^(2n + 1) - 1) - (2^(2n) - 1)*sqrt(2))((2^(2n + 1) - 1) + (2^(2n) - 1)*sqrt(2)). For example, 7 = (3 - sqrt(2))(3 + sqrt(2)), 31 = (7 - 3*sqrt(2))(7 + 3*sqrt(2)), 127 = (15 - 7*sqrt(2))(15 + 7*sqrt(2)). - Alonso del Arte, Oct 17 2017
Largest odd factors of A147590. - César Aguilera, Jan 07 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, Fig 11.
Robert Schneider, Partition zeta functions, Research in Number Theory, 2(1):9, 2016.
Eric Weisstein's World of Mathematics, Rule 220
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A083421, A000668 (primes in this sequence), A004171, A000244.
Cf. A001676, A014551, A168604, A213243, A213246, A213247, A213248, A279872.
Cf. A000302.

a083420 = subtract 1 . (* 2) . (4 ^)  -- Reinhard Zumkeller, Dec 22 2015

[2*4^n-1 : n in [0..30]]; // Wesley Ivan Hurt, Mar 14 2015

seq(2*4^n-1, n = 0..22); # Peter Luschny, Aug 17 2011

2 * 4^Range[0, 31] - 1 (* Alonso del Arte, Oct 17 2017 *)

a(n)=2*4^n-1 \\ Charles R Greathouse IV, Sep 24 2015

def A083420(n): return (1<<(n<<1|1))-1 # Chai Wah Wu, Mar 10 2025

G.f.: (1+2*x)/((1-x)*(1-4*x)).
E.g.f.: 2*exp(4*x)-exp(x).
With a leading zero, this is a(n) = (4^n - 2 + 0^n)/2, the binomial transform of A080925. - Paul Barry, May 19 2003
From Benoit Cloitre, Jun 18 2004: (Start)
a(n) = (-16^n/2)*B(2n, 1/4)/B(2n) where B(n, x) is the n-th Bernoulli polynomial and B(k) = B(k, 0) is the k-th Bernoulli number.
a(n) = 5*a(n-1) - 4*a(n-2).
a(n) = (-4^n/2)*B(2*n, 1/2)/B(2*n). (End)
a(n) = A099393(n) + A020522(n) = A000302(n) + A024036(n). - Reinhard Zumkeller, Feb 07 2006
a(n) = Stirling2(2*(n+1), 2). - Zerinvary Lajos, Dec 06 2006
a(n) = 4*a(n-1) + 3 with n > 0, a(0) = 1. - Vincenzo Librandi, Dec 30 2010
a(n) = A001576(n+1) - 2*A001576(n). - Brad Clardy, Mar 26 2011
a(n) = 6*A002450(n) + 1. - Roderick MacPhee, Jul 06 2012
a(n) = A000203(A000302(n)). - Michel Marcus, Jan 20 2014
a(n) = Sum_{i = 0..n} binomial(2n+2, 2i). - Wesley Ivan Hurt, Mar 14 2015
a(n) = (1/4^n) * Sum_{k = 0..n} binomial(2*n+1,2*k)*9^k. - Peter Bala, Feb 06 2019
a(n) = A147590(n)/A000079(n). - César Aguilera, Jan 07 2020
a(n) = numerator(zeta_star({2}(n + 1))/zeta(2*n + 2)) where zeta_star is the multiple zeta star values and ({2}_n) represents (2, ..., 2) where the multiplicity of 2 is n. - _Roudy El Haddad, Feb 22 2022

A147537 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n digits 0.

Original entry on oeis.org

2, 28, 248, 2032, 16352, 131008, 1048448, 8388352, 67108352, 536869888, 4294965248, 34359734272, 274877898752, 2199023239168, 17592186011648, 140737488289792, 1125899906711552, 9007199254478848, 72057594037403648, 576460752302374912, 4611686018425290752

Offset: 1

Omar E. Pol, Nov 06 2008

a(n) is the number whose binary representation is A138118(n).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A138118.

List([1..20], n-> 2^n*(2^(2*n-1)-1)); # G. C. Greubel, Jan 12 2020

[2^n*(2^(2*n-1)-1): n in [1..20]] // G. C. Greubel, Jan 12 2020

seq(2^n*(2^(2*n-1)-1), n = 1..20); # G. C. Greubel, Jan 12 2020

Table[FromDigits[Join[Table[1, {2n - 1}], Table[0, {n}]], 2], {n, 1, 20}] (* Stefan Steinerberger, Nov 11 2008 *)

vector(20, n, 2^n*(2^(2*n-1)-1)) \\ G. C. Greubel, Jan 12 2020

def a(n): return ((1 << (2*n-1)) - 1) << n
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Feb 28 2021

[2^n*(2^(2*n-1)-1) for n in (1..20)] # G. C. Greubel, Jan 12 2020

From Colin Barker, Nov 04 2012: (Start)
a(n) = 2^(n-1)*(4^n - 2) = 2*A147590(n).
a(n) = 10*a(n-1) - 16*a(n-2).
G.f.: 2*x*(1+4*x)/((1-2*x)*(1-8*x)). (End)

A147595 a(n) is the number whose binary representation is A138144(n).

Original entry on oeis.org

1, 3, 7, 15, 27, 51, 99, 195, 387, 771, 1539, 3075, 6147, 12291, 24579, 49155, 98307, 196611, 393219, 786435, 1572867, 3145731, 6291459, 12582915, 25165827, 50331651, 100663299, 201326595, 402653187, 805306371, 1610612739, 3221225475

Offset: 1

Omar E. Pol, Nov 08 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A138144, A145641, A147537, A147538, A147539, A147540, A147590, A147596, A147597.

[1,3,7] cat [3*(1+2^(n-2)): n in [4..40]]; // G. C. Greubel, Oct 25 2022

LinearRecurrence[{3,-2},{1,3,7,15,27},40] (* Harvey P. Dale, Nov 30 2020 *)

Vec(-x*(2*x^2-1)*(2*x^2+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

[1,3,7]+[3*(1+2^(n-2)) for n in range(4,40)] # G. C. Greubel, Oct 25 2022

a(n) = A060013(n+2), n > 3. - R. J. Mathar, Feb 05 2010
a(n+4) = 3*(2^(n+2) + 1), n >= 0. - Brad Clardy, Apr 03 2013
From Colin Barker, Sep 15 2013: (Start)
a(n) = 3*(4 + 2^n)/4 for n>3.
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(1-2*x^2)*(1+2*x^2) / ((1-x)*(1-2*x)). (End)
E.g.f.: (3/4)*(4*exp(x) + exp(2*x)) - (15/4) - 7*x/2 - 3*x^2/2 - x^3/3. - G. C. Greubel, Oct 25 2022

Extended by R. J. Mathar, Feb 05 2010

A083332 a(n) = 10a(n-2) - 16a(n-4) for n > 3, a(0) = 1, a(1) = 5, a(2) = 14, a(3) = 34.

Original entry on oeis.org

1, 5, 14, 34, 124, 260, 1016, 2056, 8176, 16400, 65504, 131104, 524224, 1048640, 4194176, 8388736, 33554176, 67109120, 268434944, 536871424, 2147482624, 4294968320, 17179867136, 34359740416, 137438949376, 274877911040

Offset: 0

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

a(n)/A083333(n) converges to 3.

Index entries for linear recurrences with constant coefficients, signature (0, 10, 0, -16).

Cf. A147590, A081342 (bisections). [R. J. Mathar, Jul 13 2009]

Cf. A199710. [Bruno Berselli, Nov 11 2011]

CoefficientList[Series[(1+5x+4x^2-16x^3)/(1-10x^2+16x^4), {x, 0, 30}], x]

(a[0] : 1, a[1] : 5, a[2] : 14, a[3] : 34, a[n] := 10*a[n - 2] - 16*a[n - 4], makelist(a[n], n, 0, 50));/* Franck Maminirina Ramaharo, Nov 12 2018 */

G.f.: (1 + 5*x + 4*x^2 - 16*x^3)/(1 - 10*x^2 + 16*x^4).
a(n) = A016116(n)*A014551(n+1). - R. J. Mathar, Jul 08 2009
From Franck Maminirina Ramaharo, Nov 12 2018: (Start)
a(n) = sqrt(2)^(3*n - 1)*(1 + sqrt(2) + (-1)^n*(sqrt(2) - 1)) + sqrt(2)^(n - 3)*(1 - sqrt(2) - (-1)^n*(sqrt(2) + 1)).
E.g.f.: (sinh(sqrt(2)*x) + 2*sinh(2*sqrt(2)*x))/sqrt(2) - cosh(sqrt(2)*x) + 2*cosh(2*sqrt(2)*x). (End)

A147596 a(n) is the number whose binary representation is A138145(n).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 119, 231, 455, 903, 1799, 3591, 7175, 14343, 28679, 57351, 114695, 229383, 458759, 917511, 1835015, 3670023, 7340039, 14680071, 29360135, 58720263, 117440519, 234881031, 469762055, 939524103, 1879048199, 3758096391

Offset: 1

Omar E. Pol, Nov 08 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A138145, A145641, A147537, A147538, A147539.

Cf. A147540, A147590, A147595, A147597.

[1,3,7,15,31] cat [7*(1+2^(n-3)): n in [6..40]]; // G. C. Greubel, Oct 25 2022

Join[{1,3,7,15,31}, 7*(1+2^(Range[6, 40] -3))] (* G. C. Greubel, Oct 25 2022 *)

Vec(-x*(2*x^2-1)*(4*x^4+2*x^2+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

def A147596(n): return 7*(1+2^(n-3)) -(1/8)*(63*int(n==0) +62*int(n==1) +60*int(n ==2)) -(7*int(n==3) +6*int(n==4) +4*int(n==5))
[A147596(n) for n in range(1,40)] # G. C. Greubel, Oct 25 2022

a(n) = 7*(2^(n-3) + 1) if n >= 6. - Hagen von Eitzen, Jun 02 2009
From Colin Barker, Sep 15 2013: (Start)
a(n) = 3*a(n-1) - 2*a(n-2), for n >= 8.
G.f.: x*(1-2*x^2)*(1+2*x^2+4*x^4) / ((1-x)*(1-2*x)). (End)
E.g.f.: (7/8)*(8*exp(x) + exp(2*x)) - (1/8)*(63 + 62*x + 30*x^2) - 7*x^3/6 - x^4/4 - x^5/30. - G. C. Greubel, Oct 25 2022

More terms from Hagen von Eitzen, Jun 02 2009

A147597 a(n) is the number whose binary representation is A138146(n).

Original entry on oeis.org

1, 7, 31, 119, 455, 1799, 7175, 28679, 114695, 458759, 1835015, 7340039, 29360135, 117440519, 469762055, 1879048199, 7516192775, 30064771079, 120259084295, 481036337159, 1924145348615, 7696581394439, 30786325577735, 123145302310919, 492581209243655

Offset: 1

Omar E. Pol, Nov 08 2008

Bisection of A147596.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A138146, A145641, A147537, A147538, A147539.

Cf. A147540, A147590, A147595, A147596.

[1,7,31] cat [7*(1+4^(n-2)): n in [4..40]]; // G. C. Greubel, Oct 25 2022

Table[FromDigits[#, 2] &@ If[n < 4, ConstantArray[1, 2 n - 1], Join[#, ConstantArray[0, 2 n - 7], #]] &@ ConstantArray[1, 3], {n, 25}] (* or *)
Rest@ CoefficientList[Series[x (2 x + 1) (2 x - 1) (4 x^2 + 2 x + 1)/((4 x - 1) (1 - x)), {x, 0, 25}], x] (* Michael De Vlieger, Nov 25 2016 *)
LinearRecurrence[{5,-4},{1,7,31,119,455},30] (* Harvey P. Dale, Aug 04 2025 *)

Vec(x*(2*x+1)*(2*x-1)*(4*x^2+2*x+1)/((4*x-1)*(1-x)) + O(x^30)) \\ Colin Barker, Nov 25 2016

def A147597(n): return 7*(1+4^(n-2)) -(119/16)*int(n==0) -(31/4)*int(n==1) -7*int(n==2) -4*int(n==3)
[A147597(n) for n in range(1,41)] # G. C. Greubel, Oct 25 2022

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = 5*a(n-1) - 4*a(n-2) for n>5.
G.f.: x*(2*x+1)*(2*x-1)*(4*x^2+2*x+1)/((4*x-1)*(1-x)). (End)
a(n) = 7*4^(n-2) + 7 for n>3. - Colin Barker, Nov 25 2016
E.g.f.: (7/16)*(16*exp(x) + exp(4*x)) -(119/16) -31*x/4 -7*x^2/2 -2*x^3/3. - G. C. Greubel, Oct 25 2022

More terms from R. J. Mathar, Feb 05 2010

A002588 a(n) = largest noncomposite factor of 2^(2n+1) - 1.

Original entry on oeis.org

1, 7, 31, 127, 73, 89, 8191, 151, 131071, 524287, 337, 178481, 1801, 262657, 2089, 2147483647, 599479, 122921, 616318177, 121369, 164511353, 2099863, 23311, 13264529, 4432676798593, 131071, 20394401, 201961, 1212847, 3203431780337

Offset: 0

N. J. A. Sloane

nonn

a(n) is also the largest noncomposite factor of A147590(n). - César Aguilera, Jul 31 2019

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 2, p. 84.
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).

Max Alekseyev, Table of n, a(n) for n = 0..602 (terms 1..494 from Sean A. Irvine)
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A002184, A005420, A147590.

[1] cat [Maximum(PrimeDivisors(2^(2*n+1) - 1)): n in [1..60]]; // Vincenzo Librandi, Aug 02 2019

Table[FactorInteger[2^(2 n + 1) - 1] [[-1, 1]], {n, 0, 30}] (* Vincenzo Librandi, Aug 02 2019 *)

a(n) = if (n==0, 1, vecmax(factor(2^(2*n+1) - 1)[, 1])); \\ Michel Marcus, Aug 03 2019

More terms from Don Reble, Nov 14 2006

A147589 Concatenation of 2n-1 digits 1 and n-1 digits 0.

Original entry on oeis.org

1, 1110, 1111100, 1111111000, 1111111110000, 1111111111100000, 1111111111111000000, 1111111111111110000000, 1111111111111111100000000, 1111111111111111111000000000, 1111111111111111111110000000000

Offset: 1

Omar E. Pol, Nov 08 2008

a(n) is also A147590(n) written in base 2.

			n .......... a(n)
1 ........... 1
2 ......... 1110
3 ....... 1111100
4 ..... 1111111000
5 ... 1111111110000

Index entries for linear recurrences with constant coefficients, signature (1010,-10000).

Cf. A138118, A147537, A147590.

vector(100, n, 10^(-2+n)*(-10+100^n)/9) \\ Colin Barker, Jul 08 2014

Vec(x*(100*x+1)/((10*x-1)*(1000*x-1)) + O(x^100)) \\ Colin Barker, Jul 08 2014

a(n) = A138118(n)/10.
a(n) = {[10^(2*n-1)-1]*10^(n-1)}/9, with n>=1. - Paolo P. Lava, Nov 26 2008
G.f.: x*(100*x+1) / ((10*x-1)*(1000*x-1)). - Colin Barker, Jul 08 2014

Keyword:base added by Charles R Greathouse IV, Apr 28 2010

cp's OEIS Frontend

A083420 a(n) = 2*4^n - 1.

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A147537 Numbers whose binary representation is the concatenation of 2n-1 digits 1 and n digits 0.

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A147595 a(n) is the number whose binary representation is A138144(n).

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A083332 a(n) = 10*a(n-2) - 16*a(n-4) for n > 3, a(0) = 1, a(1) = 5, a(2) = 14, a(3) = 34.

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A147596 a(n) is the number whose binary representation is A138145(n).

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A147597 a(n) is the number whose binary representation is A138146(n).

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A083332 a(n) = 10a(n-2) - 16a(n-4) for n > 3, a(0) = 1, a(1) = 5, a(2) = 14, a(3) = 34.