A020737 -id:A020737 - cp's OEIS frontend

A181565 a(n) = 3*2^n + 1.

4, 7, 13, 25, 49, 97, 193, 385, 769, 1537, 3073, 6145, 12289, 24577, 49153, 98305, 196609, 393217, 786433, 1572865, 3145729, 6291457, 12582913, 25165825, 50331649, 100663297, 201326593, 402653185, 805306369, 1610612737, 3221225473

Offset: 0

M. F. Hasler, Oct 30 2010

From Peter Bala, Oct 28 2013: (Start)
Let x and b be positive real numbers. We define an Engel expansion of x to the base b to be a (possibly infinite) nondecreasing sequence of positive integers [a(1), a(2), a(3), ...] such that we have the series representation x = b/a(1) + b^2/(a(1)*a(2)) + b^3/(a(1)*a(2)*a(3)) + .... Depending on the values of x and b such an expansion may not exist, and if it does exist it may not be unique. When b = 1 we recover the ordinary Engel expansion of x.
This sequence gives an Engel expansion of 2/3 to the base 2, with the associated series expansion 2/3 = 2/4 + 2^2/(4*7) + 2^3/(4*7*13) + 2^4/(4*7*13*25) + ....
More generally, for n and m positive integers, the sequence [m + 1, n*m + 1, n^2*m + 1, ...] gives an Engel expansion of the rational number n/m to the base n. See the cross references for several examples. (End)
The only squares in this sequence are 4, 25, 49. - Antti Karttunen, Sep 24 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]
Hunar Sherzad Taher and Saroj Kumar Dash, On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 448-459. See p. 2.
Wikipedia, Engel Expansion
Index entries for linear recurrences with constant coefficients, signature (3,-2)

Cf. A007283, A020737, A083575, A083683, A083686, A168596, A083705, A195744.
Essentially a duplicate of A004119.
A002253 and A039687 give the primes in this sequence, and A181492 is the subsequence of twin primes.

[3*2^n + 1: n in [0..30]]; // Vincenzo Librandi, May 19 2011

3*2^Range[0,50]+1 (* Vladimir Joseph Stephan Orlovsky, Mar 24 2011 *)
LinearRecurrence[{3,-2},{4,7},40] (* Harvey P. Dale, Sep 19 2024 *)

PARI
```
A181565(n)=3<
				
```

a(n) = A004119(n+1) = A103204(n+1) for all n >= 0.
From Ilya Gutkovskiy, Jun 01 2016: (Start)
O.g.f.: (4 - 5*x)/((1 - x)*(1 - 2*x)).
E.g.f.: (1 + 3*exp(x))*exp(x).
a(n) = 3*a(n-1) - 2*a(n-2). (End)
a(n) = 2*a(n-1) - 1. - Miquel Cerda, Aug 16 2016
For n >= 0, A005940(a(n)) = A001248(1+n). - Antti Karttunen, Sep 24 2023

A083575 a(0) = 6; for n>0, a(n) = 2*a(n-1) - 1.

Original entry on oeis.org

6, 11, 21, 41, 81, 161, 321, 641, 1281, 2561, 5121, 10241, 20481, 40961, 81921, 163841, 327681, 655361, 1310721, 2621441, 5242881, 10485761, 20971521, 41943041, 83886081, 167772161, 335544321, 671088641, 1342177281, 2684354561, 5368709121, 10737418241

Offset: 0

N. J. A. Sloane, Jun 15 2003

The primes in this sequence are listed in A050526. - M. F. Hasler, Oct 30 2010

An Engel expansion of 2/5 to the base 2 as defined in A181565, with the associated series expansion 2/5 = 2/6 + 2^2/(6*11) + 2^3/(6*11*21) + 2^4/(6*11*21*41) + ... . - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A020737, A050526, A083683, A083686, A083705, A168596, A181565, A195744.

[5*2^n+1 : n in [0..30]]; // Vincenzo Librandi, Nov 03 2011

NestList[2#-1&,6,40] (* Harvey P. Dale, Jun 23 2017 *)

PARI
```
a(n)=5<M. F. Hasler, Oct 30 2010
```

Vec((6-7*x)/((1-x)*(1-2*x)) + O(x^40)) \\ Colin Barker, Sep 20 2016

a(n) = 5*2^n + 1. - M. F. Hasler, Oct 30 2010
a(n) = 3*a(n-1) - 2*a(n-2), n>1. - Vincenzo Librandi, Nov 03 2011
G.f.: (6-7*x) / ((1-x)*(1-2*x)). - Colin Barker, Sep 20 2016
E.g.f.: exp(x)*(1 + 5*exp(x)). - Stefano Spezia, Oct 08 2022
Product_{n>=0} (1 + 1/a(n)) = 7/5. - Amiram Eldar, Aug 04 2024

A083686 a(0) = 8; for n>0, a(n) = 2*a(n-1) - 1.

Original entry on oeis.org

8, 15, 29, 57, 113, 225, 449, 897, 1793, 3585, 7169, 14337, 28673, 57345, 114689, 229377, 458753, 917505, 1835009, 3670017, 7340033, 14680065, 29360129, 58720257, 117440513, 234881025, 469762049, 939524097, 1879048193, 3758096385, 7516192769, 15032385537

Offset: 0

N. J. A. Sloane, Jun 15 2003

An Engel expansion of 2/7 to the base 2 as defined in A181565, with the associated series expansion 2/7 = 2/8 + 2^2/(8*15) + 2^3/(8*15*29) + 2^4/(8*15*29*57) + ... . - Peter Bala, Oct 29 2013

The initial 8 is the only cube in this sequence. - Antti Karttunen, Sep 24 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A020737, A083575, A083683, A083705, A168596, A181565, A195744.

Cf. A005940, A030078.

[7*2^n+1 : n in [0..30]]; // Vincenzo Librandi, Nov 03 2011

7*2^Range[0, 50] + 1 (* Paolo Xausa, Apr 02 2024 *)

Vec((8-9*x)/((1-x)*(1-2*x)) + O(x^40)) \\ Colin Barker, Sep 20 2016

a(n)=7<Charles R Greathouse IV, Sep 20 2016

a(n) = 7*2^n + 1. - David Brotherton (dbroth01(AT)aol.com), Jul 29 2003
a(n) = 3*a(n-1) - 2*a(n-2), n>1. - Vincenzo Librandi, Nov 03 2011
G.f.: (8-9*x) / ((1-x)*(1-2*x)). - Colin Barker, Sep 20 2016
E.g.f.: exp(x)*(1 + 7*exp(x)). - Stefano Spezia, Oct 08 2022
For n >= 0, A005940(a(n)) = A030078(1+n). - Antti Karttunen, Sep 24 2023

A195744 a(n) = 15*2^(n+1) + 1.

Original entry on oeis.org

31, 61, 121, 241, 481, 961, 1921, 3841, 7681, 15361, 30721, 61441, 122881, 245761, 491521, 983041, 1966081, 3932161, 7864321, 15728641, 31457281, 62914561, 125829121, 251658241, 503316481, 1006632961, 2013265921, 4026531841, 8053063681, 16106127361

Offset: 0

Brad Clardy, Sep 23 2011

Binary numbers of form 1111(0^n)1 where n is the index and number of 0's.
Base 10 numbers of this sequence always end in 1.
An Engel expansion of 1/15 to the base 2 as defined in A181565, with the associated series expansion 1/15 = 2/31 + 2^2/(31*61) + 2^3/(31*61*121) + 2^4/(31*61*121*241) + ... . - Peter Bala, Oct 29 2013
The only squares in this sequence are 121 = 11^2 and 961 = 31^2. - Antti Karttunen, Sep 24 2023

			First few terms in binary are 11111, 111101, 1111001, 11110001, 111100001.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A052996, A164094.
Cf. A020737, A083575, A083683, A083686, A083705, A168596, A181565.
Cf. A110286, A128470.
Cf. A005940, A030514.

[15*2^(n+1) + 1: n in [0..30]]; // Vincenzo Librandi, Sep 24 2011

15*2^Range[50] + 1 (* Paolo Xausa, Apr 02 2024 *)

a(n)=30*2^n+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A052996(n+3) + A164094(n+2).
From Bruno Berselli, Sep 23 2011: (Start)
G.f.: (31-32*x)/(1-3*x+2*x^2).
a(n) = 2*a(n-1)-1.
a(n) = A110286(n+1)+1 = A128470(2^n). (End)
E.g.f.: exp(x)*(1 + 30*exp(x)). - Stefano Spezia, Oct 08 2022
For n >= 0, A005940(a(n)) = A030514(2+n). - Antti Karttunen, Sep 24 2023

Corrected by Arkadiusz Wesolowski, Sep 23 2011

A083683 a(n) = 11*2^n + 1.

Original entry on oeis.org

12, 23, 45, 89, 177, 353, 705, 1409, 2817, 5633, 11265, 22529, 45057, 90113, 180225, 360449, 720897, 1441793, 2883585, 5767169, 11534337, 23068673, 46137345, 92274689, 184549377, 369098753, 738197505, 1476395009, 2952790017, 5905580033, 11811160065

Offset: 0

N. J. A. Sloane, Jun 15 2003

An Engel expansion of 2/11 to the base 2 as defined in A181565, with the associated series expansion 2/11 = 2/12 + 2^2/(12*23) + 2^3/(12*23*45) + 2^4/(12*23*45*89) + ... . - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A020737, A083575, A083686, A083705, A168596, A181565, A195744.

[11*2^n+1 : n in [0..30]]; // Vincenzo Librandi, Nov 03 2011

11*2^Range[0,30]+1 (* or *) LinearRecurrence[{3,-2},{12,23},40] (* Harvey P. Dale, Aug 17 2017 *)

a(n)=11*2^n+1 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = 2*a(n-1) - 1.
a(n) = 3*a(n-1) - 2*a(n-2), n>1. - Vincenzo Librandi, Nov 03 2011
G.f. (12-13*x)/((2*x-1)*(x-1)). - R. J. Mathar, Nov 03 2011
E.g.f.: exp(x)*(1 + 11*exp(x)). - Stefano Spezia, Oct 08 2022

A083705 a(n) = 2*a(n-1) - 1 with a(0) = 10.

Original entry on oeis.org

10, 19, 37, 73, 145, 289, 577, 1153, 2305, 4609, 9217, 18433, 36865, 73729, 147457, 294913, 589825, 1179649, 2359297, 4718593, 9437185, 18874369, 37748737, 75497473, 150994945, 301989889, 603979777, 1207959553, 2415919105, 4831838209, 9663676417, 19327352833

Offset: 0

N. J. A. Sloane, Jun 15 2003

An Engel expansion of 2/9 to the base 2 as defined in A181565, with the associated series expansion 2/9 = 2/10 + 2^2/(10*19) + 2^3/(10*19*37) + 2^4/(10*19*37*73) + ... . - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

A020737, A083575, A083683, A083686, A168596, A181565, A195744.

[9*2^n+1 : n in [0..30]]; // Vincenzo Librandi, Nov 03 2011

s=10;lst={s};Do[s=s+(s-1);AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 30 2009 *)
NestList[2#-1&,10,40]  (* Harvey P. Dale, Mar 13 2011 *)

from itertools import accumulate
def f(an, _): return 2*an - 1
print(list(accumulate([10]*32, f))) # Michael S. Branicky, Oct 19 2021

From R. J. Mathar, Aug 01 2009: (Start)
a(n) = 1 + 9*2^n = 3*a(n-1) - 2*a(n-2).
G.f.: -(-10+11*x)/((2*x-1)*(x-1)). (End)
E.g.f.: exp(x)*(1 + 9*exp(x)). - Stefano Spezia, Oct 08 2022

A168596 a(n) = 2*a(n-1) - 1 with a(0)=14.

Original entry on oeis.org

14, 27, 53, 105, 209, 417, 833, 1665, 3329, 6657, 13313, 26625, 53249, 106497, 212993, 425985, 851969, 1703937, 3407873, 6815745, 13631489, 27262977, 54525953, 109051905, 218103809, 436207617, 872415233, 1744830465, 3489660929

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 30 2009

An Engel expansion of 2/13 to the base 2 as defined in A181565, with the associated series expansion 2/13 = 2/14 + 2^2/(14*27) + 2^3/(14*27*53) + 2^4/(14*27*53*105) + .... - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A020737, A083575, A083683, A083686, A083705, A181565, A195744.

[13*2^n+1 : n in [0..30]]; // Vincenzo Librandi, Nov 03 2011

s=14;lst={s};Do[s=s+(s-1);AppendTo[lst,s],{n,5!}];lst
NestList[2#-1&,14,30] (* Harvey P. Dale, Jul 22 2014 *)

From Vincenzo Librandi, Nov 03 2011: (Start)
a(n) = 13*2^n + 1.
a(n) = 3*a(n-1) - 2*a(n-2). (End)
From G. C. Greubel, Jul 27 2016: (Start)
G.f.: (14 - 15*x)/((1-x)*(1-2*x)).
E.g.f.: exp(x) + 13*exp(2*x). (End)

Offset changed from 1 to 0 by Vincenzo Librandi, Nov 03 2011

A048578 Pisot sequence L(3,5).

Original entry on oeis.org

3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 8589934593, 17179869185

Offset: 0

David W. Wilson

Lexicographically earliest (when ordered) minimal set of generators for A001969 (numbers with an even number of binary 1's) as a group under A003987(.,.) the XOR operation. - Peter Munn, Aug 21 2019

Odd numbers with binary weight = 2. - David James Sycamore, Feb 02 2025

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Josef Eschgfäller and Andrea Scarpante, Dichotomic random number generators, arXiv:1603.08500 [math.CO], 2016.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Group.
Wikipedia, Generating set of a group.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Subsequence of A000051.
See A008776 for definitions of Pisot sequences.
Cf. A001969, A003987.
Essentially the same as A020737 and A000051.

[2^(n+1)+1 : n in [0..40]]; // Vincenzo Librandi, Sep 01 2011

LinearRecurrence[{3,-2},{3,5},40] (* Harvey P. Dale, Sep 10 2017 *)

my(x='x+O('x^99)); Vec(1/(1-x)+2/(1-2*x)) \\ Altug Alkan, Mar 29 2016

a(n) = 2^(n+1)+1.
a(n) = 3*a(n-1) - 2*a(n-2).
O.g.f.: (3-4*x)/(1-3*x+2*x^2). - R. J. Mathar, Nov 23 2007
E.g.f.: exp(x)*(1 + 2*exp(x)). - Elmo R. Oliveira, Dec 06 2024

A088039 Smallest k such that k^3 == 1 (mod some n-th power), k > 1.

Original entry on oeis.org

2, 4, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649

Offset: 1

Amarnath Murthy, Sep 19 2003

nonn

In most of the cases the n-th power that divides a(n)^3-1 is 2^n.

Cf. A088038.

Cf. A020737. [From R. J. Mathar, Sep 11 2008]

a(n) = A088038(n)^(1/3).

Conjecture: a(n) = 2^n+1 for n>2. a(n) = 3*a(n-1)-2*a(n-2) for n>4. G.f.: -x*(2*x^3-x^2+2*x-2) / ((x-1)*(2*x-1)). [Colin Barker, Feb 05 2013]

Corrected and extended by Ray Chandler, Oct 04 2003

A277088 Pisot sequences L(5,12), S(5,12).

Original entry on oeis.org

5, 12, 29, 71, 174, 427, 1048, 2573, 6318, 15514, 38095, 93544, 229702, 564045, 1385042, 3401044, 8351444, 20507414, 50357044, 123654396, 303639937, 745603993, 1830870208, 4495799044, 11039673351, 27108504296, 66566372193, 163457262657, 401377990645

Offset: 0

Ilya Gutkovskiy, Sep 29 2016

Ilya Gutkovskiy, Pisot sequences L(x,y)
Index entries for Pisot sequences

Cf. A008776 for definitions of Pisot sequences.
Cf. A000129 (with offset 3 appears to be Pisot sequences E(5,12), P(5,12)).
Cf. A020736, A020737, A048583.

RecurrenceTable[{a[0] == 5, a[1] == 12, a[n] == Ceiling[a[n - 1]^2/a[n - 2]]}, a, {n, 28}]
RecurrenceTable[{a[0] == 5, a[1] == 12, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1]}, a, {n, 28}]

a(n) = ceiling(a(n-1)^2/a(n-2)), a(0) = 5, a(1) = 12.
a(n) = floor(a(n-1)^2/a(n-2)+1), a(0) = 5, a(1) = 12.
Conjectures: (Start)
G.f.: (5 - 3*x + 3*x^2 - 2*x^3 + x^5 - 3*x^6 - x^7 - 2*x^8)/((1 - x)*(1 - 2*x - 2*x^3 - x^4 - x^5 - 2*x^6 - x^7 - x^8)).
a(n) = 3*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) - a(n-7) - a(n-9). (End)

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A181565 a(n) = 3*2^n + 1.

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A083575 a(0) = 6; for n>0, a(n) = 2*a(n-1) - 1.

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A083686 a(0) = 8; for n>0, a(n) = 2*a(n-1) - 1.

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A195744 a(n) = 15*2^(n+1) + 1.

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A083683 a(n) = 11*2^n + 1.

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A083705 a(n) = 2*a(n-1) - 1 with a(0) = 10.

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A168596 a(n) = 2*a(n-1) - 1 with a(0)=14.

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