A083575 -id:A083575 - 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

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

A020737 Pisot sequence L(5,9).

Original entry on oeis.org

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

Offset: 0

David W. Wilson

An Engel expansion of 1/2 to the base 2 as defined in A181565, with the associated series expansion 1/2 = 2/5 + 2^2/(5*9) + 2^3/(5*9*17) + 2^4/(5*9*17*33) + ... . - Peter Bala, Oct 28 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..238
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Subsequence of A000051. See A008776 for definitions of Pisot sequences.

Cf. A083575, A083683, A083686, A083705, A168596, A181565, A195744.

[2^(n+2)+1: n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011

LinearRecurrence[{3,-2},{5,9},40] (* Harvey P. Dale, Jun 10 2015 *)

a(n)=2^(n+2)+1 \\ Charles R Greathouse IV, Jun 05 2013

Vec(-(6*x-5)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Jun 21 2014

a(n) = 2^(n+2) + 1.
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: -(6*x-5) / ((x-1)*(2*x-1)). - Colin Barker, Jun 21 2014
E.g.f.: exp(x)*(1 + 4*exp(x)). - Stefano Spezia, Oct 08 2022

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

A050526 Primes of form 5*2^n+1.

Original entry on oeis.org

11, 41, 641, 40961, 163841, 167772161, 2748779069441, 180143985094819841, 188894659314785808547841, 193428131138340667952988161, 850705917302346158658436518579420528641

Offset: 1

N. J. A. Sloane, Dec 29 1999

nonn

All terms are odd since if n is even, then 5*2^n+1 is divisible by 3. - Michele Fabbrini, Jun 06 2021

Muniru A Asiru, Table of n, a(n) for n = 1..12
Ray Ballinger, Wilfried Keller, List of primes k*2^n + 1 for k < 300
Wikipedia, Proth's theorem
Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime

For the corresponding exponents n see A002254.

Cf. A019434, A039687, A050527, A050528, A050529, A195745, A300406, A300407, A300408.

Filtered(List([1..270], n->5*2^n + 1), IsPrime); # Muniru A Asiru, Mar 06 2018

[a: n in [1..200] | IsPrime(a) where a is 5*2^n + 1]; // Vincenzo Librandi, Mar 06 2018

a:=(n, k)->`if`(isprime(k*2^n+1), k*2^n+1, NULL):
seq(a(n, 5), n=1..127); # Martin Renner, Mar 05 2018

lista(nn) = {for(k=1, nn, if(ispseudoprime(p=5*2^k+1), print1(p, ", "))); } \\ Altug Alkan, Mar 29 2018

a(n) = A083575(A002254(n)). - Michel Marcus, Mar 29 2018

A346702 The a(n)-th composition in standard order is the odd bisection of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 5, 1, 3, 6, 3, 16, 8, 4, 9, 2, 5, 10, 5, 1, 3, 6, 3, 12, 6, 3, 7, 32, 16, 8, 17, 4, 9, 18, 9, 2, 5, 10, 5, 20, 10, 5, 11, 1, 3, 6, 3, 12, 6, 3, 7, 24, 12, 6, 13, 3, 7, 14, 7, 64, 32, 16, 33, 8, 17, 34, 17, 4, 9, 18, 9, 36, 18

Offset: 0

Gus Wiseman, Aug 12 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

a(n) is the row number in A066099 of the odd bisection of the n-th row of A066099.

			Composition number 741 in standard order is (2,1,1,3,2,1), with odd bisection (2,1,2), which is composition number 22 in standard order, hence a(741) = 22.

Length of the a(n)-th standard composition is A000120(n)/2 rounded up.
Positions of 1's are A003945.
Positions of 2's (and zero) are A083575.
Sum of the a(n)-th standard composition is A209281(n+1).
Positions of first appearances are A290259.
The version for prime indices is A346703.
The version for even bisection is A346705, with sums A346633.
A000120 and A080791 count binary digits 1 and 0, with difference A145037.
A011782 counts compositions.
A029837 gives length of binary expansion.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A345197 counts compositions by sum, length, and alternating sum.
Cf. A000009, A000302, A000346, A025047, A088218, A124754, A294175, A346697 (reverse: A346699).

Table[Total[2^Accumulate[Reverse[First/@Partition[Append[ Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse,0],2]]]]/2,{n,0,100}]

A029837(a(n)) = A209281(n).

A364963 Odd numbers k such that k is a multiple of A163511(k).

Original entry on oeis.org

3, 16383, 536870895, 2147482623

Offset: 1

Antti Karttunen, Sep 02 2023

Sequence A243071(A364498(n)), for n > 1, sorted into ascending order, therefore terms 151115727451794287099901, 60708402882054033466233184588234965832575213720379360039119137804340758912662765515 (and many others that do not fit in this space) are also present.
Consider the sequence 1 + 5*2^k (with k>=1): 11, 21, 41, 81, 161, 321, etc, (A083575(n) from n>=1), and compare to the sequence A163511(1 + 5*2^k): 25, 75, 225, 675, 2025, 6075, etc (= 3^(k-1) * 25). Clearly, the first sequence does not contain any multiples of 5, while all the terms in the second one are multiples of 25, and thus of 5 also.
Then consider sequences 1 + 2*(1 + 11*2^k): 47, 91, 179, 355, 707, 1411, etc., and A163511(1 + 2*(1 + 11*2^k)): 121, 605, 3025, 15125, 75625, 378125, etc. The terms in the first one are never multiples of 11, while the terms of second one are all multiples of 121, thus of 11 also.
Consider also sequences 1 + (2^k)*(1+2*11): 47, 93, 185, 369, 737, 1473, 2945, 5889, 11777, 23553, 47105, 94209, 188417, 376833, 753665, 1507329,  etc, and 1 + (2^k)*(1+4*11): 91, 181, 361, 721, 1441, 2881, 5761, 11521, 23041, 46081, 92161, 184321, 368641, 737281, 1474561, 2949121, etc. The only time their terms are multiples of 11 is when k = 5, 15, 25, ..., 5 + 10*j, j>= 0, while for sequences A163511(1 + (2^k)*(1+2*11)): 121, 363, 1089, 3267, 9801, 29403, etc, and A163511(1 + (2^k)*(1+4*11)): 605, 1815, 5445, 16335, 49005, 147015, etc, all the terms are multiples of 121, thus of 11 also.
There are numerous other such correspondences that forbid the occurrence of factor x in n, when n is a member of a certain subset of odd numbers, while on the other hand, force the same factor x to be present in A163511(n), thus making it impossible that n were a multiple of A163511(n) in those cases. However, this sequence shows that such subsets do not completely cover all odd numbers. Similar observation applies to Doudna sequence (see A364547).

			        Term [in binary]                         Factorization         A163511(Term)
           3 [11]                                (prime)             -> 3
       16383 [11111111111111]                  = 3*43*127            -> 43
   536870895 [11111111111111111111111101111]   = 3*5*11*47*107*647   -> 1177 = 11*107
  2147482623 [1111111111111111111101111111111] = 3*11*13*31*113*1429 -> 3503 = 31*113

Odd terms in A364496.
Cf. A083575, A163511, A243071, A364498.
Cf. also A364495, A364547.

cp's OEIS Frontend

A181565 a(n) = 3*2^n + 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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A020737 Pisot sequence L(5,9).

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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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