A181565 -id:A181565 - cp's OEIS frontend

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

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

A083065 4th row of number array A083064.

Original entry on oeis.org

1, 4, 19, 94, 469, 2344, 11719, 58594, 292969, 1464844, 7324219, 36621094, 183105469, 915527344, 4577636719, 22888183594, 114440917969, 572204589844, 2861022949219, 14305114746094, 71525573730469, 357627868652344

Offset: 0

Paul Barry, Apr 21 2003

Inverse binomial transform of A090040. [Paul Curtz, Jan 11 2009]
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=7, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). [Milan Janjic, Feb 21 2010]
For an integer x, consider the sequence P(x) of polynomials p_{1}, p_{2}, p_{3}, . . . defined by p_{1} = x-1, p_{n+1} = x*p_{1} - 1. P(5) = This sequence. P(1), P(2), P(3), P(4) are A000004, A123412, A007051, A007583 resp. [K.V.Iyer, Jun 22 2010]
It appears that if s(n) is a first order rational sequence of the form s(0)=2, s(n)= (3*s(n-1)+2)/(2*s(n-1)+3), n>0, then s(n)=2*a(n)/(2*a(n)-1), n>0.
An Engel expansion of 5/3 to the base b := 5/4 as defined in A181565, with the associated series expansion 5/3 = b + b^2/4 + b^3/(4*19) + b^4/(4*19*94) + .... Cf. A007051. - 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 (6,-5).

Cf. A007583, A083066, A007051.

[(3*5^n+1)/4: n in [0..30]]; // Vincenzo Librandi, Nov 04 2011

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]*5-1 od: seq(a[n], n=1..22); # Zerinvary Lajos, Feb 22 2008

CoefficientList[Series[(1-2x)/((1-5x)(1-x)),{x,0,30}],x] (* or *) LinearRecurrence[{6,-5},{1,4},30] (* Harvey P. Dale, Jul 27 2022 *)

a(n) = (3*5^n+1)/4.
G.f.: (1-2*x)/((1-5*x)(1-x)).
E.g.f.: (3*exp(5*x) + exp(x))/4.
a(n) = 5*a(n-1)-1 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 6*a(n-1)-5*a(n-2). - Vincenzo Librandi, Nov 04 2011
a(n) = 5^n - Sum_{i=0..n-1} 5^i. - Bruno Berselli, Jun 20 2013

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

A230892 Terms of A230891 written in base 10: the binary expansions of a(n) and a(n+1) taken together can be rearranged to form a palindrome.

Original entry on oeis.org

0, 3, 1, 2, 4, 7, 8, 5, 6, 9, 10, 12, 15, 16, 11, 13, 14, 17, 18, 20, 23, 24, 27, 29, 30, 32, 19, 21, 22, 25, 26, 28, 31, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60, 63, 64, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55, 56, 59, 61, 62, 65, 66, 68, 71

Offset: 0

N. J. A. Sloane, Nov 11 2013

See A230891 for precise definition.
Just as for A228407, we can ask: does every number appear? The answer is yes - see the Comments in A228407.
The difference d(n)=a(n)-n increases from d(3*2^(k-2)+2) = 1-2^(k-2) to d(3*2^(k-1)+1) = 1-2^(k-1), going through 0 at n=2^k+1 and n=2^k+2, cf. examples. - M. F. Hasler, Nov 12 2013
From Robert G. Wilson v, Nov 15 2013: (Start)
Beginning with k=3, each "grouping" of ever increasing terms, begins at 2^k + 3 and runs up to 2^(k+2) and includes 3*2^(k-1) terms.
Indices of powers of 2 occur at: 2, 3, 4, 6, 13, 25, 49, 97, 193, 385, 769, 1537, ..., which, except for 2, 3 & 6, is A181565: 3*2^n + 1.
When the index equals the term: 0, 4, 9, 10, 17, 18, 33, 34, 65, 66, 129, 130, 257, 258, 513, 514, 1025, 1026, 2049, 2050, ..., .
Parity of a(n) beginning at n=0: 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, ..., . (End)

			From _M. F. Hasler_, Nov 12 2013: (Start)
Let d(n)=a(n)-n, i.e., a(n)=n+d(n). Then we have, after initial values d(0..8)=(0, 2, -1, -1, 0, 2, 2, -2, -2), the
following pattern: d(9) = d(10) = 0, ..., d(13) = 3,
d(14) = -3, ..., d(17) = d(18) = 0, ..., d(25) = 7;
d(26) = -7, ..., d(33) = d(34) = 0, ..., d(49) = 15,
d(50) = -15, ..., d(65) = d(66) = 0, ..., d(97) = 31,
d(98) = -31, ..., d(129) = d(130) = 0, ..., d(193) = 63,
d(194) = -63,..., d(257) = d(258) = 0, ... (End)

Chai Wah Wu, Table of n, a(n) for n = 0..30000 (first 2051 terms from Robert G. Wilson v)
Robert G. Wilson v, Graph of the first 1025 terms

Cf. A230891, A228407.

{u=0; a=0; La=1; ha=0/*hack*/; for(n=1, 99, u+=1<=2^L,L++); bittest(ha+h=hammingweight(k),0)&&!bittest(La+L,0)&&next; !a&&k<3&&next; a=k; ha=h; La=L; break))} \\ M. F. Hasler, Nov 11 2013

from collections import Counter
A230892_list, l, s, b = [0, 3], Counter('11'), 1, {3}
for _ in range(30001):
    i = s
    while True:
        if i not in b:
            li, o = Counter(bin(i)[2:]), 0
            for d in (l+li).values():
                if d % 2:
                    if o > 0:
                        break
                    o += 1
            else:
                A230892_list.append(i)
                l = li
                b.add(i)
                while s in b:
                    b.remove(s)
                    s += 1
                break
        i += 1 # Chai Wah Wu, Jun 19 2016

A140660 a(n) = 3*4^n + 1.

Original entry on oeis.org

4, 13, 49, 193, 769, 3073, 12289, 49153, 196609, 786433, 3145729, 12582913, 50331649, 201326593, 805306369, 3221225473, 12884901889, 51539607553, 206158430209, 824633720833, 3298534883329, 13194139533313, 52776558133249

Offset: 0

Paul Curtz, Jul 10 2008

nonn

An Engel expansion of 4/3 to the base 4 as defined in A181565, with the associated series expansion 4/3 = 4/4 + 4^2/(4*13) + 4^3/(4*13*49) + 4^4/(4*13*49*193) + .... Cf. A199115. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A181565, A199115.

[3*4^n+1: n in [0..30] ]; // Vincenzo Librandi, May 23 2011

LinearRecurrence[{5,-4}, {4,13}, 50] (* or *) CoefficientList[Series[ (7*x-4)/((1-x)*(4*x-1)), {x,0,50}], x] (* G. C. Greubel, Sep 15 2017 *)

x='x+O('x^50); Vec((7*x-4)/((1-x)*(4*x-1))) \\ G. C. Greubel, Sep 15 2017

a(n) = A002001(n+1) + 1.
a(n) = 4*a(n-1) - 3.
First differences: a(n+1) - a(n) = A002063(n).
a(n+k) - a(n) = 3*(4^k - 1)*A000302(n) = 9*A002450(k)*A000302(n).
a(n) = A140529(n) - A096045(n).
O.g.f.: (7*x - 4)/((1 - x)*(4*x - 1)). - R. J. Mathar, Jul 14 2008
From G. C. Greubel, Sep 15 2017: (Start)
E.g.f.: 3*exp(4*x) + exp(x).
a(n) = 5*a(n-1) - 4*a(n-2). (End)

Edited and extended R. J. Mathar, Jul 14 2008

A207262 a(n) = 2^(4n - 2) + 1.

Original entry on oeis.org

5, 65, 1025, 16385, 262145, 4194305, 67108865, 1073741825, 17179869185, 274877906945, 4398046511105, 70368744177665, 1125899906842625, 18014398509481985, 288230376151711745, 4611686018427387905, 73786976294838206465, 1180591620717411303425, 18889465931478580854785, 302231454903657293676545

Offset: 1

Alonso del Arte, Feb 16 2012

With the exception of the first term, all these numbers are composite, and in fact are all multiples of 5. The other factors can be considerably larger than 5, as is the case with say, 2^158 + 1. These numbers can be factored as (2^(2n - 1) + 2^n + 1)(2^(2n - 1) - 2^n + 1). For example, 2^6 + 1 = 65 = (2^3 + 2^2 + 1)(2^3 - 2^2 + 1) = 13 * 5.
This formula was discovered by Leon-Francois-Antoine Aurifeuille in 1873. Wells (2005) remarks that knowledge of this formula would have saved Fortune Landry years of work he spent factoring 2^58 + 1.
Aurifeuille actually rediscovered a very special case of the identity 4x^4+1 = (2x^2-2x+1)(2x^2+2x+1), which Euler communicated to Goldbach in 1742. (The Fuss reference is in my book Seminumerical Algorithms, 3rd ed., p. 392; I had cited Aurifeuille in the 1st and 2nd editions.) - Don Knuth, Feb 09 2013
An Engel expansion of 4 to the base 16 as defined in A181565, with the associated series expansion 4 = 16/5 + 16^2/(5*65) + 16^3/(5*65*1025) + 16^4/(5*65*1025*16385) + .... Cf. A087289 and A199561. - Peter Bala, Oct 29 2013
Conjecture: Let m = 4n - 2. a(n) equals the sum of the m-th powers of the divisors of m divided by the sum of the m-th powers of the odd divisors of m. - Ivan N. Ianakiev, Jan 29 2020

David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005) p. 15

Vincenzo Librandi, Table of n, a(n) for n = 1..200
FactorDB, Factorizations of 2^(4*n-2)+1
P. H. Fuss, Correspondance math. et physique, 1 (1843) p. 145.
Primenumbers Yahoo Group, Aurifeuille and factoring, search results.
Eric Weisstein's World of Mathematics, Aurifeuillean Factorization.
Yahoo Groups, Aurifeuille and factoring
Index entries for linear recurrences with constant coefficients, signature (17,-16).

Cf. A000051, A052539 (supersets). A016825. A087289, A199561.

2^(4*Range[20] - 2) + 1
LinearRecurrence[{17, -16}, {5, 65}, 50] (* Vincenzo Librandi, Mar 03 2012 *)

a(n)=4^(2*n-1)+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 4^(2n - 1) + 1.
G.f.: 5*x*(1-4*x)/((1-x)*(1-16*x)). - Bruno Berselli, Feb 17 2012
a(1) = 5, a(n) = 16*(a(n-1) - 1) + 1. - Arkadiusz Wesolowski, Feb 17 2012
a(n) = 5*A299960(n-1). - R. J. Mathar, Feb 28 2018
E.g.f.: exp(x) + (exp(16*x) - 5)/4. - Stefano Spezia, Jan 30 2020

cp's OEIS Frontend

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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A083065 4th row of number array A083064.

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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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