A005126 -id:A005126 - cp's OEIS frontend

A271573 Numerator of (0 followed by A005126(n)= 2, 4, 7, ...)/2^n.

0, 1, 1, 7, 3, 21, 19, 71, 17, 265, 261, 1035, 515, 4109, 4103, 16399, 2049, 65553, 65545, 262163, 131077, 1048597, 1048587, 4194327, 1048579, 16777241, 16777229, 67108891, 33554439, 268435485, 268435471, 1073741855, 67108865, 4294967329, 4294967313

Offset: 0

Paul Curtz, Apr 10 2016

Reduced fractions: f(n) = 0, 1, 1, 7/8, 3/4, 21/32, 19/32, 71/128, 17/32, 265/512, 261/512, ... .

f(n) is an autosequence of the first kind.

			a(0), a(1), a(2), a(3), a(4), are the numerators of reduced fractions 0/1, 2/2, 4/4, 7/8, 12/16, ... .

Robert Price, Table of n, a(n) for n = 0..101

Cf. A000004, A000079, A005126, A006519, A060576(n+1), A075101, A198631, A279635 (denominator).

[0] cat [Numerator((2^(n-1)+n)/2^n): n in [1..40]]; // Vincenzo Librandi, Oct 13 2017

Prepend[Table[Numerator[(2^n + n + 1)/2^(n + 1)], {n, 0, 100}], 0] (* Robert Price, Apr 10 2016 *)
(* Computation from Oresme numbers n/2^n: *) a[n_] := Numerator[n/2^n + If[n < 2, 0, 1]/2]; (* Jean-François Alcover, Apr 28 2016, after Paul Curtz *)

a(n) = if(n==0, 0, numerator((2^(n-1)+n)/2^n)); \\ Altug Alkan, Apr 10 2016

a(n) = numerator(n/2^n + (if n<2 0 else 1)/2), a formula using Oresme numbers n/2^n. - Jean-François Alcover, Apr 28 2016 after Paul Curtz

A279635 Denominator of (0 followed by A005126(n)= 2, 4, 7, ...)/2^n, a sequence corresponding to A271573.

Original entry on oeis.org

1, 1, 1, 8, 4, 32, 32, 128, 32, 512, 512, 2048, 1024, 8192, 8192, 32768, 4096, 131072, 131072, 524288, 262144, 2097152, 2097152, 8388608, 2097152, 33554432, 33554432, 134217728, 67108864, 536870912, 536870912, 2147483648, 134217728, 8589934592, 8589934592, 34359738368, 17179869184, 137438953472, 137438953472, 549755813888, 137438953472

Offset: 0

Jean-François Alcover and Paul Curtz, Dec 16 2016

Cf. A005126, A075101, A271573.

a[0] = 1; a[n_] := Denominator[(2^(n-1)+n)/2^n]; Table[a[n], {n, 0, 40}]
(* or *)
a[0] = 1; a[n_] := 2^(n-IntegerExponent[2^(n-1)+n, 2]); Table[a[n], {n, 0, 40}]

a(n) = 2^(n-valuation(2^(n-1)+n,2)), with a(0) = 1.

A000051 a(n) = 2^n + 1.

Original entry on oeis.org

2, 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

Offset: 0

N. J. A. Sloane

Same as Pisot sequence L(2,3).
Length of the continued fraction for Sum_{k=0..n} 1/3^(2^k). - Benoit Cloitre, Nov 12 2003
See also A004119 for a(n) = 2a(n-1)-1 with first term = 1. - Philippe Deléham, Feb 20 2004
From the second term on (n>=1), in base 2, these numbers present the pattern 1000...0001 (with n-1 zeros), which is the "opposite" of the binary 2^n-2: (0)111...1110 (cf. A000918). - Alexandre Wajnberg, May 31 2005
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (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,3). - Milan Janjic, Jan 27 2010
First differences of A006127. - Reinhard Zumkeller, Apr 14 2011
The odd prime numbers in this sequence form A019434, the Fermat primes. - David W. Wilson, Nov 16 2011
Pisano period lengths: 1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 10, 2, 12, 3, 4, 1, 8, 6, 18, 4, ... . - R. J. Mathar, Aug 10 2012
Is the mentioned Pisano period lengths (see above) the same as A007733? - Omar E. Pol, Aug 10 2012
Only positive integers that are not 1 mod (2k+1) for any k>1. - Jon Perry, Oct 16 2012
For n >= 1, a(n) is the total length of the segments of the Hilbert curve after n iterations. - Kival Ngaokrajang, Mar 30 2014
Frénicle de Bessy (1657) proved that a(3) = 9 is the only square in this sequence. - Charles R Greathouse IV, May 13 2014
a(n) is the number of distinct possible sums made with at most two elements in {1,...,a(n-1)} for n > 0. - Derek Orr, Dec 13 2014
For n > 0, given any set of a(n) lattice points in R^n, there exist 2 distinct members in this set whose midpoint is also a lattice point. - Melvin Peralta, Jan 28 2017
Also the number of independent vertex sets, irredundant sets, and vertex covers in the (n+1)-star graph. - Eric W. Weisstein, Aug 04 and Sep 21 2017
Also the number of maximum matchings in the 2(n-1)-crossed prism graph. - Eric W. Weisstein, Dec 31 2017
Conjecture: For any integer n >= 0, a(n) is the permanent of the (n+1) X (n+1) matrix with M(j, k) = -floor((j - k - 1)/(n + 1)). This conjecture is inspired by the conjecture of Zhi-Wei Sun in A036968. - Peter Luschny, Sep 07 2021

Paul Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 75.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 46, 60, 244.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 141.

Ivan Panchenko, Table of n, a(n) for n = 0..100
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
Bakir Farhi, Summation of Certain Infinite Lucas-Related Series, J. Int. Seq., Vol. 22 (2019), Article 19.1.6.
Massimiliano Fasi and Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019).
Bartomeu Fiol, Jairo Martínez-Montoya, and Alan Rios Fukelman, The planar limit of N=2 superconformal field theories, arXiv:2003.02879 [hep-th], 2020.
Bernard Frénicle de Bessy, Solutio duorum problematum circa numeros cubos et quadratos, (1657). Bibliothèque Nationale de Paris.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 114
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 362
Edouard Lucas, The Theory of Simply Periodic Numerical Functions, Fibonacci Association, 1969. English translation of article "Théorie des Fonctions Numériques Simplement Périodiques, I", Amer. J. Math., 1 (1878), 184-240.
Kival Ngaokrajang, Illustration of Hilbert curve for n = 1..5
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
D. C. Santos, E. A. Costa, and P. M. M. C. Catarino, On Gersenne Sequence: A Study of One Family in the Horadam-Type Sequence, Axioms 14, 203, (2025). See p. 1.
Amelia Carolina Sparavigna, On the generalized sums of Mersenne, Fermat, Cullen and Woodall Numbers, Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, Composition Operations of Generalized Entropies Applied to the Study of Numbers, International Journal of Sciences (2019) Vol. 8, No. 4, 87-92.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Eric Weisstein's World of Mathematics, Crossed Prism Graph.
Eric Weisstein's World of Mathematics, Cunningham Number.
Eric Weisstein's World of Mathematics, Fermat-Lucas Number.
Eric Weisstein's World of Mathematics, Hilbert curve.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Irredundant Set.
Eric Weisstein's World of Mathematics, Matching Number.
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set.
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence.
Eric Weisstein's World of Mathematics, Star Graph.
Eric Weisstein's World of Mathematics, Vertex Cover.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Apart from the initial 1, identical to A094373.
See A008776 for definitions of Pisot sequences.
Column 2 of array A103438.
Cf. A007583 (a((n-1)/2)/3 for odd n).
Cf. A000079, A000225, A000918, A004119, A005126, A006217, A006521, A007583.
Cf. A007689, A007733, A019434, A024036, A027641, A027642, A034472, A034474.
Cf. A034491, A034524, A036968, A052539, A052944, A062394, A062395, A062396.
Cf. A062397, A063376, A063481, A074600 - A074624, A127904, A173786, A176691.
Cf. A178248, A194455, A228081, A323482.

a000051 = (+ 1) . a000079
a000051_list = iterate ((subtract 1) . (* 2)) 2
-- Reinhard Zumkeller, May 03 2012

[2^n+1: n in [0..40]]; // G. C. Greubel, Jan 18 2025

A000051:=-(-2+3*z)/(2*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation
a := n -> add(binomial(n,k)*bernoulli(n-k,1)*2^(k+1)/(k+1),k=0..n); # Peter Luschny, Apr 20 2009

Table[2^n + 1, {n,0,40}]
2^Range[0,40] + 1 (* Eric W. Weisstein, Jul 17 2017 *)
LinearRecurrence[{3, -2}, {2, 3}, 40] (* Eric W. Weisstein, Sep 21 2017 *)

PARI
```
a(n)=2^n+1
```

first(n) = Vec((2 - 3*x)/((1 - x)*(1 - 2*x)) + O(x^n)) \\ Iain Fox, Dec 31 2017

def A000051(n): return (1<Chai Wah Wu, Dec 21 2022

a(n) = 2*a(n-1) - 1 = 3*a(n-1) - 2*a(n-2).
G.f.: (2-3*x)/((1-x)*(1-2*x)).
First differences of A052944. - Emeric Deutsch, Mar 04 2004
a(0) = 1, then a(n) = (Sum_{i=0..n-1} a(i)) - (n-2). - Gerald McGarvey, Jul 10 2004
Inverse binomial transform of A007689. Also, V sequence in Lucas sequence L(3, 2). - Ross La Haye, Feb 07 2005
a(n) = A127904(n+1) for n>0. - Reinhard Zumkeller, Feb 05 2007
Equals binomial transform of [2, 1, 1, 1, ...]. - Gary W. Adamson, Apr 23 2008
a(n) = A000079(n)+1. - Omar E. Pol, May 18 2008
E.g.f.: exp(x) + exp(2*x). - Mohammad K. Azarian, Jan 02 2009
a(n) = A024036(n)/A000225(n). - Reinhard Zumkeller, Feb 14 2009
From Peter Luschny, Apr 20 2009: (Start)
A weighted binomial sum of the Bernoulli numbers A027641/A027642 with A027641(1)=1 (which amounts to the definition B_{n} = B_{n}(1)).
a(n) = Sum_{k=0..n} C(n,k)*B_{n-k}*2^(k+1)/(k+1). (See also A052584.) (End)
a(n) is the a(n-1)-th odd number for n >= 1. - Jaroslav Krizek, Apr 25 2009
From Reinhard Zumkeller, Feb 28 2010: (Start)
a(n)*A000225(n) = A000225(2*n).
a(n) = A173786(n,0). (End)
If p[i]=Fibonacci(i-4) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise, then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
a(n+2) = a(n) + a(n+1) + A000225(n). - Ivan N. Ianakiev, Jun 24 2012
a(A006521(n)) mod A006521(n) = 0. - Reinhard Zumkeller, Jul 17 2014
a(n) = 3*A007583((n-1)/2) for n odd. - Eric W. Weisstein, Jul 17 2017
Sum_{n>=0} 1/a(n) = A323482. - Amiram Eldar, Nov 11 2020

A176691 a(n) = 2^n + 2*n + 1.

Original entry on oeis.org

2, 5, 9, 15, 25, 43, 77, 143, 273, 531, 1045, 2071, 4121, 8219, 16413, 32799, 65569, 131107, 262181, 524327, 1048617, 2097195, 4194349, 8388655, 16777265, 33554483, 67108917, 134217783, 268435513, 536870971, 1073741885, 2147483711, 4294967361, 8589934659, 17179869253

Offset: 0

Jonathan Vos Post, Apr 23 2010

The subsequence of primes in this sequence is A163115.

Also the number of connected dominating sets in the (n+1)-wheel graph. - Eric W. Weisstein, Aug 30 2017

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Eric Weisstein's World of Mathematics, Connected Dominating Set
Eric Weisstein's World of Mathematics, Wheel Graph
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000051, A000079, A000290, A005126, A005843, A163115, A194455.

List([0..35],n->2^n+2*n+1); # Muniru A Asiru, Mar 25 2018

[2^n + 2*n + 1: n in [0..40]]; // Vincenzo Librandi, Aug 12 2015

seq(2^n+2*n+1,n=0..35); # Muniru A Asiru, Mar 25 2018

Table[2^n + 2 n + 1, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)
LinearRecurrence[{4, -5, 2}, {2, 5, 9}, 40] (* Vincenzo Librandi, Aug 12 2015 *)
CoefficientList[Series[(-2 + 3 x + x^2)/((-1 + x)^2 (-1 + 2 x)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 30 2017 *)

vector(40, n, n--; 2^n + 2*n + 1) \\ Michel Marcus, Aug 12 2015

a(n) = 2^n + 2*n + 1 = A000079(n) + A005843(n) + 1 = A000051(n) + A005843(n).
From R. J. Mathar, Apr 28 2010: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: (-2 + 3*x + x^2)/((2*x - 1)*(x - 1)^2). (End)
E.g.f.: exp(x)*(1 + exp(x) + 2*x). - Stefano Spezia, May 06 2023

Corrected (one 1048617 replaced by 2097195) by R. J. Mathar, Apr 28 2010

A267471 T(n,k)=Number of length-n 0..k arrays with no following elements larger than the first repeated value.

Original entry on oeis.org

2, 3, 4, 4, 9, 7, 5, 16, 24, 12, 6, 25, 58, 62, 21, 7, 36, 115, 204, 160, 38, 8, 49, 201, 515, 712, 418, 71, 9, 64, 322, 1096, 2285, 2490, 1112, 136, 10, 81, 484, 2072, 5921, 10119, 8770, 3018, 265, 11, 100, 693, 3592, 13216, 31880, 44901, 31200, 8352, 522, 12, 121

Offset: 1

R. H. Hardin, Jan 15 2016

Table starts
...2.....3......4.......5........6.........7.........8..........9.........10
...4.....9.....16......25.......36........49........64.........81........100
...7....24.....58.....115......201.......322.......484........693........955
..12....62....204.....515.....1096......2072......3592.......5829.......8980
..21...160....712....2285.....5921.....13216.....26440......48657......83845
..38...418...2490...10119....31880.....83972....193852.....404589.....779938
..71..1112...8770...44901...171601....532840...1418740....3357537....7240267
.136..3018..31200..200119...925176...3381860..10378144...27838701...67140808
.265..8352.112300..897301..5002641..21491464..75944464..230790033..622347697
.522.23522.409254.4052183.27155800.136856180.556295860.1914051597.5768860606

			Some solutions for n=6 k=4
..0....1....1....4....0....1....0....4....1....2....3....4....4....1....0....3
..4....0....4....0....4....0....3....0....2....3....1....4....3....4....1....0
..4....2....2....4....3....4....2....4....0....0....2....4....2....0....3....2
..0....3....4....0....3....4....3....3....1....1....4....2....4....4....4....3
..2....4....1....1....0....0....1....1....2....1....3....1....3....2....1....3
..4....4....1....1....3....2....0....2....0....0....0....1....4....3....1....1

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A005126(n-1).
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A081436.

Empirical for column k:
k=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3)
k=2: a(n) = 8*a(n-1) -23*a(n-2) +28*a(n-3) -12*a(n-4)
k=3: a(n) = 13*a(n-1) -65*a(n-2) +155*a(n-3) -174*a(n-4) +72*a(n-5)
k=4: a(n) = 19*a(n-1) -145*a(n-2) +565*a(n-3) -1174*a(n-4) +1216*a(n-5) -480*a(n-6)
k=5: [order 7]
k=6: [order 8]
k=7: [order 9]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + (5/2)*n^2 + (5/2)*n + 1
n=4: a(n) = n^4 + (17/6)*n^3 + 4*n^2 + (19/6)*n + 1
n=5: a(n) = n^5 + (37/12)*n^4 + (11/2)*n^3 + (77/12)*n^2 + 4*n + 1
n=6: a(n) = n^6 + (197/60)*n^5 + 7*n^4 + (43/4)*n^3 + 10*n^2 + (149/30)*n + 1
n=7: a(n) = n^7 + (69/20)*n^6 + (17/2)*n^5 + (97/6)*n^4 + 20*n^3 + (893/60)*n^2 + 6*n + 1

A356784 Inventory of positions as an irregular table; row 0 contains 0, subsequent rows contain the 0-based positions of 0's, followed by the position of 1's, of 2's, etc. in prior rows flattened.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 4, 3, 5, 6, 7, 0, 1, 2, 4, 8, 3, 5, 9, 6, 10, 7, 12, 11, 13, 14, 15, 0, 1, 2, 4, 8, 16, 3, 5, 9, 17, 6, 10, 18, 7, 12, 21, 11, 19, 13, 22, 14, 24, 15, 26, 20, 23, 25, 28, 27, 29, 30, 31, 0, 1, 2, 4, 8, 16, 32, 3, 5, 9, 17, 33

Offset: 0

Rémy Sigrist, Oct 01 2022

The n-th row contains A011782(n) terms, and is a permutation of 0..A011782(n)-1.

The leading term of each row is 0, and is followed by powers of 2, and then by positive nonpowers of 2.

			Table begins:
   0,
   0,
   0, 1,
   0, 1, 2, 3,
   0, 1, 2, 4, 3, 5, 6, 7,
   0, 1, 2, 4, 8, 3, 5, 9, 6, 10, 7, 12, 11, 13, 14, 15,
   ...
For n = 5:
- the terms in rows 0..4 are: 0, 0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 4, 3, 5, 6, 7,
- we have 0's at positions 0, 1, 2, 4, 8,
- we have 1's at positions 3, 5, 9,
- we have 2's at positions 6, 10,
- we have 3's at positions 7, 12,
- we have one 4 at position 11,
- we have one 5 at position 13,
- we have one 6 at position 14,
- we have one 7 at position 15,
- so row 5 is: 0, 1, 2, 4, 8, 3, 5, 9, 6, 10, 7, 12, 11, 13, 14, 15.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Scatterplot of first 2^22 terms
Rémy Sigrist, C++ program

Cf. A000051, A005126, A011782 (row lengths), A052548, A131577, A342585, A357317, A357491.

terms = [0,]
for i in range(1,10):
    new_terms = []
    for j in range(max(terms)+1):
        for k in range(len(terms)):
            if terms[k] == j: new_terms.append(k)
    terms.extend(new_terms)
print(terms) # Gleb Ivanov, Nov 01 2022

a(n) = 0 iff n belongs to A131577.
a(n) = 1 iff n belongs to A000051 \ {2}.
a(n) = 2 iff n belongs to A052548 \ {3, 4}.
a(n) = 3 iff n belongs to A005126 \ {2, 4}.
T(n, 0) = 0.
T(n, k) = 2^(k-1) for k = 1..n-1.

A100314 Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

Original entry on oeis.org

1, 4, 8, 14, 24, 42, 76, 142, 272, 530, 1044, 2070, 4120, 8218, 16412, 32798, 65568, 131106, 262180, 524326, 1048616, 2097194, 4194348, 8388654, 16777264, 33554482, 67108916, 134217782, 268435512, 536870970, 1073741884, 2147483710, 4294967360, 8589934658

Offset: 0

Sergey Kitaev, Nov 13 2004

An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^m + 2^n + 2*(n*m-n-m).

Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Ronald Cools, Encyclopaedia of Cubature Formulas
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. this sequence (m=2), A100315 (m=3), A100316 (m=4).
Cf. A000051, A000079, A001787, A002064, A005126, A005843.
Cf. A052548, A058331, A099003, A132750, A176691.
Row sums of A131830.

List([0..40],n->2^n+2*n); # Muniru A Asiru, Dec 21 2018

[2^n+2*n: n in [1..40]]; // Vincenzo Librandi, Oct 22 2011

a:= proc(n) 2^n + 2*n: end: seq(a(n),n=0..50); # Gary W. Adamson, Jul 20 2007

LinearRecurrence[{4,-5,2}, {1,4,8}, 34] (* Jean-François Alcover, Mar 19 2020 *)

makelist(2^n + 2*n, n, 0, 50); /* Franck Maminirina Ramaharo, Dec 19 2018 */

[2^n +2*n for n in range(41)] # G. C. Greubel, Feb 01 2023

a(n) = 2^n + 2*n.
From Gary W. Adamson, Jul 20 2007: (Start)
Binomial transform of (1, 3, 1, 1, 1, ...).
For n > 0, a(n) = 2*A005126(n-1). (End)
From R. J. Mathar, Jun 13 2008: (Start)
G.f.: 1 + 2*x*(2 -4*x +x^2)/((1-x)^2*(1-2*x)).
a(n+1)-a(n) = A052548(n). (End)
From Colin Barker, Oct 16 2013: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: (1 - 3*x^2)/((1-x)^2*(1-2*x)). (End)
E.g.f.: exp(2*x) + 2*x*exp(x). - Franck Maminirina Ramaharo, Dec 19 2018
a(n) = A000079(n) + A005843(n). - Muniru A Asiru, Dec 21 2018

a(0)=1 prepended by Alois P. Heinz, Dec 21 2018

A228367 n-th element of the ruler function plus the highest power of 2 dividing n.

Original entry on oeis.org

2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 38, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 71, 2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 21, 2, 4, 2, 7

Offset: 1

Omar E. Pol, Aug 22 2013

a(n) is also the length of the n-th pair of orthogonal line segments in a diagram of compositions, see example.
a(n) is also the largest part plus the number of parts of the n-th region of the mentioned diagram (if the axes both "x" and "y" are included in the diagram).
a(n) is also the number of toothpicks added at n-th stage to the structure of A228366. Essentially the first differences of A228366.
The equivalent sequence for partitions is A207779.

			Illustration of initial terms (n = 1..16) using a diagram of compositions in which A001511(n) is the length of the horizontal line segment in row n and A006519(n) is the length of the vertical line segment ending in row n. Hence a(n) is the length of the n-th pair of orthogonal line segments. Also counting both the x-axis and the y-axis we have that A001511(n) is also the largest part of the n-th region of the diagram and A006519(n) is also the number of parts of the n-th region of the diagram, see below.
---------------------------------------------------------
.                Diagram of
n   A001511(n)  compositions   A006519(n)    a(n)
---------------------------------------------------------
1       1        _| | | | |        1          2
2       2        _ _| | | |        2          4
3       1        _|   | | |        1          2
4       3        _ _ _| | |        4          7
5       1        _| |   | |        1          2
6       2        _ _|   | |        2          4
7       1        _|     | |        1          2
8       4        _ _ _ _| |        8         12
9       1        _| | |   |        1          2
10      2        _ _| |   |        2          4
11      1        _|   |   |        1          2
12      3        _ _ _|   |        4          7
13      1        _| |     |        1          2
14      2        _ _|     |        2          4
15      1        _|       |        1          2
16      5        _ _ _ _ _|       16         21
...
If written as an irregular triangle the sequence begins:
  2;
  4;
  2, 7;
  2, 4, 2, 12;
  2, 4, 2, 7, 2, 4, 2, 21;
  2, 4, 2, 7, 2, 4, 2, 12, 2, 4, 2, 7, 2, 4, 2, 38;
  ...
Row lengths is A011782. Right border gives A005126.
Counting both the x-axis and the y-axis we have that A038712(n) is the area (or the number of cells) of the n-th region of the diagram. Note that adding only the x-axis to the diagram we have a tree. - _Omar E. Pol_, Nov 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..16383
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A001511, A001792, A005126, A006519, A011782, A038712, A139250, A139251, A207779, A228366.

Array[1 + # + 2^# &[IntegerExponent[#, 2]] &, 84] (* Michael De Vlieger, Nov 06 2018 *)

A228367(n) = (1 + valuation(n,2) + 2^valuation(n,2)); \\ Antti Karttunen, Nov 06 2018

def A228367(n): return (m:=n&-n)+m.bit_length() # Chai Wah Wu, Jul 14 2022

a(n) = A001511(n) + A006519(n).

A194455 a(n) = 2^n + 3n + 1.

Original entry on oeis.org

2, 6, 11, 18, 29, 48, 83, 150, 281, 540, 1055, 2082, 4133, 8232, 16427, 32814, 65585, 131124, 262199, 524346, 1048637, 2097216, 4194371, 8388678, 16777289, 33554508, 67108943, 134217810, 268435541, 536871000, 1073741915, 2147483742, 4294967393, 8589934692, 17179869287

Offset: 0

Bruno Berselli, Sep 01 2011

Inverse binomial transform of this sequence: 2,4,1,1 (1 continued).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000051, A005126, A176691; A120845.
Cf. A062709 (first differences), A000079 (second and successive differences).
Cf. A146529 (differences between alternate terms, for n>2).
Cf. A086653, A115067.

Magma
```
[2^n+3*n+1: n in [0..31]];
```

Table[2^n + 3 n + 1, {n, 0, 40}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{4,-5,2},{2,6,11},40] (* Harvey P. Dale, Oct 01 2014 *)

PARI
```
for(n=0, 31, print1(2^n+3*n+1", "));
```

G.f.: (2 - 2*x - 3*x^2)/((1 - 2*x)*(1 - x)^2).
a(n) = A086653(n) - 1  for n > 0.
Sum_{i=0..n} a(i) = A115067(n+1) + 2^(n+1).
a(n) = 3*a(n-1) - 2*a(n-2) - 3  for n > 1.
a(n)^2 = 2^(n+1)*(a(n-1) + 3) + (3*n + 1)^2  for n > 2.
E.g.f.: exp(x)*(1 + exp(x) + 3*x). - Stefano Spezia, May 06 2023

A288133 Positions of 0 in A288132; complement of A288134.

Original entry on oeis.org

1, 2, 4, 7, 12, 21, 38, 71, 136, 265, 522, 1035, 2060, 4109, 8206, 16399, 32784, 65553, 131090, 262163, 524308, 1048597, 2097174, 4194327, 8388632, 16777241, 33554458, 67108891, 134217756, 268435485

Offset: 1

Clark Kimberling, Jun 07 2017

a(n+1)/a(n) -> 2. It appears that a(n) = A005126(n-2) for n >= 2.

This conjecture by Kimberling is proved in A288132. - Michel Dekking, Feb 18 2021

Cf. A288132, A288134.

s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "1" -> "11"}]
Table[w[n], {n, 0, 8}]
st = ToCharacterCode[w[11]] - 48   (* A288132 *)
Flatten[Position[st, 0]]  (* A288133 *)
Flatten[Position[st, 1]]  (* A288134 *)

Conjectures from Colin Barker, Jun 09 2017: (Start)
G.f.: x*(1 - 2*x + x^2 - x^3) / ((1 - x)^2*(1 - 2*x)).
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3) for n>4.
(End)
Colin Barker's conjectures are a consequence of
a(n) = 2^{n-2} + n - 1 = A005126(n-2) for n >= 2. - Michel Dekking, Feb 18 2021

cp's OEIS Frontend

A271573 Numerator of (0 followed by A005126(n)= 2, 4, 7, ...)/2^n.

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A279635 Denominator of (0 followed by A005126(n)= 2, 4, 7, ...)/2^n, a sequence corresponding to A271573.

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A000051 a(n) = 2^n + 1.

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A176691 a(n) = 2^n + 2*n + 1.

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A267471 T(n,k)=Number of length-n 0..k arrays with no following elements larger than the first repeated value.

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A356784 Inventory of positions as an irregular table; row 0 contains 0, subsequent rows contain the 0-based positions of 0's, followed by the position of 1's, of 2's, etc. in prior rows flattened.

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A100314 Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

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