A186947 -id:A186947 - cp's OEIS frontend

A000325 a(n) = 2^n - n.

1, 1, 2, 5, 12, 27, 58, 121, 248, 503, 1014, 2037, 4084, 8179, 16370, 32753, 65520, 131055, 262126, 524269, 1048556, 2097131, 4194282, 8388585, 16777192, 33554407, 67108838, 134217701, 268435428, 536870883, 1073741794, 2147483617

Offset: 0

Rosario Salamone (Rosario.Salamone(AT)risc.uni-linz.ac.at)

Number of permutations of degree n with at most one fall; called "Grassmannian permutations" by Lascoux and Schützenberger. - Axel Kohnert (Axel.Kohnert(AT)uni-bayreuth.de)
Number of different permutations of a deck of n cards that can be produced by a single shuffle. [DeSario]
Number of Dyck paths of semilength n having at most one long ascent (i.e., ascent of length at least two). Example: a(4)=12 because among the 14 Dyck paths of semilength 4, the only paths that have more than one long ascent are UUDDUUDD and UUDUUDDD (each with two long ascents). Here U = (1, 1) and D = (1, -1). Also number of ordered trees with n edges having at most one branch node (i.e., vertex of outdegree at least two). - Emeric Deutsch, Feb 22 2004
Number of {12,1*2*,21*}-avoiding signed permutations in the hyperoctahedral group.
Number of 1342-avoiding circular permutations on [n+1].
2^n - n is the number of ways to partition {1, 2, ..., n} into arithmetic progressions, where in each partition all the progressions have the same common difference and have lengths at least 1. - Marty Getz (ffmpg1(AT)uaf.edu) and Dixon Jones (fndjj(AT)uaf.edu), May 21 2005
if b(0) = x and b(n) = b(n-1) + b(n-1)^2*x^(n-2) for n > 0, then b(n) is a polynomial of degree a(n). - Michael Somos, Nov 04 2006
The chromatic invariant of the Mobius ladder graph M_n for n >= 2. - Jonathan Vos Post, Aug 29 2008
Dimension sequence of the dual alternative operad (i.e., associative and satisfying the identity xyz + yxz + zxy + xzy + yzx + zyx = 0) over the field of characteristic 3. - Pasha Zusmanovich, Jun 09 2009
An elephant sequence, see A175654. For the corner squares six A[5] vectors, with decimal values between 26 and 176, lead to this sequence (without the first leading 1). For the central square these vectors lead to the companion sequence A168604. - Johannes W. Meijer, Aug 15 2010
a(n+1) is also the number of order-preserving and order-decreasing partial isometries (of an n-chain). - Abdullahi Umar, Jan 13 2011
A040001(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
A130103(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, May 12 2012
The number of labeled graphs with n vertices whose vertex set can be partitioned into a clique and a set of isolated points. - Alex J. Best, Nov 20 2012
For n > 0, a(n) is a B_2 sequence. - Thomas Ordowski, Sep 23 2014
See coefficients of the linear terms of the polynomials of the table on p. 10 of the Getzler link. - Tom Copeland, Mar 24 2016
Consider n points lying on a circle, then for n>=2 a(n-1) is the maximum number of ways to connect two points with non-intersecting chords. - Anton Zakharov, Dec 31 2016
Also the number of cliques in the (n-1)-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
From Eric M. Schmidt, Jul 17 2017: (Start)
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) < e(k). [Martinez and Savage, 2.7]
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i), e(j), e(k) pairwise distinct. [Martinez and Savage, 2.7]
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(j) >= e(k) and e(i) != e(k) pairwise distinct. [Martinez and Savage, 2.7]
(End)
Number of F-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are F-equivalent iff the positions of pattern F are identical in these paths. -  Sergey Kirgizov, Apr 08 2018
From Gus Wiseman, Feb 10 2019: (Start)
Also the number of connected partitions of an n-cycle. For example, the a(1) = 1 through a(4) = 12 connected partitions are:
  {{1}}  {{12}}    {{123}}      {{1234}}
         {{1}{2}}  {{1}{23}}    {{1}{234}}
                   {{12}{3}}    {{12}{34}}
                   {{13}{2}}    {{123}{4}}
                   {{1}{2}{3}}  {{124}{3}}
                                {{134}{2}}
                                {{14}{23}}
                                {{1}{2}{34}}
                                {{1}{23}{4}}
                                {{12}{3}{4}}
                                {{14}{2}{3}}
                                {{1}{2}{3}{4}}
(End)
Number of subsets of n-set without the single-element subsets. - Yuchun Ji, Jul 16 2019
For every prime p, there are infinitely many terms of this sequence that are divisible by p (see IMO Compendium link and Doob reference). Corresponding indices n are: for p = 2, even numbers A299174; for p = 3, A047257; for p = 5, A349767. - Bernard Schott, Dec 10 2021
Primes are in A081296 and corresponding indices in A048744. - Bernard Schott, Dec 12 2021

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 27*x^5 + 58*x^6 + 121*x^7 + ...

Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1983, page 158, 1993.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Kassie Archer and Aaron Geary, Descents in powers of permutations, arXiv:2406.09369 [math.CO], 2024.
Cory B. H. Ball, The Apprentices' Tower of Hanoi, Electronic Theses and Dissertations, East Tennessee State University, Paper 2512, 2015.
Jean-Luc Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178;
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
David Callan, Pattern avoidance in circular permutations, arXiv:math/0210014 [math.CO], 2002.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
Robert DeSario and LeRoy Wenstrom, Invertible shuffles, Problem 10931, Amer. Math. Monthly, 111 (No. 2, 2004), 169-170.
Askar Dzhumadil'daev and Pasha Zusmanovich, The alternative operad is not Koszul, arXiv:0906.1272 [math.RA], 2009-2013.
Marty Getz, Dixon Jones and Ken Dutch, Partitioning by Arithmetic Progressions: Problem 11005, American Math. Monthly, Vol. 112, 2005, p. 89. (The published solution is incomplete. Letting d be the common difference of the arithmetic progressions, the solver's expression q_1(n,d)=2^(n-d) must be summed over all d=1,...,n and duplicate partitions must be removed.)
E. Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996.
Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, arXiv:2112.03338 [math.CO], 2021.
Juan B. Gil and Michael D. Weiner, On pattern-avoiding Fishburn permutations, arXiv:1812.01682 [math.CO], 2018.
The IMO Compendium, Problem 4, 15th Canadian Mathematical Olympiad 1983.
R. Kehinde, S. O. Makanjuola and A. Umar, On the semigroup of order-decreasing partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.
Alain Lascoux and Marcel-Paul Schützenberger, Schubert polynomials and the Littlewood Richardson rule, Letters in Math. Physics 10 (1985) 111-124.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Eric Weisstein's World of Mathematics, Chromatic Invariant
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Moebius Ladder
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).
Index to sequences related to Olympiads.

Column 1 of triangle A008518.
Row sum of triangles A184049 and A184050.
Cf. A000108, A002064, A133116, A160692, A005803, A006127, A186947.
Cf. A001610, A066982, A323950, A323951.
Cf. A047257, A299174, A349767.
Cf.  A048744, A081296.

a000325 n = 2 ^ n - n
a000325_list = zipWith (-) a000079_list [0..]
-- Reinhard Zumkeller, Jul 17 2012

[2^n - n: n in [0..35]]; // Vincenzo Librandi, May 13 2011

A000325 := proc(n) option remember; if n <=1 then n+1 else 2*A000325(n-1)+n-1; fi; end;
g:=1/(1-2*z): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-n, n=0..31); # Zerinvary Lajos, Jan 09 2009

Table[2^n - n, {n, 0, 39}] (* Alonso del Arte, Sep 15 2014 *)
LinearRecurrence[{4, -5, 2}, {1, 2, 5}, {0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)

{a(n) = 2^n - n}; /* Michael Somos, Nov 04 2006 */

def A000325(n): return (1<Chai Wah Wu, Jan 11 2023

a(n+1) = 2*a(n) + n - 1, a(0) = 1. - Reinhard Zumkeller, Apr 12 2003
Binomial transform of 1, 0, 1, 1, 1, .... The sequence starting 1, 2, 5, ... has a(n) = 1 + n + 2*Sum_{k=2..n} binomial(n, k) = 2^(n+1) - n - 1. This is the binomial transform of 1, 1, 2, 2, 2, 2, .... a(n) = 1 + Sum_{k=2..n} C(n, k). - Paul Barry, Jun 06 2003
G.f.: (1-3x+3x^2)/((1-2x)*(1-x)^2). - Emeric Deutsch, Feb 22 2004
A107907(a(n+2)) = A000051(n+2) for n > 0. - Reinhard Zumkeller, May 28 2005
a(n+1) = sum of n-th row of the triangle in A109128. - Reinhard Zumkeller, Jun 20 2005
Row sums of triangle A133116. - Gary W. Adamson, Sep 14 2007
G.f.: 1 / (1 - x / (1 - x / ( 1 - x / (1 + x / (1 - 2*x))))). - Michael Somos, May 12 2012
First difference is A000225. PSUM transform is A084634. - Michael Somos, May 12 2012
a(n) = [x^n](B(x)^n-B(x)^(n-1)), n>0, a(0)=1, where B(x) = (1+2*x+sqrt(1+4*x^2))/2. - Vladimir Kruchinin, Mar 07 2014
E.g.f.: (exp(x) - x)*exp(x). - Ilya Gutkovskiy, Aug 07 2016
a(n) = A125128(n) - A000225(n) + 1. - Miquel Cerda, Aug 12 2016
a(n) = 2*A125128(n) - A095151(n) + 1. - Miquel Cerda, Aug 12 2016
a(n) = A079583(n-1) - A000225(n-1). - Miquel Cerda, Aug 15 2016
a(n)^2 - 4*a(n-1)^2 = (n-2)*(a(n)+2*a(n-1)). - Yuchun Ji, Jul 13 2018
a(n) = 2^(-n) * A186947(n) = 2^n * A002064(-n) for all n in Z. - Michael Somos, Jul 18 2018
a(2^n) = (2^a(n) - 1)*2^n. - Lorenzo Sauras Altuzarra, Feb 01 2022

A002064 Cullen numbers: a(n) = n*2^n + 1.

Original entry on oeis.org

1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521, 44040193, 92274689, 192937985, 402653185, 838860801, 1744830465, 3623878657, 7516192769, 15569256449, 32212254721, 66571993089

Offset: 0

N. J. A. Sloane

Binomial transform is A084859. Inverse binomial transform is A004277. - Paul Barry, Jun 12 2003
Let A be the Hessenberg matrix of order n defined by: A[1,j]=1, A[i,i]:=2,(i>1), A[i,i-1] =-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= (-1)^(n-1)*coeff(charpoly(A,x),x). - Milan Janjic, Jan 26 2010
Indices of primes are listed in A005849. - M. F. Hasler, Jan 18 2015
Add the list of fractions beginning with 1/2 + 3/4 + 7/8 + ... + (2^n - 1)/2^n and take the sums pairwise from left to right. For 1/2 + 3/4 = 5/4, 5 + 4 = 9 = a(2); for 5/4 + 7/8 = 17/8, 17 + 8 = 25 = a(3); for 17/8 + 15/16 = 49/16, 49 + 16 = 65 = a(4); for 49/16 + 31/32 = 129/32, 129 + 32 = 161 = a(5). For each pairwise sum a/b, a + b = n*2^(n+1). - J. M. Bergot, May 06 2015
Number of divisors of (2^n)^(2^n). - Gus Wiseman, May 03 2021
Named after the Irish Jesuit priest James Cullen (1867-1933), who checked the primality of the terms up to n=100. - Amiram Eldar, Jun 05 2021

			G.f. = 1 + 3*x + 9*x^2 + 25*x^3 + 65*x^4 + 161*x^5 + 385*x^6 + 897*x^7 + ... - _Michael Somos_, Jul 18 2018

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
R. K. Guy, Unsolved Problems in Number Theory, B20.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 240-242.
W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 346.
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).

T. D. Noe, Table of n, a(n) for n=0..300
Ray Ballinger, Cullen Primes: Definition and Status.
Attila Bérczes, István Pink, and Paul Thomas Young, Cullen numbers and Woodall numbers in generalized Fibonacci sequences, J. Num. Theor. (2024) Vol. 262, 86-102.
Yuri Bilu, Diego Marques, and Alain Togbé, Generalized Cullen numbers in linear recurrence sequences, Journal of Number Theory, Vol. 202 (2019), pp. 412-425; arXiv preprint, arXiv:1806.09441 [math.NT], 2018.
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
C. K. Caldwell, The Top Twenty: Cullen Primes.
James Cullen, Question 15897, Educational Times, Vol. 58 (December 1905), p. 534.
Orhan Eren and Yüksel Soykan, Gaussian Generalized Woodall Numbers, Arch. Current Res. Int'l (2023) Vol. 23, Iss. 8, Art. No. ACRI.108618, 48-68. See p. 50.
Orhan Eren and Yüksel Soykan, On Dual Hyperbolic Generalized Woodall Numbers, Archives Current Res. Int'l (2024) Vol. 24, Iss. 11, Art. No. ACRI.126420, 398-423. See p. 401.
Jon Grantham and Hester Graves, The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits, arXiv:2009.04052 [math.NT], 2020.
José María Grau and Florian Luca, Cullen numbers with the Lehmer property, Proceedings of the American Mathematical Society, Vol. 140, No. 1 (2012), pp. 129-134; arXiv preprint, arXiv:1103.3578 [math.NT], Mar 18 2011.
Paul Leyland, Factors of Cullen and Woodall numbers.
Paul Leyland, Generalized Cullen and Woodall numbers.
Diego Marques, On Generalized Cullen and Woodall Numbers That are Also Fibonacci Numbers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.4.
Hisanori Mishima, Factorizations of many number sequences, Cullen numbers (n = 1 to 100), (n = 101 to 200), (n = 201 to 300), (n = 301 to 323).
Simon Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Wacław Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
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, Vol. 8, No. 4 (2019), pp. 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, Vol. 8, No. 10 (2019).
Eric Weisstein's World of Mathematics, Cullen Number.
Wikipedia, Cullen number.
Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Cf. A005849, A003261, A050914, A130197, A134081, A001787, A143038, A156708, A181527.
Cf. A000325, A186947.
Diagonal k = n + 1 of A046688.
A000005 counts divisors of n.
A000312 = n^n.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
A173339 lists positions of squares in A062319.
A188385 gives the highest prime exponent in n^n.
A249784 counts divisors of n^n^n.
Cf. A000169, A000272, A036289, A066959, A176029, A340841, A343656.

a002064 n = n * 2 ^ n + 1
a002064_list = 1 : 3 : (map (+ 1) $ zipWith (-) (tail xs) xs)
   where xs = map (* 4) a002064_list
-- Reinhard Zumkeller, Mar 16 2013

[n*2^n + 1: n in [0..40]]; // Vincenzo Librandi, May 07 2015

A002064:=-(1-2*z+2*z**2)/((z-1)*(-1+2*z)**2); # conjectured by Simon Plouffe in his 1992 dissertation

Table[n*2^n+1,{n,0,2*4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 25 2010 *)
LinearRecurrence[{5,-8,4},{1,3,9},51] (* Harvey P. Dale, Oct 13 2011 *)
CoefficientList[Series[(1 - 2 x + 2 x^2)/((1 - x) (2 x - 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, May 07 2015 *)

A002064(n)=n*2^n+1  \\ M. F. Hasler, Oct 31 2012

a(n) = 4a(n-1) - 4a(n-2) + 1. - Paul Barry, Jun 12 2003
a(n) = sum of row (n+1) of triangle A130197. Example: a(3) = 25 = (12 + 8 + 4 + 1), row 4 of A130197. - Gary W. Adamson, May 16 2007
Row sums of triangle A134081. - Gary W. Adamson, Oct 07 2007
Equals row sums of triangle A143038. - Gary W. Adamson, Jul 18 2008
Equals row sums of triangle A156708. - Gary W. Adamson, Feb 13 2009
G.f.: -(1-2*x+2*x^2)/((-1+x)*(2*x-1)^2). a(n) = A001787(n+1)+1-A000079(n). - R. J. Mathar, Nov 16 2007
a(n) = 1 + 2^(n + log_2(n)) ~ 1 + A000079(n+A004257(n)). a(n) ~ A000051(n+A004257(n)). - Jonathan Vos Post, Jul 20 2008
a(0)=1, a(1)=3, a(2)=9, a(n) = 5*a(n-1)-8*a(n-2)+4*a(n-3). - Harvey P. Dale, Oct 13 2011
a(n) = A036289(n) + 1 = A003261(n) + 2. - Reinhard Zumkeller, Mar 16 2013
E.g.f.: 2*x*exp(2*x) + exp(x). - Robert Israel, Dec 12 2014
a(n) = 2^n * A000325(n) = 4^n * A186947(-n) for all n in Z. - Michael Somos, Jul 18 2018
a(n) = Sum_{i=0..n-1} a(i) + A000325(n+1). - Ivan N. Ianakiev, Aug 07 2019
a(n) = sigma((2^n)^(2^n)) = A000005(A057156(n)) = A062319(2^n). - Gus Wiseman, May 03 2021
Sum_{n>=0} 1/a(n) = A340841. - Amiram Eldar, Jun 05 2021

Edited by M. F. Hasler, Oct 31 2012

cp's OEIS Frontend

A000325 a(n) = 2^n - n.

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A002064 Cullen numbers: a(n) = n*2^n + 1.

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A186948 a(n) = 3^n - 2*n.

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A186949 a(n) = 2^n - 2*(binomial(1,n) - binomial(0,n)).

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