A000244 -id:A000244 - cp's OEIS frontend

A140298 a(0)=1; a(3n+1) = a(3n)+1, a(3n+2) = a(3n+1) + a(3n) (=3*A000244), a(3n+3) = a(3n+2) + a(3n) (=A003462(n+2)).

1, 2, 3, 4, 5, 9, 13, 14, 27, 40, 41, 81, 121, 122, 243, 364, 365, 729, 1093, 1094, 2187, 3280, 3281, 6561, 9841, 9842, 19683, 29524, 29525, 59049, 88573, 88574, 177147, 265720, 265721, 531441, 797161, 797162, 1594323, 2391484, 2391485, 4782969, 7174453, 7174454

Offset: 0

Paul Curtz, May 25 2008

Note period 12 for a(n) mod 10.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (-1,-1,3,3,3)

Cf. A107365.

Vec((1+3*x+6*x^2+6*x^3+3*x^4)/((1+x+x^2)*(1-3*x^3))+O(x^99)) \\ Charles R Greathouse IV, Jun 20 2011

From R. J. Mathar, Jan 17 2009: (Start)
G.f.: (1 + 3*x + 6*x^2 + 6*x^3 + 3*x^4)/((1 + x + x^2)*(1 - 3*x^3)).
a(n) = -a(n-1) - a(n-2) + 3*a(n-3) + 3*a(n-4) + 3*a(n-5).
a(n) = (3*b(n)-A049347(n))/2 where b(n)=1,1,2,3,3,6,9,9,18,27,27,54,.. = 3*b(n-3).
(End)

a(27)-a(43) added by Andrew Howroyd, Nov 03 2018

A248163 Chebyshev's S polynomials (A049310) evaluated at 34/3 and multiplied by powers of 3 (A000244).

Original entry on oeis.org

1, 34, 1147, 38692, 1305205, 44028742, 1485230383, 50101574344, 1690086454249, 57012025275370, 1923198081274339, 64875626535849196, 2188462519487403613, 73823845023749080078, 2490314568132082090135, 84006280711277049343888, 2833800713070230938880977, 95593167717986358477858226

Offset: 0

Wolfdieter Lang, Nov 07 2014

This sequence appears in the solution for the curvature sequence of the touching circle and chord example given in A249457. See also the pair A249862(n) and a(n-1), with a(-1) = 0, for which details are given in A249862.

G. C. Greubel, Table of n, a(n) for n = 0..650
Index entries for linear recurrences with constant coefficients, signature (34,-9).

Cf. A000244, A049310, A249457, A249862.

I:=[1, 34]; [n le 2 select I[n] else 34*Self(n-1) - 9*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2014

CoefficientList[Series[1/(1-34 x +(3 x)^2), {x,0,40}], x] (* Vincenzo Librandi, Nov 08 2014 *)
Table[3^n*ChebyshevU[n,17/3], {n,0,40}] (* G. C. Greubel, May 31 2025 *)

a(n) = 3^n*polchebyshev(n, 2, 17/3); \\ Michel Marcus, May 31 2025

def A248163(n): return 3^n*chebyshev_U(n,17/3)
print([A248163(n) for n in range(41)]) # G. C. Greubel, May 31 2025

a(n) = 3^n*S(n, 34/3) with Chebyshev's S polynomial (for S see the coefficient triangle A049310).
O.g.f.: 1/(1 - 34*x + 9*x^2).
a(n) = 34*a(n-1) - 9*a(n-2), a(-1) = 0, a(0) = 1 .
E.g.f.: exp(17*x)*(140*cosh(2*sqrt(70)*x) + 17*sqrt(70)*sinh(2*sqrt(70)*x))/140. - Stefano Spezia, Mar 24 2023

a(16)-a(17) from Stefano Spezia, Mar 24 2023

A133647 A133566 * A000244.

Original entry on oeis.org

1, 3, 12, 27, 108, 243, 972, 2187, 8748, 19683, 78732, 177147, 708588, 1594323, 6377292, 14348907, 57395628, 129140163, 516560652, 1162261467, 4649045868, 10460353203, 41841412812, 94143178827, 376572715308, 847288609443, 3389154437772, 7625597484987, 30502389939948

Offset: 0

Gary W. Adamson, Sep 19 2007

A133646 is a companion case.

			a(3) = 27 = 3^3.
a(4) = 108 = 4 * 3^3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,9).

Cf. A000244, A133566, A133646.

Join[{1},Flatten[NestList[9#&,{3,12},20]]] (* or *) LinearRecurrence[{0,9},{1,3,12},40] (* Harvey P. Dale, Aug 01 2019 *)

A133566 * A000244, where A133566 = an infinite lower triangular matrix and A000244 = [3^0, 3^1, 3^2, ...]. For odd n, a(n) = 3^n. For even n, a(n) = 4 * 3^(n-1).
From Philippe Deléham, Apr 06 2012: (Start)
G.f.: (1+3*x+3*x^2)/(1-9*x^2).
a(n+2) = 9*a(n), a(0) = 1, a(1) = 3, a(2) = 12. (End)
From Amiram Eldar, Jun 02 2025: (Start)
Sum_{n>=0} 1/a(n) = 47/32.
Sum_{n>=0} (-1)^n/a(n) = 23/32. (End)

More terms from Philippe Deléham, Apr 06 2012

A101555 Convolution of A010060 and A000244.

Original entry on oeis.org

0, 1, 4, 12, 37, 111, 333, 1000, 3001, 9003, 27009, 81028, 243084, 729253, 2187760, 6563280, 19689841, 59069523, 177208569, 531625708, 1594877124, 4784631373, 14353894120, 43061682360, 129185047080, 387555141241, 1162665423724

Offset: 0

Paul Barry, Dec 06 2004

A Thue-Morse convolution.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000244, A010060.

Table[Sum[ThueMorse[n - k]*3^k, {k, 0, n}], {n, 0, 100}] (* G. C. Greubel, May 23 2018 *)

a(n) = Sum_{k=0..n} A010060(n-k) * 3^k.

A141317 a(n) = A000244(n) - A010684(n).

Original entry on oeis.org

0, 0, 8, 24, 80, 240, 728, 2184, 6560, 19680, 59048, 177144, 531440, 1594320, 4782968, 14348904, 43046720, 129140160, 387420488, 1162261464, 3486784400, 10460353200, 31381059608, 94143178824, 282429536480, 847288609440

Offset: 0

Paul Curtz, Aug 02 2008

nonn

Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

A010684 := proc(n) 1+2*(n mod 2) ; end: A141317 := proc(n) 3^n-A010684(n) ; end: for n from 0 to 50 do printf("%d,",A141317(n)) ; od: # R. J. Mathar, Aug 09 2008

CoefficientList[Series[8x^2/((1-3x)(1-x)(1+x)),{x,0,30}],x] (* Harvey P. Dale, Sep 16 2019 *)

O.g.f.: 8*x^2/((1-3*x)*(1-x)*(1+x)). - R. J. Mathar, Aug 09 2008

Extended by R. J. Mathar, Aug 09 2008

A203197 (n-1)-st elementary symmetric function of the first n terms of (1,3,9,27,...)=A000244.

Original entry on oeis.org

1, 4, 39, 1080, 88209, 21493836, 15683355351, 34309958505840, 225130514549271201, 4431394012508602048404, 261672339357326993189906439, 46354644349343413982791427120040, 24634789450813795903041020740742981169

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A000244, A003462 (1st symm. func.), A203243 (2nd symm. func.).

f[k_] := 3^(k - 1); t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}]  (* A203197 *)
Table[1/2 (3 - 1/3^(n-1)) 3^Binomial[n, 2], {n, 1, 20}] (* Emanuele Munarini, Sep 14 2017 *)

a(n) = (1/2)*(3-1/3^(n-1))*3^(binomial(n,2)). - Emanuele Munarini, Sep 14 2017

A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

Original entry on oeis.org

1, 3, 4, 9, 12, 16, 27, 36, 48, 64, 81, 108, 144, 192, 256, 243, 324, 432, 576, 768, 1024, 729, 972, 1296, 1728, 2304, 3072, 4096, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 6561, 8748, 11664, 15552, 20736, 27648, 36864, 49152, 65536, 19683, 26244, 34992, 46656, 62208, 82944, 110592, 147456, 196608, 262144

Offset: 0

Wolfdieter Lang, Aug 01 2022

This is Boethius's triangle, with rows read as columns. See the link and reference.

			The triangle T begins:
n\k     0     1      2      3      4      5      6      7      8      9  ...
0:      1
1:      3     4
2:      9    12     16
3:     27    36     48     64
4:     81   108    144    192    256
5:    243   324    432    576    768   1024
6:    729   972   1296   1728   2304   3072   4096
7:   2187  2916   3888   5184   6912   9216  12288  16384
8:   6561  8748  11664  15552  20736  27648  36864  49152  65536
9:  19683 26244  34992  46656  62208  82944 110592 147456 196608 262144
...

Thomas Sonar, 3000 Jahre Analysis, 2. Auflage, Springer Spektrum, 2016, p.94, Abb. 3.1.2 und Abb. 3.1.3.

Anicius Manlius Severinus Boethius, De Institutione Arithmetica, 1488, p. 55 of 104, top of left column.

Columns: A000244, A003946, A257970, ...
Diagonals: A000302, A002001(n+1), A002063, A002063(n+3), A118265(n+4), ...
Row sums: A005061(n+1).

T[n_, k_] := 3^(n - k) * 4^k; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 05 2022 *)

T(n, k) = 3^(n-k)*4^k, for n >= 0, and k = 1, 2, ..., n.

G.f. of row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k: G(x, y) = 1/((1 - 3*x)*(1 - 4*x*y)).

A000079 Powers of 2: a(n) = 2^n.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

N. J. A. Sloane

2^0 = 1 is the only odd power of 2.
Number of subsets of an n-set.
There are 2^(n-1) compositions (ordered partitions) of n (see for example Riordan). This is the unlabeled analog of the preferential labelings sequence A000670.
This is also the number of weakly unimodal permutations of 1..n + 1, that is, permutations with exactly one local maximum. E.g., a(4) = 16: 12345, 12354, 12453, 12543, 13452, 13542, 14532 and 15432 and their reversals. - Jon Perry, Jul 27 2003 [Proof: see next line! See also A087783.]
Proof: n must appear somewhere and there are 2^(n-1) possible choices for the subset that precedes it. These must appear in increasing order and the rest must follow n in decreasing order. QED. - N. J. A. Sloane, Oct 26 2003
a(n+1) is the smallest number that is not the sum of any number of (distinct) earlier terms.
Same as Pisot sequences E(1, 2), L(1, 2), P(1, 2), T(1, 2). See A008776 for definitions of Pisot sequences.
With initial 1 omitted, same as Pisot sequences E(2, 4), L(2, 4), P(2, 4), T(2, 4). - David W. Wilson
Not the sum of two or more consecutive numbers. - Lekraj Beedassy, May 14 2004
Least deficient or near-perfect numbers (i.e., n such that sigma(n) = A000203(n) = 2n - 1). - Lekraj Beedassy, Jun 03 2004. [Comment from Max Alekseyev, Jan 26 2005: All the powers of 2 are least deficient numbers but it is not known if there exists a least deficient number that is not a power of 2.]
Almost-perfect numbers referred to as least deficient or slightly defective (Singh 1997) numbers. Does "near-perfect numbers" refer to both almost-perfect numbers (sigma(n) = 2n - 1) and quasi-perfect numbers (sigma(n) = 2n + 1)? There are no known quasi-perfect or least abundant or slightly excessive (Singh 1997) numbers.
The sum of the numbers in the n-th row of Pascal's triangle; the sum of the coefficients of x in the expansion of (x+1)^n.
The Collatz conjecture (the hailstone sequence will eventually reach the number 1, regardless of which positive integer is chosen initially) may be restated as (the hailstone sequence will eventually reach a power of 2, regardless of which positive integer is chosen initially).
The only hailstone sequence which doesn't rebound (except "on the ground"). - Alexandre Wajnberg, Jan 29 2005
With p(n) as the number of integer partitions of n, p(i) is the number of parts of the i-th partition of n, d(i) is the number of different parts of the i-th partition of n, m(i,j) is the multiplicity of the j-th part of the i-th partition of n, one has: a(n) = Sum_{i = 1..p(n)} (p(i)! / (Product_{j=1..d(i)} m(i,j)!)). - Thomas Wieder, May 18 2005
The number of binary relations on an n-element set that are both symmetric and antisymmetric. Also the number of binary relations on an n-element set that are symmetric, antisymmetric and transitive.
The first differences are the sequence itself. - Alexandre Wajnberg and Eric Angelini, Sep 07 2005
a(n) is the largest number with shortest addition chain involving n additions. - David W. Wilson, Apr 23 2006
Beginning with a(1) = 0, numbers not equal to the sum of previous distinct natural numbers. - Giovanni Teofilatto, Aug 06 2006
For n >= 1, a(n) is equal to the number of functions f:{1, 2, ..., n} -> {1, 2} such that for a fixed x in {1, 2, ..., n} and a fixed y in {1, 2} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
Let P(A) be the power set of an n-element set A. Then a(n) is the number of pairs of elements {x,y} of P(A) for which x = y. - Ross La Haye, Jan 09 2008
a(n) is the number of permutations on [n+1] such that every initial segment is an interval of integers. Example: a(3) counts 1234, 2134, 2314, 2341, 3214, 3241, 3421, 4321. The map "p -> ascents of p" is a bijection from these permutations to subsets of [n]. An ascent of a permutation p is a position i such that p(i) < p(i+1). The permutations shown map to 123, 23, 13, 12, 3, 2, 1 and the empty set respectively. - David Callan, Jul 25 2008
2^(n-1) is the largest number having n divisors (in the sense of A077569); A005179(n) is the smallest. - T. D. Noe, Sep 02 2008
a(n) appears to match the number of divisors of the modified primorials (excluding 2, 3 and 5). Very limited range examined, PARI example shown. - Bill McEachen, Oct 29 2008
Successive k such that phi(k)/k = 1/2, where phi is Euler's totient function. - Artur Jasinski, Nov 07 2008
A classical transform consists (for general a(n)) in swapping a(2n) and a(2n+1); examples for Jacobsthal A001045 and successive differences: A092808, A094359, A140505. a(n) = A000079 leads to 2, 1, 8, 4, 32, 16, ... = A135520. - Paul Curtz, Jan 05 2009
This is also the (L)-sieve transform of {2, 4, 6, 8, ..., 2n, ...} = A005843. (See A152009 for the definition of the (L)-sieve transform.) - John W. Layman, Jan 23 2009
a(n) = a(n-1)-th even natural number (A005843) for n > 1. - Jaroslav Krizek, Apr 25 2009
For n >= 0, a(n) is the number of leaves in a complete binary tree of height n. For n > 0, a(n) is the number of nodes in an n-cube. - K.V.Iyer, May 04 2009
Permutations of n+1 elements where no element is more than one position right of its original place. For example, there are 4 such permutations of three elements: 123, 132, 213, and 312. The 8 such permutations of four elements are 1234, 1243, 1324, 1423, 2134, 2143, 3124, and 4123. - Joerg Arndt, Jun 24 2009
Catalan transform of A099087. - R. J. Mathar, Jun 29 2009
a(n) written in base 2: 1,10,100,1000,10000,..., i.e., (n+1) times 1, n times 0 (A011557(n)). - Jaroslav Krizek, Aug 02 2009
Or, phi(n) is equal to the number of perfect partitions of n. - Juri-Stepan Gerasimov, Oct 10 2009
These are the 2-smooth numbers, positive integers with no prime factors greater than 2. - Michael B. Porter, Oct 04 2009
A064614(a(n)) = A000244(n) and A064614(m) < A000244(n) for m < a(n). - Reinhard Zumkeller, Feb 08 2010
a(n) is the largest number m such that the number of steps of iterations of {r - (largest divisor d < r)} needed to reach 1 starting at r = m is equal to n. Example (a(5) = 32): 32 - 16 = 16; 16 - 8 = 8; 8 - 4 = 4; 4 - 2 = 2; 2 - 1 = 1; number 32 has 5 steps and is the largest such number. See A105017, A064097, A175125. - Jaroslav Krizek, Feb 15 2010
a(n) is the smallest proper multiple of a(n-1). - Dominick Cancilla, Aug 09 2010
The powers-of-2 triangle T(n, k), n >= 0 and 0 <= k <= n, begins with: {1}; {2, 4}; {8, 16, 32}; {64, 128, 256, 512}; ... . The first left hand diagonal T(n, 0) = A006125(n + 1), the first right hand diagonal T(n, n) = A036442(n + 1) and the center diagonal T(2*n, n) = A053765(n + 1). Some triangle sums, see A180662, are: Row1(n) = A122743(n), Row2(n) = A181174(n), Fi1(n) = A181175(n), Fi2(2*n) = A181175(2*n) and Fi2(2*n + 1) = 2*A181175(2*n + 1). - Johannes W. Meijer, Oct 10 2010
Records in the number of prime factors. - Juri-Stepan Gerasimov, Mar 12 2011
Row sums of A152538. - Gary W. Adamson, Dec 10 2008
A078719(a(n)) = 1; A006667(a(n)) = 0. - Reinhard Zumkeller, Oct 08 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 2-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Equals A001405 convolved with its right-shifted variant: (1 + 2x + 4x^2 + ...) = (1 + x + 2x^2 + 3x^3 + 6x^4 + 10x^5 + ...) * (1 + x + x^2 + 2x^3 + 3x^4 + 6x^5 + ...). - Gary W. Adamson, Nov 23 2011
The number of odd-sized subsets of an n+1-set. For example, there are 2^3 odd-sized subsets of {1, 2, 3, 4}, namely {1}, {2}, {3}, {4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, and {2, 3, 4}. Also, note that 2^n = Sum_{k=1..floor((n+1)/2)} C(n+1, 2k-1). - Dennis P. Walsh, Dec 15 2011
a(n) is the number of 1's in any row of Pascal's triangle (mod 2) whose row number has exactly n 1's in its binary expansion (see A007318 and A047999). (The result of putting together A001316 and A000120.) - Marcus Jaiclin, Jan 31 2012
A204455(k) = 1 if and only if k is in this sequence. - Wolfdieter Lang, Feb 04 2012
For n>=1 apparently the number of distinct finite languages over a unary alphabet, whose minimum regular expression has alphabetic width n (verified up to n=17), see the Gruber/Lee/Shallit link. - Hermann Gruber, May 09 2012
First differences of A000225. - Omar E. Pol, Feb 19 2013
This is the lexicographically earliest sequence which contains no arithmetic progression of length 3. - Daniel E. Frohardt, Apr 03 2013
a(n-2) is the number of bipartitions of {1..n} (i.e., set partitions into two parts) such that 1 and 2 are not in the same subset. - Jon Perry, May 19 2013
Numbers n such that the n-th cyclotomic polynomial has a root mod 2; numbers n such that the n-th cyclotomic polynomial has an even number of odd coefficients. - Eric M. Schmidt, Jul 31 2013
More is known now about non-power-of-2 "Almost Perfect Numbers" as described in Dagal. - Jonathan Vos Post, Sep 01 2013
Number of symmetric Ferrers diagrams that fit into an n X n box. - Graham H. Hawkes, Oct 18 2013
Numbers n such that sigma(2n) = 2n + sigma(n). - Jahangeer Kholdi, Nov 23 2013
a(1), ..., a(floor(n/2)) are all values of permanent on set of square (0,1)-matrices of order n>=2 with row and column sums 2. - Vladimir Shevelev, Nov 26 2013
Numbers whose base-2 expansion has exactly one bit set to 1, and thus has base-2 sum of digits equal to one. - Stanislav Sykora, Nov 29 2013
A072219(a(n)) = 1. - Reinhard Zumkeller, Feb 20 2014
a(n) is the largest number k such that (k^n-2)/(k-2) is an integer (for n > 1); (k^a(n)+1)/(k+1) is never an integer (for k > 1 and n > 0). - Derek Orr, May 22 2014
If x = A083420(n), y = a(n+1) and z = A087289(n), then x^2 + 2*y^2 = z^2. - Vincenzo Librandi, Jun 09 2014
The mini-sequence b(n) = least number k > 0 such that 2^k ends in n identical digits is given by {1, 18, 39}. The repeating digits are {2, 4, 8} respectively. Note that these are consecutive powers of 2 (2^1, 2^2, 2^3), and these are the only powers of 2 (2^k, k > 0) that are only one digit. Further, this sequence is finite. The number of n-digit endings for a power of 2 with n or more digits id 4*5^(n-1). Thus, for b(4) to exist, one only needs to check exponents up to 4*5^3 = 500. Since b(4) does not exist, it is clear that no other number will exist. - Derek Orr, Jun 14 2014
The least number k > 0 such that 2^k ends in n consecutive decreasing digits is a 3-number sequence given by {1, 5, 25}. The consecutive decreasing digits are {2, 32, 432}. There are 100 different 3-digit endings for 2^k. There are no k-values such that 2^k ends in '987', '876', '765', '654', '543', '321', or '210'. The k-values for which 2^k ends in '432' are given by 25 mod 100. For k = 25 + 100*x, the digit immediately before the run of '432' is {4, 6, 8, 0, 2, 4, 6, 8, 0, 2, ...} for x = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...}, respectively. Thus, we see the digit before '432' will never be a 5. So, this sequence is complete. - Derek Orr, Jul 03 2014
a(n) is the number of permutations of length n avoiding both 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014
Numbers n such that sigma(n) = sigma(2n) - phi(4n). - Farideh Firoozbakht, Aug 14 2014
This is a B_2 sequence: for i < j, differences a(j) - a(i) are all distinct. Here 2*a(n) < a(n+1) + 1, so a(n) - a(0) < a(n+1) - a(n). - Thomas Ordowski, Sep 23 2014
a(n) counts n-walks (closed) on the graph G(1-vertex; 1-loop, 1-loop). - David Neil McGrath, Dec 11 2014
a(n-1) counts walks (closed) on the graph G(1-vertex; 1-loop, 2-loop, 3-loop, 4-loop, ...). - David Neil McGrath, Jan 01 2015
b(0) = 4; b(n+1) is the smallest number not in the sequence such that b(n+1) - Prod_{i=0..n} b(i) divides b(n+1) - Sum_{i=0..n} b(i). Then b(n) = a(n) for n > 2. - Derek Orr, Jan 15 2015
a(n) counts the permutations of length n+2 whose first element is 2 such that the permutation has exactly one descent. - Ran Pan, Apr 17 2015
a(0)-a(30) appear, with a(26)-a(30) in error, in tablet M 08613 (see CDLI link) from the Old Babylonian period (c. 1900-1600 BC). - Charles R Greathouse IV, Sep 03 2015
Subsequence of A028982 (the squares or twice squares sequence). - Timothy L. Tiffin, Jul 18 2016
A000120(a(n)) = 1. A000265(a(n)) = 1. A000593(a(n)) = 1. - Juri-Stepan Gerasimov, Aug 16 2016
Number of monotone maps f : [0..n] -> [0..n] which are order-increasing (i <= f(i)) and idempotent (f(f(i)) = f(i)). In other words, monads on the n-th ordinal (seen as a posetal category). Any monad f determines a subset of [0..n] that contains n, by considering its set of monad algebras = fixed points { i | f(i) = i }. Conversely, any subset S of [0..n] containing n determines a monad on [0..n], by the function i |-> min { j | i <= j, j in S }. - Noam Zeilberger, Dec 11 2016
Consider n points lying on a circle. Then for n>=2 a(n-2) gives the number of ways to connect two adjacent points with nonintersecting chords. - Anton Zakharov, Dec 31 2016
Satisfies Benford's law [Diaconis, 1977; Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017
Also the number of independent vertex sets and vertex covers in the n-empty graph. - Eric W. Weisstein, Sep 21 2017
Also the number of maximum cliques in the n-halved cube graph for n > 4. - Eric W. Weisstein, Dec 04 2017
Number of pairs of compositions of n corresponding to a seaweed algebra of index n-1. - Nick Mayers, Jun 25 2018
The multiplicative group of integers modulo a(n) is cyclic if and only if n = 0, 1, 2. For n >= 3, it is a product of two cyclic groups. - Jianing Song, Jun 27 2018
k^n is the determinant of n X n matrix M_(i, j) = binomial(k + i + j - 2, j) - binomial(i+j-2, j), in this case k=2. - Tony Foster III, May 12 2019
Solutions to the equation Phi(2n + 2*Phi(2n)) = 2n. - M. Farrokhi D. G., Jan 03 2020
a(n-1) is the number of subsets of {1,2,...,n} which have an element that is the size of the set. For example, for n = 4, a(3) = 8 and the subsets are {1}, {1,2}, {2,3}, {2,4}, {1,2,3}, {1,3,4}, {2,3,4}, {1,2,3,4}. - Enrique Navarrete, Nov 21 2020
a(n) is the number of self-inverse (n+1)-order permutations with 231-avoiding. E.g., a(3) = 8: [1234, 1243, 1324, 1432, 2134, 2143, 3214, 4321]. - Yuchun Ji, Feb 26 2021
For any fixed k > 0, a(n) is the number of ways to tile a strip of length n+1 with tiles of length 1, 2, ... k, where the tile of length k can be black or white, with the restriction that the first tile cannot be black. - Greg Dresden and Bora Bursalı, Aug 31 2023

			There are 2^3 = 8 subsets of a 3-element set {1,2,3}, namely { -, 1, 2, 3, 12, 13, 23, 123 }.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 1016.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education Journal, Vol. 31, No. 1, pp. 24-28, Winter 1997.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 73, 84.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.5 Logarithms and §8.1 Terminology, pp. 150, 264.
Paul J. Nahin, An Imaginary Tale: The Story of sqrt(-1), Princeton University Press, Princeton, NJ. 1998, pp. 69-70.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 273.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 124.
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).
V. E. Tarakanov, Combinatorial problems on binary matrices, Combin. Analysis, MSU, 5 (1980), 4-15. (Russian)
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 141.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

N. J. A. Sloane, Table of n, 2^n for n = 0..1000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
Arno Berger and Theodore P. Hill, What is Benford's Law?, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
Tobias Boege and Thomas Kahle, Construction Methods for Gaussoids, arXiv:1902.11260 [math.CO], 2019.
Anicius Manlius Severinus Boethius, De arithmetica, Book 1, section 9.
Henry Bottomley, Illustration of initial terms
Douglas Butler, Powers of Two up to 2^222
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, Notes on Counting, Peter Cameron's Blog, 15/05/2017.
CDLI, M 08613.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
V. Coll, M. Hyatt, C. Magnant, and H. Wang, Meander graphs and Frobenius seaweed Lie algebras II, Journal of Generalized Lie Theory and Applications 9 (1) (2015) 227.
M. Coons and H. Winning, Powers of Two Modulo Powers of Three, J. Int. Seq. 18 (2015) # 15.6.1.
Keneth Adrian P. Dagal and Jose Arnaldo B. Dris, A Criterion for Almost Perfect Numbers in Terms of the Abundancy Index, arXiv:1308.6767v1 [math.NT], Aug 14 2013.
V. Dergachev and A. Kirillov, Index of Lie algebras of seaweed type, J. Lie Theory 10 (2) (2000) 331-343.
Persi Diaconis, The distribution of leading digits and uniform distribution mod 1, Ann. Probability, 5, 1977, 72--81.
Kenneth Edwards and Michael A. Allen, New combinatorial interpretations of the Fibonacci numbers squared, golden rectangle numbers, and Jacobsthal numbers using two types of tile, J. Int. Seq. 24 (2021) Article 21.3.8.
David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 18
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Hermann Gruber, Jonathan Lee and Jeffrey Shallit, Enumerating regular expressions and their languages, arXiv:1204.4982v1 [cs.FL], 2012.
R. K. Guy, Letter to N. J. A. Sloane
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 6
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 68
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 72
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 267
Marcus Jaiclin, et al. Pascal's Triangle, Mod 2,3,5
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
P. A. MacMahon, Memoir on the Theory of the Compositions of Numbers, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901.
Victor Meally, Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
R. Ondrejka, Exact values of 2^n, n=1(1)4000, Math. Comp., 23 (1969), 456.
Permutation Pattern Avoidance Library (PermPAL), Av(012_021)
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Robert Price, Comments on A000079 concerning Elementary Cellular Automata, Feb 26 2016.
Shingo Saito, Tatsushi Tanaka, and Noriko Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values, J. Int. Seq. 14 (2011) # 11.2.4, Table 1.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
J. Tanton, A Dozen Questions about the Powers of Two, Math Horizons, Vol. 8, pp. 5-10, September 2001.
G. Villemin's Almanac of Numbers, Puissances de 2
Sage Weil, 1058 powers of two
Eric Weisstein's World of Mathematics, Abundance
Eric Weisstein's World of Mathematics, Binomial Sums
Eric Weisstein's World of Mathematics, Binomial Transform
Eric Weisstein's World of Mathematics, Hailstone Number (Collatz Problem)
Eric Weisstein's World of Mathematics, Composition
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Eric Weisstein's World of Mathematics, Empty Graph
Eric Weisstein's World of Mathematics, Erf
Eric Weisstein's World of Mathematics, Fractional Part
Eric Weisstein's World of Mathematics, Halved Cube Graph
Eric Weisstein's World of Mathematics, Hypercube
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Least Deficient Number
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, PowerFractional Parts
Eric Weisstein's World of Mathematics, Subset
Eric Weisstein's World of Mathematics, Vertex Cover
Wikipedia, Almost perfect number
S. Wolfram, A New Kind of Science
Index entries for sequences related to binary expansion of n
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata
Index entries for "core" sequences
Index to divisibility sequences
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (2).
Index entries for sequences related to Benford's law

Cf. A000225, A038754, A133464, A140730, A037124, A001787, A001788, A001789, A003472, A054849, A002409, A054851, A140325, A140354, A000041, A152537, A001405, A007318, A000120, A000265, A000593, A001227, A077020, A077021.
This is the Hankel transform (see A001906 for the definition) of A000984, A002426, A026375, A026387, A026569, A026585, A026671 and A032351. - John W. Layman, Jul 31 2000
Euler transform of A001037, A209406 (multisets), inverse binomial transform of A000244, binomial transform of A000012.
Complement of A057716.
Boustrophedon transforms: A000734, A000752.
Range of values of A006519, A007875, A011782, A030001, A034444, A037445, A053644, and A054243.
Cf. A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (sum of 2, ..., 9 distinct powers of 2).
Cf. A090129.
The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).

a000079 = (2 ^)
a000079_list = iterate (* 2) 1
-- Reinhard Zumkeller, Jan 22 2014, Mar 05 2012, Dec 29 2011

[2^n: n in [0..40]]; // Vincenzo Librandi, Feb 17 2014

[n le 2 select n else 5*Self(n-1)-6*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 17 2014

A000079 := n->2^n; [ seq(2^n,n=0..50) ];
isA000079 := proc(n)
    local fs;
    fs := numtheory[factorset](n) ;
    if n = 1 then
        true ;
    elif nops(fs) <> 1 then
        false;
    elif op(1,fs) = 2 then
        true;
    else
        false ;
    end if;
end proc: # R. J. Mathar, Jan 09 2017

Table[2^n, {n, 0, 50}]
2^Range[0, 50] (* Wesley Ivan Hurt, Jun 14 2014 *)
LinearRecurrence[{2}, {2}, {0, 20}] (* Eric W. Weisstein, Sep 21 2017 *)
CoefficientList[Series[1/(1 - 2 x), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)
NestList[2# &, 1, 40] (* Harvey P. Dale, Oct 07 2019 *)

A000079(n):=2^n$ makelist(A000079(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

A000079(n)=2^n \\ Edited by M. F. Hasler, Aug 27 2014

unimodal(n)=local(x,d,um,umc); umc=0; for (c=0,n!-1, x=numtoperm(n,c); d=0; um=1; for (j=2,n,if (x[j]x[j-1] && d==1,um=0); if (um==0,break)); if (um==1,print(x)); umc+=um); umc

def a(n): return 1<Michael S. Branicky, Jul 28 2022

def is_powerof2(n) -> bool: return n and (n & (n - 1)) == 0  # Peter Luschny, Apr 10 2025

(List.fill(20)(2: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, Jan 16 2020

(define (A000079 n) (expt 2 n)) ;; Antti Karttunen, Mar 21 2017

a(n) = 2^n.
a(0) = 1; a(n) = 2*a(n-1).
G.f.: 1/(1 - 2*x).
E.g.f.: exp(2*x).
a(n)= Sum_{k = 0..n} binomial(n, k).
a(n) is the number of occurrences of n in A000523. a(n) = A001045(n) + A001045(n+1). a(n) = 1 + Sum_{k = 0..(n - 1)} a(k). The Hankel transform of this sequence gives A000007 = [1, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Feb 25 2004
n such that phi(n) = n/2, for n > 1, where phi is Euler's totient (A000010). - Lekraj Beedassy, Sep 07 2004
a(n + 1) = a(n) XOR 3*a(n) where XOR is the binary exclusive OR operator. - Philippe Deléham, Jun 19 2005
a(n) = StirlingS2(n + 1, 2) + 1. - Ross La Haye, Jan 09 2008
a(n+2) = 6a(n+1) - 8a(n), n = 1, 2, 3, ... with a(1) = 1, a(2) = 2. - Yosu Yurramendi, Aug 06 2008
a(n) = ka(n-1) + (4 - 2k)a(n-2) for any integer k and n > 1, with a(0) = 1, a(1) = 2. - Jaume Oliver Lafont, Dec 05 2008
a(n) = Sum_{l_1 = 0..n + 1} Sum_{l_2 = 0..n}...Sum_{l_i = 0..n - i}...Sum_{l_n = 0..1} delta(l_1, l_2, ..., l_i, ..., l_n) where delta(l_1, l_2, ..., l_i, ..., l_n) = 0 if any l_i <= l_(i+1) and l_(i+1) != 0 and delta(l_1, l_2, ..., l_i, ..., l_n) = 1 otherwise. - Thomas Wieder, Feb 25 2009
a(0) = 1, a(1) = 2; a(n) = a(n-1)^2/a(n-2), n >= 2. - Jaume Oliver Lafont, Sep 22 2009
a(n) = A173786(n, n)/2 = A173787(n + 1, n). - Reinhard Zumkeller, Feb 28 2010
If p[i] = i - 1 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 02 2010
If p[i] = Fibonacci(i-2) 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 >= 2, a(n-2) = det A. - Milan Janjic, May 08 2010
The sum of reciprocals, 1/1 + 1/2 + 1/4 + 1/8 + ... + 1/(2^n) + ... = 2. - Mohammad K. Azarian, Dec 29 2010
a(n) = 2*A001045(n) + A078008(n) = 3*A001045(n) + (-1)^n. - Paul Barry, Feb 20 2003
a(n) = A118654(n, 2).
a(n) = A140740(n+1, 1).
a(n) = A131577(n) + A011782(n) = A024495(n) + A131708(n) + A024493(n) = A000749(n) + A038503(n) + A038504(n) + A038505(n) = A139761(n) + A139748(n) + A139714(n) + A133476(n) + A139398(n). - Paul Curtz, Jul 25 2011
a(n) = row sums of A007318. - Susanne Wienand, Oct 21 2011
a(n) = Hypergeometric([-n], [], -1). - Peter Luschny, Nov 01 2011
G.f.: A(x) = B(x)/x, B(x) satisfies B(B(x)) = x/(1 - x)^2. - Vladimir Kruchinin, Nov 10 2011
a(n) = Sum_{k = 0..n} A201730(n, k)*(-1)^k. - Philippe Deléham, Dec 06 2011
2^n = Sum_{k = 1..floor((n+1)/2)} C(n+1, 2k-1). - Dennis P. Walsh, Dec 15 2011
A209229(a(n)) = 1. - Reinhard Zumkeller, Mar 07 2012
A001227(a(n)) = 1. - Reinhard Zumkeller, May 01 2012
Sum_{n >= 1} mobius(n)/a(n) = 0.1020113348178103647430363939318... - R. J. Mathar, Aug 12 2012
E.g.f.: 1 + 2*x/(U(0) - x) where U(k) = 6*k + 1 + x^2/(6*k+3 + x^2/(6*k + 5 + x^2/U(k+1) )); (continued fraction, 3-step). - Sergei N. Gladkovskii, Dec 04 2012
a(n) = det(|s(i+2,j)|, 1 <= i,j <= n), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 04 2013
a(n) = det(|ps(i+1,j)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). - Mircea Merca, Apr 06 2013
G.f.: W(0), where W(k) = 1 + 2*x*(k+1)/(1 - 2*x*(k+1)/( 2*x*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013
a(n-1) = Sum_{t_1 + 2*t_2 + ... + n*t_n = n} multinomial(t_1 + t_2 + ... + t_n; t_1, t_2, ..., t_n). - Mircea Merca, Dec 06 2013
Construct the power matrix T(n,j) = [A^*j]*[S^*(j-1)] where A(n)=(1,1,1,...) and S(n)=(0,1,0,0,...) (where * is convolution operation). Then a(n-1) = Sum_{j=1..n} T(n,j). - David Neil McGrath, Jan 01 2015
a(n) = A000005(A002110(n)). - Ivan N. Ianakiev, May 23 2016
From Ilya Gutkovskiy, Jul 18 2016: (Start)
Exponential convolution of A000012 with themselves.
a(n) = Sum_{k=0..n} A011782(k).
Sum_{n>=0} a(n)/n! = exp(2) = A072334.
Sum_{n>=0} (-1)^n*a(n)/n! = exp(-2) = A092553. (End)
G.f.: (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) = A090129(x) = (1 + 2x + 2x^2 + 4x^3 + 8x^4 + ...). - Gary W. Adamson, Sep 13 2016
a(n) = A000045(n + 1) + A000045(n) + Sum_{k = 0..n - 2} A000045(k + 1)*2^(n - 2 - k). - Melvin Peralta, Dec 22 2017
a(n) = 7*A077020(n)^2 + A077021(n)^2, n>=3. - Ralf Steiner, Aug 08 2021
a(n)= n + 1 + Sum_{k=3..n+1} (2*k-5)*J(n+2-k), where Jacobsthal number J(n) = A001045(n). - Michael A. Allen, Jan 12 2022
Integral_{x=0..Pi} cos(x)^n*cos(n*x) dx = Pi/a(n) (see Nahin, pp. 69-70). - Stefano Spezia, May 17 2023

Clarified a comment T. D. Noe, Aug 30 2009
Edited by Daniel Forgues, May 12 2010
Incorrect comment deleted by Matthew Vandermast, May 17 2014
Comment corrected to match offset by Geoffrey Critzer, Nov 28 2014

A000007 The characteristic function of {0}: a(n) = 0^n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Changing the offset to 1 gives the arithmetical function a(1) = 1, a(n) = 0 for n > 1, the identity function for Dirichlet multiplication (see Apostol). - N. J. A. Sloane
Changing the offset to 1 makes this the decimal expansion of 1. - N. J. A. Sloane, Nov 13 2014
Hankel transform (see A001906 for definition) of A000007 (powers of 0), A000012 (powers of 1), A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001018 (powers of 8), A001019 (powers of 9), A011557 (powers of 10), A001020 (powers of 11), etc. - Philippe Deléham, Jul 07 2005
This is the identity sequence with respect to convolution. - David W. Wilson, Oct 30 2006
a(A000004(n)) = 1; a(A000027(n)) = 0. - Reinhard Zumkeller, Oct 12 2008
The alternating sum of the n-th row of Pascal's triangle gives the characteristic function of 0, a(n) = 0^n. - Daniel Forgues, May 25 2010
The number of maximal self-avoiding walks from the NW to SW corners of a 1 X n grid. - Sean A. Irvine, Nov 19 2010
Historically there has been some disagreement as to whether 0^0 = 1. Graphing x^0 seems to support that conclusion, but graphing 0^x instead suggests that 0^0 = 0. Euler and Knuth have argued in favor of 0^0 = 1. For some calculators, 0^0 triggers an error, while in Mathematica, 0^0 is Indeterminate. - Alonso del Arte, Nov 15 2011
Another consequence of changing the offset to 1 is that then this sequence can be described as the sum of Moebius mu(d) for the divisors d of n. - Alonso del Arte, Nov 28 2011
With the convention 0^0 = 1, 0^n = 0 for n > 0, the sequence a(n) = 0^|n-k|, which equals 1 when n = k and is 0 for n >= 0, has g.f. x^k. A000007 is the case k = 0. - George F. Johnson, Mar 08 2013
A fixed point of the run length transform. - Chai Wah Wu, Oct 21 2016

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

David Wasserman, Table of n, a(n) for n = 0..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Dr. Math, 0^0 (zero to the zero power)
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Donald E. Knuth, Two notes on notation, arXiv:math/9205211 [math.HO], 1992. See page 6 on 0^0.
Robert Price, Comments on A000007, Jan 27 2016
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for "core" sequences
Index entries for characteristic functions
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (1).

Characteristic function of {g}: this sequence (g = 0), A063524 (g = 1), A185012 (g = 2), A185013 (g = 3), A185014 (g = 4), A185015 (g = 5), A185016 (g = 6), A185017 (g = 7). - Jason Kimberley, Oct 14 2011
Characteristic function of multiples of g: this sequence (g = 0), A000012 (g = 1), A059841 (g = 2), A079978 (g = 3), A121262 (g = 4), A079998 (g = 5), A079979 (g = 6), A082784 (g = 7). - Jason Kimberley, Oct 14 2011
Cf. A074909, A027641, A057427.

a000007 = (0 ^)
a000007_list = 1 : repeat 0
-- Reinhard Zumkeller, May 07 2012, Mar 27 2012

[1] cat [0:n in [1..100]]; // Sergei Haller, Dec 21 2006

A000007 := proc(n) if n = 0 then 1 else 0 fi end: seq(A000007(n), n=0..20);
spec := [A, {A=Z} ]: seq(combstruct[count](spec, size=n+1), n=0..20);

Table[If[n == 0, 1, 0], {n, 0, 99}]
Table[Boole[n == 0], {n, 0, 99}] (* Michael Somos, Aug 25 2012 *)
Join[{1},LinearRecurrence[{1},{0},102]] (* Ray Chandler, Jul 30 2015 *)
PadRight[{1},120,0] (* Harvey P. Dale, Jul 18 2024 *)

PARI
```
{a(n) = !n};
```

def A000007(n): return int(n==0) # Chai Wah Wu, Feb 04 2022

Multiplicative with a(p^e) = 0. - David W. Wilson, Sep 01 2001
a(n) = floor(1/(n + 1)). - Franz Vrabec, Aug 24 2005
As a function of Bernoulli numbers (cf. A027641: (1, -1/2, 1/6, 0, -1/30, ...)), triangle A074909 (the beheaded Pascal's triangle) * B_n as a vector = [1, 0, 0, 0, 0, ...]. - Gary W. Adamson, Mar 05 2012
a(n) = Sum_{k = 0..n} exp(2*Pi*i*k/(n+1)) is the sum of the (n+1)th roots of unity. - Franz Vrabec, Nov 09 2012
a(n) = (1-(-1)^(2^n))/2. - Luce ETIENNE, May 05 2015
a(n) = 1 - A057427(n). - Alois P. Heinz, Jan 20 2016
From Ilya Gutkovskiy, Sep 02 2016: (Start)
Binomial transform of A033999.
Inverse binomial transform of A000012. (End)

A000302 Powers of 4: a(n) = 4^n.

Original entry on oeis.org

1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, 68719476736, 274877906944, 1099511627776, 4398046511104, 17592186044416, 70368744177664, 281474976710656

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 4), L(1, 4), P(1, 4), T(1, 4). Essentially same as Pisot sequences E(4, 16), L(4, 16), P(4, 16), T(4, 16). See A008776 for definitions of Pisot sequences.
The convolution square root of this sequence is A000984, the central binomial coefficients: C(2n,n). - T. D. Noe, Jun 11 2002
With P(n) being the number of integer partitions of n, p(i) as the number of parts of the i-th partition of n, d(i) as the number of different parts of the i-th partition of n, m(i, j) the multiplicity of the j-th part of the i-th partition of n, one has a(n) = Sum_{i = 1..P(n)} p(i)!/(Product_{j = 1..d(i)} m(i, j)!) * 2^(n-1). - Thomas Wieder, May 18 2005
Sums of rows of the triangle in A122366. - Reinhard Zumkeller, Aug 30 2006
Hankel transform of A076035. - Philippe Deléham, Feb 28 2009
Equals the Catalan sequence: (1, 1, 2, 5, 14, ...), convolved with A032443: (1, 3, 11, 42, ...). - Gary W. Adamson, May 15 2009
Sum of coefficients of expansion of (1 + x + x^2 + x^3)^n.
a(n) is number of compositions of natural numbers into n parts less than 4. For example, a(2) = 16 since there are 16 compositions of natural numbers into 2 parts less than 4.
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 4-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Squares in A002984. - Reinhard Zumkeller, Dec 28 2011
Row sums of Pascal's triangle using the rule that going left increases the value by a factor of k = 3. For example, the first three rows are {1}, {3, 1}, and {9, 6, 1}. Using this rule gives row sums as (k+1)^n. - Jon Perry, Oct 11 2012
First differences of A002450. - Omar E. Pol, Feb 20 2013
Sum of all peak heights in Dyck paths of semilength n+1. - David Scambler, Apr 22 2013
Powers of 4 exceed powers of 2 by A020522 which is the m-th oblong number A002378(m), m being the n-th Mersenne number A000225(n); hence, we may write, a(n) = A000079(n) + A002378(A000225(n)). - Lekraj Beedassy, Jan 17 2014
a(n) is equal to 1 plus the sum for 0 < k < 2^n of the numerators and denominators of the reduced fractions k/2^n. - J. M. Bergot, Jul 13 2015
Binomial transform of A000244. - Tony Foster III, Oct 01 2016
From Ilya Gutkovskiy, Oct 01 2016: (Start)
Number of nodes at level n regular 4-ary tree.
Partial sums of A002001. (End)
Satisfies Benford's law [Berger-Hill, 2011]. - N. J. A. Sloane, Feb 08 2017
Also the number of connected dominating sets in the (n+1)-barbell graph. - Eric W. Weisstein, Jun 29 2017
Side length of the cells at level n in a pyramid scheme where a square grid is decomposed into overlapping 2 X 2 blocks (cf. Kropatsch, 1985). - Felix Fröhlich, Jul 04 2019
a(n-1) is the number of 3-compositions of n; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 15 2020

H. W. Gould, Combinatorial Identities, 1972, eq. (1.93), p. 12.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, eq. (5.39), p. 187.
D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
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).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

T. D. Noe, Table of n, a(n) for n = 0..100
Arno Berger and Theodore P. Hill, Benford's law strikes back: no simple explanation in sight for mathematical gem, The Mathematical Intelligencer 33.1 (2011): 85-91.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
G. Dresden and Y. Li, Periodic Weighted Sums of Binomial Coefficients, arXiv:2210.04322 [math.NT], 2022.
R. Duarte and A. G. de Oliveira, Short note on the convolution of binomial coefficients, arXiv preprint arXiv:1302.2100 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.7.6.
Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See p. 5.
R. K. Guy, Letter to N. J. A. Sloane
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 8
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 269
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Tanya Khovanova, Recursive Sequences
Craig Knecht, Number of tilings for a 6 sphinx tile repetitive unit.
Walter G. Kropatsch, A pyramid that grows by powers of 2, Pattern Recognition Letters, Vol. 3, No. 5 (1985), 315-322 [Subscription required].
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
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
Robert Price, Comments on A000302 concerning Elementary Cellular Automata, Feb 26 2016.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Robert Schneider, Partition zeta functions, Research in Number Theory, 2(1):9., 2016.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv preprint arXiv:1504.04404 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Barbell Graph
Eric Weisstein's World of Mathematics, Cantor Dust
Eric Weisstein's World of Mathematics, Connected Dominating Set
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata
Index entries for "core" sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (4).
Index entries for sequences related to Benford's law

Cf. A024036, A052539, A032443, A000351 (Binomial transform).
Cf. A249307.
Cf. A083420.

a000302 = (4 ^)
a000302_list = iterate (* 4) 1  -- Reinhard Zumkeller, Apr 04 2012

A000302 := n->4^n;
for n from 0 to 10 do sum(2^(n-j)*binomial(n+j,j),j=0..n); od; # Peter C. Heinig (algorithms(AT)gmx.de), Apr 06 2007
A000302:=-1/(-1+4*z); # Simon Plouffe in his 1992 dissertation.

Table[4^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 01 2006 *)
CoefficientList[Series[1/(1 - 4 x), {x, 0, 50}], x]  (* Vincenzo Librandi, May 29 2014 *)
NestList[4 # &, 1, 30] (* Harvey P. Dale, Mar 26 2015 *)
4^Range[0, 30] (* Eric W. Weisstein, Jun 29 2017 *)
LinearRecurrence[{4}, {1}, 31] (* Robert A. Russell, Nov 08 2018 *)

A000302(n):=4^n$ makelist(A000302(n),n,0,30); /* Martin Ettl, Oct 24 2012 */

A000302(n)=4^n \\ Michael B. Porter, Nov 06 2009

print([4**n for n in range(25)]) # Michael S. Branicky, Jan 04 2021

is_A000302 = lambda n: n.bit_count()==1 and n.bit_length()&1 # M. F. Hasler, Nov 25 2024

[4**n for n in range(0,25)] # Stefano Spezia, Jul 23 2025

(List.fill(20)(4: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, Jun 22 2019

a(n) = 4^n.
a(0) = 1; a(n) = 4*a(n-1).
G.f.: 1/(1-4*x).
E.g.f.: exp(4*x).
a(n) = Sum_{k = 0..n} binomial(2k, k) * binomial(2(n - k), n - k). - Benoit Cloitre, Jan 26 2003 [See Graham et al., eq. (5.39), p. 187. - Wolfdieter Lang, Aug 16 2019]
1 = Sum_{n >= 1} 3/a(n) = 3/4 + 3/16 + 3/64 + 3/256 + 3/1024, ...; with partial sums: 3/4, 15/16, 63/64, 255/256, 1023/1024, ... - Gary W. Adamson, Jun 16 2003
a(n) = A001045(2*n) + A001045(2*n+1). - Paul Barry, Apr 27 2004
A000005(a(n)) = A005408(n+1). - Reinhard Zumkeller, Mar 04 2007
a(n) = Sum_{j = 0..n} 2^(n - j)*binomial(n + j, j). - Peter C. Heinig (algorithms(AT)gmx.de), Apr 06 2007
Hankel transform of A115967. - Philippe Deléham, Jun 22 2007
a(n) = 6*Stirling2(n+1, 4) + 6*Stirling2(n+1, 3) + 3*Stirling2(n+1, 2) + 1 = 2*Stirling2(2^n, 2^n - 1) + Stirling2(n+1, 2) + 1. - Ross La Haye, Jun 26 2008
a(n) = A159991(n)/A001024(n) = A047653(n) + A181765(n). A160700(a(n)) = A010685(n). - Reinhard Zumkeller, May 02 2009
a(n) = A188915(A006127(n)). - Reinhard Zumkeller, Apr 14 2011
a(n) = Sum_{k = 0..n} binomial(2*n+1, k). - Mircea Merca, Jun 25 2011
Sum_{n >= 1} Mobius(n)/a(n) = 0.1710822479183... - R. J. Mathar, Aug 12 2012
a(n) = Sum_{k = 0..n} binomial(2*k + x, k)*binomial(2*(n - k) - x, n - k) for every real number x. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013
a(n) = 5*a(n - 1) - 4*a(n - 2). - Jean-Bernard François, Sep 12 2013
a(n) = (2*n+1) * binomial(2*n,n) * Sum_{j=0..n} (-1)^j/(2*j+1)*binomial(n,j). - Vaclav Kotesovec, Sep 15 2013
a(n) = A000217(2^n - 1) + A000217(2^n). - J. M. Bergot, Dec 28 2014
a(n) = (2^n)^2 = A000079(n)^2. - Doug Bell, Jun 23 2015
a(n) = A002063(n)/3 - A004171(n). - Zhandos Mambetaliyev, Nov 19 2016
a(n) = (1/2) * Product_{k = 0..n} (1 + (2*n + 1)/(2*k + 1)). - Peter Bala, Mar 06 2018
a(n) = A001045(n+1)*A001045(n+2) + A001045(n)^2. - Ezhilarasu Velayutham, Aug 30 2019
a(n) = denominator(zeta_star({2}(n + 1))/zeta(2*n + 2)) where zeta_star is the multiple zeta star values and ({2}_n) represents (2, ..., 2) where the multiplicity of 2 is n. - _Roudy El Haddad, Feb 22 2022
a(n) = 1 + 3*Sum_{k=0..n} binomial(2*n, n+k)*(k|9), where (k|9) is the Jacobi symbol. - Greg Dresden, Oct 11 2022
a(n) = Sum_{k = 0..n} binomial(2*n+1, 2*k) = Sum_{k = 0..n} binomial(2*n+1, 2*k+1). - Sela Fried, Mar 23 2023

Partially edited by Joerg Arndt, Mar 11 2010

cp's OEIS Frontend

A140298 a(0)=1; a(3n+1) = a(3n)+1, a(3n+2) = a(3n+1) + a(3n) (=3*A000244), a(3n+3) = a(3n+2) + a(3n) (=A003462(n+2)).

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A248163 Chebyshev's S polynomials (A049310) evaluated at 34/3 and multiplied by powers of 3 (A000244).

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A133647 A133566 * A000244.

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A101555 Convolution of A010060 and A000244.

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A141317 a(n) = A000244(n) - A010684(n).

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Maple

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A203197 (n-1)-st elementary symmetric function of the first n terms of (1,3,9,27,...)=A000244.

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A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

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A000079 Powers of 2: a(n) = 2^n.

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