A007283 -id:A007283 - cp's OEIS frontend

A078840 Table of n-almost-primes T(n,k) (n >= 0, k > 0), read by antidiagonals, starting at T(0,1)=1 followed by T(1,1)=2.

1, 2, 3, 4, 5, 6, 8, 7, 9, 12, 16, 11, 10, 18, 24, 32, 13, 14, 20, 36, 48, 64, 17, 15, 27, 40, 72, 96, 128, 19, 21, 28, 54, 80, 144, 192, 256, 23, 22, 30, 56, 108, 160, 288, 384, 512, 29, 25, 42, 60, 112, 216, 320, 576, 768, 1024, 31, 26, 44, 81, 120, 224, 432, 640, 1152

Offset: 0

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

An n-almost-prime is a positive integer that has exactly n prime factors.
This sequence is a rearrangement of the natural numbers. - Robert G. Wilson v, Feb 11 2006
Each antidiagonal begins with the n-th prime and ends with 2^n.
From Eric Desbiaux, Jun 27 2009: (Start)
(A001222 gives this sequence)
A001221 gives another table:
  1
  -    2    3    4    5    7    8    9   11 ... A000961
  -    6   10   12   14   15   18   20   21 ... A007774
  -   30   42   60   66   70   78   84   90 ... A033992
  -  210  330  390  420  462  510  546  570 ... A033993
  - 2310 2730 3570 3990 4290 4620 4830 5460 ... A051270
Antidiagonals begin with A000961 and end with A002110.
Diagonal is A073329 which is last term in n-th row of A048692. (End)

			Table begins:
  1
  -  2  3   5   7  11  13  17  19  23  29 ...
  -  4  6   9  10  14  15  21  22  25  26 ...
  -  8 12  18  20  27  28  30  42  44  45 ...
  - 16 24  36  40  54  56  60  81  84  88 ...
  - 32 48  72  80 108 112 120 162 168 176 ...
  - 64 96 144 160 216 224 240 324 336 352 ...

Robert G. Wilson v, Table of n, a(n) for n = 0..10011 (corrected by Ivan Neretin).
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A078840, A078841, A078842, A078843, A078844, A078445, A078846, A109636.
T(1, k)=A000040(k), T(2, k)=A001358(k), T(3, k)=A014612(k), T(4, k)=A014613(k), T(5, k)=A014614(k), T(6, k)=A046306(k), T(7, k)=A046308(k), T(8, k)=A046310(k), T(9, k)=A046312(k), T(10, k)=A046314(k).
T(11, k)=A069272(k), T(12, k)=A069273(k), T(13, k)=A069274(k), T(14, k)=A069275(k), T(15, k)=A069276(k), T(16, k)=A069277(k), T(17, k)=A069278(k), T(18, k)=A069279(k), T(19, k)=A069280(k), T(20, k)=A069281(k).
T(k, 1)=A000079(k), T(k, 2)=A007283(k), T(k, 3)=A116453(k), T(k, k)=A101695(k), T(k, k+1)=A078841(k).
A091538 is this sequence with zeros inserted, making a square array.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[ Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]] ]]]; (* Eric W. Weisstein, Feb 07 2006 *)
AlmostPrime[k_, n_] := Block[{e = Floor[Log[2, n]+k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; Table[ AlmostPrime[k, n - k + 1], {n, 11}, {k, n}] // Flatten (* Robert G. Wilson v *)
mx = 11; arr = NestList[Take[Union@Flatten@Outer[Times, #, primes], mx] &, primes = Prime@Range@mx, mx]; Prepend[Flatten@Table[arr[[k, n - k + 1]], {n, mx}, {k, n}], 1] (* Ivan Neretin, Apr 30 2016 *)
(* The next code skips the initial 1. *)
width = 15; (seq = Table[
  Rest[NestList[1 + NestWhile[# + 1 &, #, ! PrimeOmega[#] == z &] &,
  2^z, width - z + 1]] - 1, {z, width}]) // TableForm
Flatten[Map[Reverse[Diagonal[Reverse[seq], -width + #]] &, Range[width]]]
(* Peter J. C. Moses, Jun 05 2019 *)
Grid[Table[Select[Range[200], PrimeOmega[#] == n &], {n, 0, 7}]]
(* Clark Kimberling, Nov 17 2024 *)

T(n,k)=if(k<0,0,s=1; while(sum(i=1,s,if(bigomega(i)-n,0,1))

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi, prime
def A078840_T(n,k):
    if n == 1: return prime(k)
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(k-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Edited by Robert G. Wilson v, Feb 11 2006

A054886 Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3,Pi/3,0) (this is the classical modular tessellation).

Original entry on oeis.org

1, 3, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

The layer sequence is the sequence of the cardinalities of the layers accumulating around a ( finite-sided ) polygon of the tessellation under successive side-reflections; see the illustration accompanying A054888.
Equivalently, coordination sequence for (3,3,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
Equivalently, spherical growth series for modular group.
Also, number of sequences of length n with terms 1, 2, and 3, with no adjacent terms equal, and no three consecutive terms (1, 2, 3) or (3, 2, 1). - Pontus von Brömssen, Jan 03 2022

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 156.

G. C. Greubel, Table of n, a(n) for n = 0..999 [Offset changed to 0 by _Georg Fischer_, Mar 01 2022]
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for Coordination Sequences [A layer sequence is a kind of coordination sequence. - _N. J. A. Sloane_, Nov 20 2022]
Index entries for sequences related to modular groups
Index entries for linear recurrences with constant coefficients, signature (1,1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.
Essentially the same as A006355.
Cf. A000045, A054888.

Join[{1,3},2Fibonacci[Range[4,40]]] (* Harvey P. Dale, Jan 06 2012 *)

my(x='x+O('x^50)); Vec((1+2*x+2*x^2+x^3)/(1-x-x^2)) \\ G. C. Greubel, Aug 06 2017

G.f.: (1+2*x+2*x^2+x^3)/(1-x-x^2) = (x^2+x+1)*(1+x)/(1-x-x^2).
a(n) = 2*F(n+2) for n >= 2, with F(n) the n-th Fibonacci number (cf. A000045).
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 1 - x. - Stefano Spezia, Apr 18 2022

Offset changed to 0 by N. J. A. Sloane, Jan 03 2022 at the suggestion of Pontus von Brömssen

A029858 a(n) = (3^n - 3)/2.

Original entry on oeis.org

0, 3, 12, 39, 120, 363, 1092, 3279, 9840, 29523, 88572, 265719, 797160, 2391483, 7174452, 21523359, 64570080, 193710243, 581130732, 1743392199, 5230176600, 15690529803, 47071589412, 141214768239

Offset: 1

Christian G. Bower

Also the number of 2-block covers of a labeled n-set. a(n) = A055154(n,2). Generally, number of k-block covers of a labeled n-set is T(n,k) = (1/k!)*Sum_{i = 1..k + 1} Stirling1(k + 1,i)*(2^(i - 1) - 1)^n. In particular, T(n,2) = (1/2!)*(3^n - 3), T(n,3) = (1/3!)*(7^n - 6*3^n + 11), T(n,4) = (1/4)!*(15^n - 10*7^n + 35*3^n - 50), ... - Vladeta Jovovic, Jan 19 2001
Conjectured to be the number of integers from 0 to 10^(n-1) - 1 that lack 0, 1, 2, 3, 4, 5 and 6 as a digit. - Alexandre Wajnberg, Apr 25 2005. This is easily verified to be true. - Renzo Benedetti, Sep 25 2008
Number of monic irreducible polynomials of degree 1 in GF(3)[x1,...,xn]. - Max Alekseyev, Jan 23 2006
Also, the greatest number of identical weights among which an odd one can be identified and it can be decided if the odd one is heavier or lighter, using n weighings with a comparing balance. If the odd one only needs to be identified, the sequence starts 4, 13, 40 and is A003462 (3^n - 1)/2, n > 1. - Tanya Khovanova, Dec 11 2006; corrected by Samuel E. Rhoads, Apr 18 2016
Binomial transform yields A134057. Inverse binomial transform yields A062510 with one additional 0 in front. - R. J. Mathar, Jun 18 2008
Numbers k where the recurrence s(0)=0, if s(k-1) >= k then s(k) = s(k-1) - k otherwise s(k) = s(k-1) + k produces s(k) = 0. - Hugo Pfoertner, Jan 05 2012
For n > 1: A008344(a(n)) = a(n). - Reinhard Zumkeller, May 09 2012
Also the number of edges in the (n-1)-Hanoi graph. - Eric W. Weisstein, Jun 18 2017
A level 1 Sierpiński triangle graph is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles.  a(n) is the number of degree 4 vertices in the level n Sierpinski triangle graph. - Allan Bickle, Jul 30 2020
Also the number of minimum vertex cuts in the n-Apollonian network. - Eric W. Weisstein, Dec 20 2020
Also the minimum number of turns in n-dimensional Euclidean space needed to visit all 3^n points of the grid {0, 1, 2}^n, moving in straight lines between turns (repeated visits and direction changes at non-grid points are allowed). - Marco Ripà, Aug 06 2025

			For the Sierpiński triangle, Level 1 is a triangle, so a(1) = 0.
Level 2 has three corners (degree 2) and three degree 4 vertices, so a(2) = 3.
The level 2 Hanoi graph has 3 triangles joined by 3 edges, so a(2+1) = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
Natalia Agudelo Muñetón, Agustín Moreno Cañadas, Pedro Fernando Fernández Espinosa, and Isaías David Marín Gaviria, Brauer Configuration Algebras and Their Applications in Graph Energy Theory, Mathematics (2021) Vol. 9, 3042.
Alex Born, Cor A. J. Hukrnes, and Gerhard J. Woeginger, How to detect a counterfeit coin: adaptive versus non-adaptive solutions, Inf. Proc. Lett. 86 (2003) 137-141.
Gary Darby, The Counterfeit Coin
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. 17.
Lorenz Halbeisen and Norbert Hungerbuhler, The general counterfeit coin problem, Discr. Math 147 (1-3) (1995) 139-150, Theorem 1 with b=1.
Andreas Hinz, Sandi Klavzar, and Sara Sabrina Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Bennet Manvel, Counterfeit coin problems, Math. Mag. 50 (2) (1977) 90-92, theorem 2.
Marco Ripà, Solving the 106 years old 3^k points problem with the clockwise-algorithm, Journal of Fundamental Mathematics and Applications, 2020, 3(2), 84-97.
Allen Stenger and Jack Wert, The Twelve Coins (or Twelve bags of Gold)
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Edge Count
Eric Weisstein's World of Mathematics, Hanoi Graph
Eric Weisstein's World of Mathematics, Minimum Vertex Cut
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A055154, A003462, A007051, A034472, A024023.
Cf. A000203, A008776.
Cf. A007283, A029858, A067771, A233774, A233775, A246959 (Sierpiński triangle graphs).
Cf. A000225, A029858, A058809, A375256 (Hanoi graphs).

a029858 = (`div` 2) . (subtract 3) . (3 ^)
a029858_list = iterate ((+ 3) . (* 3)) 0
-- Reinhard Zumkeller, May 09 2012

[(3^n-3)/2: n in [1..30]]; // Vincenzo Librandi, Jun 05 2011

a:=n->sum(3^j,j=1..n): seq(a(n),n=0..23); # Zerinvary Lajos, Jun 27 2007

Table[(3^n - 3)/2, {n, 24}] (* Alonso del Arte, Dec 29 2014 *)

a(n)=(3^n-3)\2 \\ Charles R Greathouse IV, Apr 17 2012

a(n) = 3*a(n-1) + 3. - Alexandre Wajnberg, Apr 25 2005
O.g.f: 3*x^2/((1-x)*(1-3*x)). - R. J. Mathar, Jun 18 2008
a(n) = 3^(n-1) + a(n-1) (with a(1)=0). - Vincenzo Librandi, Nov 18 2010
a(n) = 3*A003462(n-1). - R. J. Mathar, Sep 10 2015
E.g.f.: 3*(-1 + exp(2*x))*exp(x)/2. - Ilya Gutkovskiy, Apr 19 2016
a(n) = A067771(n-1) - 3. - Allan Bickle, Jul 30 2020
a(n) = sigma(A008776(n-2)) for n>=2. - Flávio V. Fernandes, Apr 20 2021

Corrected by T. D. Noe, Nov 07 2006

A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81, 32, 48, 72, 108, 162, 243, 64, 96, 144, 216, 324, 486, 729, 128, 192, 288, 432, 648, 972, 1458, 2187, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

N. J. A. Sloane

The triangle pertaining to this sequence has the property that every row, every column and every diagonal contains a nontrivial geometric progression. More interestingly every line joining any two elements contains a nontrivial geometric progression. - Amarnath Murthy, Jan 02 2002
Kappraff states (pp. 148-149): "I shall refer to this as Nicomachus' table since an identical table of numbers appeared in the Arithmetic of Nicomachus of Gerasa (circa 150 A.D.)" The table was rediscovered during the Italian Renaissance by Leon Battista Alberti, who incorporated the numbers in dimensions of his buildings and in a system of musical proportions. Kappraff states "Therefore a room could exhibit a 4:6 or 6:9 ratio but not 4:9. This ensured that ratios of these lengths would embody musical ratios". - Gary W. Adamson, Aug 18 2003
After Nichomachus and Alberti several Renaissance authors described this table. See for instance Pierre de la Ramée in 1569 (facsimile of a page of his Arithmetic Treatise in Latin in the links section). - Olivier Gérard, Jul 04 2013
The triangle sums, see A180662 for their definitions, link Nicomachus's table with eleven different sequences, see the crossrefs. It is remarkable that these eleven sequences can be described with simple elegant formulas. The mirror of this triangle is A175840. - Johannes W. Meijer, Sep 22 2010
The diagonal sums Sum_{k} T(n - k, k) give A167762(n + 2). - Michael Somos, May 28 2012
Where d(n) is the divisor count function, then d(T(i,j)) = A003991, the rows of which sum to the tetrahedral numbers A000292(n+1). For example, the sum of the divisors of row 4 of this triangle (i = 4), gives d(16) + d(24) + d(36) + d(54) + d(81) = 5 + 8 + 9 + 8 + 5 = 35 = A000292(5). In fact, where p and q are distinct primes, the aforementioned relationship to the divisor function and tetrahedral numbers can be extended to any triangle of numbers in which the i-th row is of form {p^(i-j)*q^j, 0<=j<=i}; i >= 0 (e.g., A003593, A003595). - Raphie Frank, Nov 18 2012, corrected Dec 07 2012
Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then 2*x and 3*x are in S, and duplicates are deleted as they occur; see A232559. - Clark Kimberling, Nov 28 2013
Partial sums of rows produce Stirling numbers of the 2nd kind: A000392(n+2) = Sum_{m=1..(n^2+n)/2} a(m). - Fred Daniel Kline, Sep 22 2014
A permutation of A003586. - L. Edson Jeffery, Sep 22 2014
Form a word of length i by choosing a (possibly empty) word on alphabet {0,1} then concatenating a word of length j on alphabet {2,3,4}. T(i,j) is the number of such words. - Geoffrey Critzer, Jun 23 2016
Form of Zorach additive triangle (see A035312) where each number is sum of west and northwest numbers, with the additional condition that each number is GCD of the two numbers immediately below it. - Michel Lagneau, Dec 27 2018

			The start of the sequence as a triangular array read by rows:
   1
   2   3
   4   6   9
   8  12  18  27
  16  24  36  54  81
  32  48  72 108 162 243
  ...
The start of the sequence as a table T(n,k) n, k > 0:
    1    2    4    8   16   32 ...
    3    6   12   24   48   96 ...
    9   18   36   72  144  288 ...
   27   54  108  216  432  864 ...
   81  162  324  648 1296 2592 ...
  243  486  972 1944 3888 7776 ...
  ...
- _Boris Putievskiy_, Jan 08 2013

Jay Kappraff, Beyond Measure, World Scientific, 2002, p. 148.
Flora R. Levin, The Manual of Harmonics of Nicomachus the Pythagorean, Phanes Press, 1994, p. 114.

Reinhard Zumkeller and Matthew House, Rows n = 0..300 of triangle, flattened [Rows 0 through 120 were computed by Reinhard Zumkeller; rows 121 through 300 by Matthew House, Jul 09 2015]
Fred Daniel Kline, How do I convert this Nicomachus' Triangle to one with edges?
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Pierre de la Ramée (Petrus Ramus), P. Rami Arithmeticae (anno 1569) Liber 2, Cap. XVI "De inventione continue proportionalium" p.46 (leaf 0055) describes this integer triangle in a layout close to the current OEIS 'tabl' layout.
Marko Riedel, Proof of identity by Egorychev method.
Thomas Scheuerle, Version of this triangle from Boethius (480-524), Anicius Manlius Severinus Boethius, De institutione arithmetica, Medeltidshandskrift 1 (Mh 1), Lund University Library, early 10th century, page 4r.
Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.

Cf. A001047 (row sums), A000400 (central terms), A013620, A007318.
Cf. A003586, A000079, A000244, A007283, A025197, A005010, A003946, A005051.
Triangle sums (see the comments): A001047 (Row1); A015441 (Row2); A005061 (Kn1, Kn4); A016133 (Kn2, Kn3); A016153 (Fi1, Fi2); A016140 (Ca1, Ca4); A180844 (Ca2, Ca3); A180845 (Gi1, Gi4); A180846 (Gi2, Gi3); A180847 (Ze1, Ze4); A016185 (Ze2, Ze3). - Johannes W. Meijer, Sep 22 2010, Sep 10 2011
Antidiagonal cumulative sum: A000392; square arrays cumulative sum: A160869. Antidiagonal products: 6^A000217; antidiagonal cumulative products: 6^A000292; square arrays products: 6^A005449; square array cumulative products: 6^A006002.

a036561 n k = a036561_tabf !! n !! k
a036561_row n = a036561_tabf !! n
a036561_tabf = iterate (\xs@(x:_) -> x * 2 : map (* 3) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

/* As triangle: */ [[(2^(i-j)*3^j)/3: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Oct 17 2014

A036561 := proc(n,k): 2^(n-k)*3^k end:
seq(seq(A036561(n,k),k=0..n),n=0..9);
T := proc(n,k) option remember: if k=0 then 2^n elif k>=1 then procname(n,k-1) + procname(n-1,k-1) fi: end: seq(seq(T(n,k),k=0..n),n=0..9);
# Johannes W. Meijer, Sep 22 2010, Sep 10 2011

Flatten[Table[ 2^(i-j) 3^j, {i, 0, 12}, {j, 0, i} ]] (* Flatten added by Harvey P. Dale, Jun 07 2011 *)

for(i=0,9,for(j=0,i,print1(3^j<<(i-j)", "))) \\ Charles R Greathouse IV, Dec 22 2011

{T(n, k) = if( k<0 || k>n, 0, 2^(n - k) * 3^k)} /* Michael Somos, May 28 2012 */

T(n,k) = A013620(n,k)/A007318(n,k). - Reinhard Zumkeller, May 14 2006
T(n,k) = T(n,k-1) + T(n-1,k-1) for n>=1 and 1<=k<=n with T(n,0) = 2^n for n>=0. - Johannes W. Meijer, Sep 22 2010
T(n,k) = 2^(k-1)*3^(n-1), n, k > 0 read by antidiagonals. - Boris Putievskiy, Jan 08 2013
a(n) = 2^(A004736(n)-1)*3^(A002260(n)-1), n > 0, or a(n) = 2^(j-1)*3^(i-1) n > 0, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2]. - Boris Putievskiy, Jan 08 2013
G.f.: 1/((1-2x)(1-3yx)). - Geoffrey Critzer, Jun 23 2016
T(n,k) = (-1)^n * Sum_{q=0..n} (-1)^q * C(k+3*q, q) * C(n+2*q, n-q). - Marko Riedel, Jul 01 2024

A122132 Squarefree numbers multiplied by binary powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 85

Offset: 1

Reinhard Zumkeller, Aug 21 2006

These numbers are called "oddly squarefree" in Banks and Luca. - Michel Marcus, Mar 14 2016

The asymptotic density of this sequence is 8/Pi^2 (A217739). - Amiram Eldar, Sep 21 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
William D. Banks and Florian Luca, Roughly squarefree values of the Euler and Carmichael functions, Acta Arithmetica, Vol. 120, No. 3 (2005), pp. 211-230.

Subsequences: A000079, A005117, A007283, A051916, A056911, A029747.
Complement: A038838.
Cf. A217739.

a122132 n = a122132_list !! (n-1)
a122132_list = filter ((== 1) . a008966 . a000265) [1..]
-- Reinhard Zumkeller, Jan 24 2012

Select[Range@ 85, SquareFreeQ[#/2^IntegerExponent[#, 2]] &] (* Michael De Vlieger, Mar 15 2020 *)

is(n)=issquarefree(n>>valuation(n,2)); \\ Charles R Greathouse IV, Sep 02 2015

a(n) = A007947(a(n)) * A006519(a(n)) / (2 - a(n) mod 2);
A007947(a(n)) = A000265(a(n)) * (2 - a(n) mod 2).
A008966(A000265(a(n))) = 1. - Reinhard Zumkeller, Jan 24 2012
A010052(A008477(a(n))) = 1. - Reinhard Zumkeller, Feb 17 2012

A042950 Row sums of the Lucas triangle A029635.

Original entry on oeis.org

2, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944

Offset: 0

N. J. A. Sloane and J. H. Conway

Map a binary sequence b=[ b_1,...] to a binary sequence c=[ c_1,...] so that C = 1/Product((1-x^i)^c_i == 1 + Sum b_i*x^i mod 2.
This produces 2 new sequences: d={i:c_i=1} and e=[ 1,e_1,... ] where C = 1 + Sum e_i*x^i.
This sequence is d when b=[ 0,1,1,1,1,...].
Number of rises after n+1 iterations of morphism A007413.
a(n) written in base 2: a(0) = 10, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009
Row sums of the Lucas triangle A029635. - Sergio Falcon, Mar 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Kitaev and T. Mansour, Counting the occurrences of generalized patterns in words generated by a morphism, arXiv:math/0210170 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A003945, A007283.

[2] cat [2^(n+1) - 2^(n-1): n in [1..40]]; // Vincenzo Librandi, Aug 08 2015

Table[ Ceiling[3*2^(n - 1)], {n, 0, 32}] (* Robert G. Wilson v, Jul 08 2006 *)
a[0] = 2; a[1] = 3; a[n_] := 2a[n - 1]; Table[a[n], {n, 0, 32}] (* Robert G. Wilson v, Jul 08 2006 *)
f[s_] := Append[s, 1 + Plus @@ s]; Nest[f, {2}, 32] (* Robert G. Wilson v, Jul 08 2006 *)
CoefficientList[Series[(2 - x)/(1 - 2x), {x, 0, 32}], x] (* Robert G. Wilson v, Jul 08 2006 *)

PARI
```
a(n)=ceil(3*2^(n-1))
```

def A042950(n): return (3*2**n + int(n==0))//2 # G. C. Greubel, Jun 06 2025

G.f.: (2-x)/(1-2*x).
a(n) = 2*a(n-1), n > 1; a(0)=2, a(1)=3.
a(n) = A003945(n), for n > 0.
From Paul Barry, Dec 06 2004: (Start)
Binomial transform of 2, 1, 2, 1, 2, 1, ... = (3+(-1)^n)/2.
a(n) = (3*2^n + 0^n)/2. (End)
a(0) = 2, a(n) = 3*2^(n-1) = 2^n + 2^(n-1) for n >= 1. - Jaroslav Krizek, Aug 17 2009
a(n) = 2^(n+1) - 2^(n-1), for n > 0. - Ilya Gutkovskiy, Aug 08 2015
E.g.f.: (3*exp(2*x) + 1)/2. - G. C. Greubel, Jun 06 2025

A057168 Next larger integer with same binary weight (number of 1 bits) as n.

Original entry on oeis.org

2, 4, 5, 8, 6, 9, 11, 16, 10, 12, 13, 17, 14, 19, 23, 32, 18, 20, 21, 24, 22, 25, 27, 33, 26, 28, 29, 35, 30, 39, 47, 64, 34, 36, 37, 40, 38, 41, 43, 48, 42, 44, 45, 49, 46, 51, 55, 65, 50, 52, 53, 56, 54, 57, 59, 67, 58, 60, 61, 71, 62, 79, 95, 128, 66, 68, 69, 72, 70, 73, 75

Offset: 1

Marc LeBrun, Sep 14 2000

Binary weight is given by A000120.

			a(6)=9 since 6 has two one-bits (i.e., 6=2+4) and 9 is the next higher integer of binary weight two (7 is weight three and 8 is weight one).

Donald Knuth, The Art of Computer Programming, Vol. 4A, section 7.1.3, exercises 20-21.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Beeler, R. W. Gosper, and R. Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory, Memo AIM-239, February 1972, Item 175 by Gosper, page 81. Also HTML transcription.
Donald E. Knuth, The Art of Computer Programming, Pre-Fascicle 1A, Draft of Section 7.1.3, 2008. Exercises 20 and 21 page 54 and answers pages 75-76.
Marc B. Reynolds, Popcount walks: next, previous, toward and nearest (2023)

Cf. A000120, A006519, A057169, A000051, A052548, A140504, A000225, A055010, A007283, A171942.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hamming weight = 1, 2, ..., 9).

a057168 n = a057168_list !! (n-1)
a057168_list = f 2 $ tail a000120_list where
   f x (z:zs) = (x + length (takeWhile (/= z) zs)) : f (x + 1) zs
-- Reinhard Zumkeller, Aug 26 2012

a[n_] := (bw = DigitCount[n, 2, 1]; k = n+1; While[ DigitCount[k, 2, 1] != bw, k++]; k); Table[a[n], {n, 1, 71}](* Jean-François Alcover, Nov 28 2011 *)

a(n)=my(u=bitand(n,-n),v=u+n);(bitxor(v,n)/u)>>2+v \\ Charles R Greathouse IV, Oct 28 2009

A057168(n)=n+bitxor(n,n+n=bitand(n,-n))\n\4+n \\ M. F. Hasler, Aug 27 2014

def a(n): u = n&-n; v = u+n; return (((v^n)//u)>>2)+v
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Jul 10 2022 after Charles R Greathouse IV

def A057168(n): return ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b # Chai Wah Wu, Mar 06 2025

From Reinhard Zumkeller, Aug 18 2008: (Start)
a(A000079(n)) = A000079(n+1);
a(A000051(n)) = A052548(n);
a(A052548(n)) = A140504(n);
a(A000225(n)) = A055010(n);
a(A007283(n)) = A000051(n+2). (End)
a(n) = MIN{m: A000120(m)=A000120(n) and m>n}. - Reinhard Zumkeller, Aug 15 2009
For k,m>0, a((2^k-1)*2^m) = 2^(k+m)+2^(k-1)-1. - Chai Wah Wu, Mar 07 2025
If n is odd, then a(n) = XOR(n,OR(a,a/2)) where a = AND(-n,n+1). - Chai Wah Wu, Mar 08 2025

A078042 Expansion of (1-x)/(1+x-x^2+x^3).

Original entry on oeis.org

1, -2, 3, -6, 11, -20, 37, -68, 125, -230, 423, -778, 1431, -2632, 4841, -8904, 16377, -30122, 55403, -101902, 187427, -344732, 634061, -1166220, 2145013, -3945294, 7256527, -13346834, 24548655, -45152016, 83047505, -152748176, 280947697, -516743378, 950439251, -1748130326, 3215312955

Offset: 0

N. J. A. Sloane, Nov 17 2002

Absolute values give coordination sequence for (3,infinity,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015

a(n) is the upper left entry of the n-th power of the 3 X 3 matrix M = [-2, -2, 1; 1, 1, 0; 1, 0, 0]; a(n) = M^n [1, 1]. - Philippe Deléham, Apr 19 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for linear recurrences with constant coefficients, signature (-1,1,-1).

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

[n le 3 select -n*(-1)^n else -Self(n-1)+Self(n-2)-Self(n-3): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015

CoefficientList[Series[(1-x)/(1+x-x^2+x^3),{x,0,40}],x] (* or *) LinearRecurrence[{-1,1,-1},{1,-2,3},40] (* Harvey P. Dale, Jun 01 2012 *)

Vec((1-x)/(1+x-x^2+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = -a(n-1) + a(n-2) - a(n-3) for n > 2; a(0)=1, a(1)=-2, a(2)=3. - Harvey P. Dale, Jun 01 2012
a(n) = (-1)^n * A001590(n+2).
a(n) = Sum_{k=0..n} A188316(n,k)*(-2)^k. - Philippe Deléham, Apr 19 2023

A163403 a(n) = 2*a(n-2) for n > 2; a(1) = 1, a(2) = 2.

Original entry on oeis.org

1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152

Offset: 1

Klaus Brockhaus, Jul 26 2009

a(n+1) is the number of palindromic words of length n using a two-letter alphabet. - Michael Somos, Mar 20 2011

			x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 8*x^7 + 16*x^8 + 16*x^9 + 32*x^10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Equals A016116 without initial 1. Unsigned version of A152166.
Partial sums are in A136252.
Binomial transform is A078057, second binomial transform is A007070, third binomial transform is A102285, fourth binomial transform is A163350, fifth binomial transform is A163346.
Cf. A000079 (powers of 2), A009116, A009545, A051032.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

[ n le 2 select n else 2*Self(n-2): n in [1..43] ];

LinearRecurrence[{0, 2}, {1, 2}, 50] (* Paolo Xausa, Feb 02 2024 *)

{a(n) = if( n<1, 0, 2^(n\2))} /* Michael Somos, Mar 20 2011 */

def A163403():
    x, y = 1, 1
    while True:
        yield x
        x, y = x + y, x - y
a = A163403(); [next(a) for i in range(40)]  # Peter Luschny, Jul 11 2013

a(n) = 2^((1/4)*(2*n - 1 + (-1)^n)).
G.f.: x*(1 + 2*x)/(1 - 2*x^2).
a(n) = A051032(n) - 1.
G.f.: x / (1 - 2*x / (1 + x / (1 + x))) = x * (1 + 2*x / (1 - x / (1 - x / (1 + 2*x)))). - Michael Somos, Jan 03 2013
From R. J. Mathar, Aug 06 2009: (Start)
a(n) = A131572(n).
a(n) = A060546(n-1), n > 1. (End)
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = |A009116(n-1)| + |A009545(n-1)|. - Bruno Berselli, May 30 2011
E.g.f.: cosh(sqrt(2)*x) + sinh(sqrt(2)*x)/sqrt(2) - 1. - Stefano Spezia, Feb 05 2023

A163876 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 93, 180, 351, 684, 1332, 2592, 5046, 9825, 19128, 37239, 72498, 141144, 274788, 534972, 1041513, 2027676, 3947595, 7685400, 14962368, 29129580, 56711106, 110408373, 214949232, 418475259, 814711182, 1586125572, 3087958512

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

Also, coordination sequence for (6,6,6) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
The initial terms coincide with those of A003945, although the two sequences are eventually different.
Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, 1, 1, -1).

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x)*(1-x^6)/(1-2*x+2*x^6-x^7) )); // G. C. Greubel, Apr 25 2019

coxG[{6,1,-1,40}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Mar 22 2015 *)
CoefficientList[Series[(1+x)*(1-x^6)/(1-2*x+2*x^6-x^7), {x,0,40}], x] (* G. C. Greubel, Aug 06 2017, modified Apr 25 2019 *)

x='x+O('x^40); Vec((x^6+2*x^5+2*x^4+2*x^3+2*x^2+2*x+1)/(x^6-x^5- x^4-x^3-x^2-x+1)) \\ G. C. Greubel, Aug 06 2017

((1+x)*(1-x^6)/(1-2*x+2*x^6-x^7)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 25 2019

G.f.: (x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 2*x^2 + 2*x + 1)/(x^6 - x^5 - x^4 - x^3 - x^2 - x + 1).
G.f.: (1+x)*(1-x^6)/(1-2*x+2*x^6-x^7). - G. C. Greubel, Apr 25 2019
a(n) = -a(n-6) + Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 07 2021

cp's OEIS Frontend

A078840 Table of n-almost-primes T(n,k) (n >= 0, k > 0), read by antidiagonals, starting at T(0,1)=1 followed by T(1,1)=2.

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A054886 Layer counting sequence for hyperbolic tessellation by cuspidal triangles of angles (Pi/3,Pi/3,0) (this is the classical modular tessellation).

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A029858 a(n) = (3^n - 3)/2.

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A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

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A122132 Squarefree numbers multiplied by binary powers.

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A042950 Row sums of the Lucas triangle A029635.

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