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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A001787 a(n) = n*2^(n-1).

Original entry on oeis.org

0, 1, 4, 12, 32, 80, 192, 448, 1024, 2304, 5120, 11264, 24576, 53248, 114688, 245760, 524288, 1114112, 2359296, 4980736, 10485760, 22020096, 46137344, 96468992, 201326592, 419430400, 872415232, 1811939328, 3758096384, 7784628224, 16106127360, 33285996544
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of edges in an n-dimensional hypercube.
Number of 132-avoiding permutations of [n+2] containing exactly one 123 pattern. - Emeric Deutsch, Jul 13 2001
Number of ways to place n-1 nonattacking kings on a 2 X 2(n-1) chessboard for n >= 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 22 2001
Arithmetic derivative of 2^n: a(n) = A003415(A000079(n)). - Reinhard Zumkeller, Feb 26 2002
(-1) times the determinant of matrix A_{i,j} = -|i-j|, 0 <= i,j <= n.
a(n) is the number of ones in binary numbers 1 to 111...1 (n bits). a(n) = A000337(n) - A000337(n-1) for n = 2,3,... . - Emeric Deutsch, May 24 2003
The number of 2 X n 0-1 matrices containing n+1 1's and having no zero row or column. The number of spanning trees of the complete bipartite graph K(2,n). This is the case m = 2 of K(m,n). See A072590. - W. Edwin Clark, May 27 2003
Binomial transform of 0,1,2,3,4,5,... (A001477). Without the initial 0, binomial transform of odd numbers.
With an additional leading zero, [0,0,1,4,...] this is the binomial transform of the integers repeated A004526. Its formula is then (2^n*(n-1) + 0^n)/4. - Paul Barry, May 20 2003
Number of zeros in all different (n+1)-bit integers. - Ralf Stephan, Aug 02 2003
From Lekraj Beedassy, Jun 03 2004: (Start)
Final element of a summation table (as opposed to a difference table) whose first row consists of integers 0 through n (or first n+1 nonnegative integers A001477); illustrating the case n=5:
0 1 2 3 4 5
1 3 5 7 9
4 8 12 16
12 20 28
32 48
80
and the final element is a(5)=80. (End)
This sequence and A001871 arise in counting ordered trees of height at most k where only the rightmost branch at the root actually achieves this height and the count is by the number of edges, with k = 3 for this sequence and k = 4 for A001871.
Let R be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all elements x,y of P(A), xRy if x is a proper subset of y and there are no z in P(A) such that x is a proper subset of z and z is a proper subset of y. Then a(n) = |R|. - Ross La Haye, Sep 21 2004
Number of 2 X n binary matrices avoiding simultaneously the right-angled numbered polyomino patterns (ranpp) (00;1) and (10;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004
Number of subsequences 00 in all binary words of length n+1. Example: a(2)=4 because in 000,001,010,011,100,101,110,111 the sequence 00 occurs 4 times. - Emeric Deutsch, Apr 04 2005
If you expand the n-factor expression (a+1)*(b+1)*(c+1)*...*(z+1), there are a(n) variables in the result. For example, the 3-factor expression (a+1)*(b+1)*(c+1) expands to abc+ab+ac+bc+a+b+c+1 with a(3) = 12 variables. - David W. Wilson, May 08 2005
An inverse Chebyshev transform of n^2, where g(x)->(1/sqrt(1-4*x^2))*g(x*c(x^2)), c(x) the g.f. of A000108. - Paul Barry, May 13 2005
Sequences A018215 and A058962 interleaved. - Graeme McRae, Jul 12 2006
The number of never-decreasing positive integer sequences of length n with a maximum value of 2*n. - Ben Paul Thurston, Nov 13 2006
Total size of all the subsets of an n-element set. For example, a 2-element set has 1 subset of size 0, 2 subsets of size 1 and 1 of size 2. - Ross La Haye, Dec 30 2006
Convolution of the natural numbers [A000027] and A045623 beginning [0,1,2,5,...]. - Ross La Haye, Feb 03 2007
If M is the matrix (given by rows) [2,1;0,2] then the sequence gives the (1,2) entry in M^n. - Antonio M. Oller-Marcén, May 21 2007
If X_1,X_2,...,X_n is a partition of a 2n-set X into 2-blocks then, for n > 0, a(n) is equal to the number of (n+1)-subsets of X intersecting each X_i (i=1,2,...,n). - Milan Janjic, Jul 21 2007
Number of n-permutations of 3 objects u,v,w, with repetition allowed, containing exactly one u. Example: a(2)=4 because we have uv, vu, uw and wu. - Zerinvary Lajos, Dec 27 2007
A member of the family of sequences defined by a(n) = n*[c(1)*...*c(r)]^(n-1); c(i) integer. This sequence has c(1)=2, A027471 has c(1)=3. - Ctibor O. Zizka, Feb 23 2008
a(n) is the number of ways to split {1,2,...,n-1} into two (possibly empty) complementary intervals {1,2,...,i} and {i+1,i+2,...,n-1} and then select a subset from each interval. - Geoffrey Critzer, Jan 31 2009
Equals the Jacobsthal sequence A001045 convolved with A003945: (1, 3, 6, 12, ...). - Gary W. Adamson, May 23 2009
Starting with offset 1 = A059570: (1, 2, 6, 14, 34, ...) convolved with (1, 2, 2, 2, ...). - Gary W. Adamson, May 23 2009
Equals the first left hand column of A167591. - Johannes W. Meijer, Nov 12 2009
The number of tatami tilings of an n X n square with n monomers is n*2^(n-1). - Frank Ruskey, Sep 25 2010
Under T. D. Noe's variant of the hypersigma function, this sequence gives hypersigma(2^n): a(n) = A191161(A000079(n)). - Alonso del Arte, Nov 04 2011
Number of Dyck (n+2)-paths with exactly one valley at height 1 and no higher valley. - David Scambler, Nov 07 2011
Equals triangle A059260 * A016777 as a vector, where A016777 = (3n + 1): [1, 4, 7, 10, 13, ...]. - Gary W. Adamson, Mar 06 2012
Main transitions in systems of n particles with spin 1/2 (see A212697 with b=2). - Stanislav Sykora, May 25 2012
Let T(n,k) be the triangle with (first column) T(n,1) = 2*n-1 for n >= 1, otherwise T(n,k) = T(n,k-1) + T(n-1,k-1), then a(n) = T(n,n). - J. M. Bergot, Jan 17 2013
Sum of all parts of all compositions (ordered partitions) of n. The equivalent sequence for partitions is A066186. - Omar E. Pol, Aug 28 2013
Starting with a(1)=1: powers of 2 (A000079) self-convolved. - Bob Selcoe, Aug 05 2015
Coefficients of the series expansion of the normalized Schwarzian derivative -S{p(x)}/6 of the polynomial p(x) = -(x-x1)*(x-x2) with x1 + x2 = 1 (cf. A263646). - Tom Copeland, Nov 02 2015
a(n) is the number of North-East lattice paths from (0,0) to (n+1,n+1) that have exactly one east step below y = x-1 and no east steps above y = x+1. Details can be found in Pan and Remmel's link. - Ran Pan, Feb 03 2016
Also the number of maximal and maximum cliques in the n-hypercube graph for n > 0. - Eric W. Weisstein, Dec 01 2017
Let [n]={1,2,...,n}; then a(n-1) is the total number of elements missing in proper subsets of [n] that contain n to form [n]. For example, for n = 3, a(2) = 4 since the proper subsets of [3] that contain 3 are {3}, {1,3}, {2,3} and the total number of elements missing in these subsets to form [3] is 4: 2 in the first subset, 1 in the second, and 1 in the third. - Enrique Navarrete, Aug 08 2020
Number of 3-permutations of n elements avoiding the patterns 132, 231. See Bonichon and Sun. - Michel Marcus, Aug 19 2022

Examples

			a(2)=4 since 2314, 2341,3124 and 4123 are the only 132-avoiding permutations of 1234 containing exactly one increasing subsequence of length 3.
x + 4*x^2 + 12*x^3 + 32*x^4 + 80*x^5 + 192*x^6 + 448*x^7 + ...
a(5) = 1*0 + 5*1 + 10*2 + 10*3 + 5*4 + 1*5 = 80, with 1,5,10,10,5,1 the 5th row of Pascal's triangle. - _J. M. Bergot_, Apr 29 2014

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 131.
  • Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 282.
  • 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).

Links

Crossrefs

Three other versions, essentially identical, are A085750, A097067, A118442.
Partial sums of A001792.
A058922(n+1) = 4*A001787(n).
Equals A090802(n, 1).
Column k=1 of A038207.
Row sums of A003506, A322427, A322428.
Cf. A053109, A001788, A001789, A000337, A130300, A134083, A002064, A027471, A003945, A059670, A167591, A059260, A016777, A212697, A000079, A263646.

Programs

  • Haskell
    a001787 n = n * 2 ^ (n - 1)
a001787_list = zipWith (*) [0..] $ 0 : a000079_list
-- Reinhard Zumkeller, Jul 11 2014
  • Magma
    [n*2^(n-1): n in [0..40]]; // Vincenzo Librandi, Feb 04 2016
  • Maple
    spec := [S, {B=Set(Z, 0 <= card), S=Prod(Z, B, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..29); # Zerinvary Lajos, Oct 09 2006
A001787:=1/(2*z-1)^2; # Simon Plouffe in his 1992 dissertation, dropping the initial zero
  • Mathematica
    Table[Sum[Binomial[n, i] i, {i, 0, n}], {n, 0, 30}] (* Geoffrey Critzer, Mar 18 2009 *)
f[n_] := n 2^(n - 1); f[Range[0, 40]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
Array[# 2^(# - 1) &, 40, 0] (* Harvey P. Dale, Jul 26 2011 *)
Join[{0}, Table[n 2^(n - 1), {n, 20}]] (* Eric W. Weisstein, Dec 01 2017 *)
Join[{0}, LinearRecurrence[{4, -4}, {1, 4}, 20]] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[x/(-1 + 2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)
  • PARI
    {a(n) = if( n<0, 0, n * 2^(n-1))}
  • PARI
    concat(0, Vec(x/(1-2*x)^2 + O(x^50))) \\ Altug Alkan, Nov 03 2015
  • Python
    def A001787(n): return n*(1<Chai Wah Wu, Nov 14 2022

Formula

a(n) = Sum_{k=1..n} k*binomial(n, k). - Benoit Cloitre, Dec 06 2002
E.g.f.: x*exp(2x). - Paul Barry, Apr 10 2003
G.f.: x/(1-2*x)^2.
G.f.: x / (1 - 4*x / (1 + x / (1 - x))). - Michael Somos, Apr 07 2012
A108666(n) = Sum_{k=0..n} binomial(n, k)^2 * a(n). - Michael Somos, Apr 07 2012
PSumSIGN transform of A053220. PSumSIGN transform is A045883. Binomial transform is A027471(n+1). - Michael Somos, Jul 10 2003
Starting at a(1)=1, INVERT transform is A002450, INVERT transform of A049072, MOBIUS transform of A083413, PSUM transform is A000337, BINOMIAL transform is A081038, BINOMIAL transform of A005408. - Michael Somos, Apr 07 2012
a(n) = 2*a(n-1)+2^(n-1).
a(2*n) = n*4^n, a(2*n+1) = (2*n+1)4^n.
G.f.: x/det(I-x*M) where M=[1,i;i,1], i=sqrt(-1). - Paul Barry, Apr 27 2005
Starting 1, 1, 4, 12, ... this is 0^n + n2^(n-1), the binomial transform of the 'pair-reversed' natural numbers A004442. - Paul Barry, Jul 24 2003
Convolution of [1, 2, 4, 8, ...] with itself. - Jon Perry, Aug 07 2003
The signed version of this sequence, n(-2)^(n-1), is the inverse binomial transform of n(-1)^(n-1) (alternating sign natural numbers). - Paul Barry, Aug 20 2003
a(n-1) = (Sum_{k=0..n} 2^(n-k-1)*C(n-k, k)*C(1,(k+1)/2)*(1-(-1)^k)/2) - 0^n/4. - Paul Barry, Oct 15 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)(n-2k)^2. - Paul Barry, May 13 2005
a(n+2) = A049611(n+2) - A001788(n).
a(n) = n! * Sum_{k=0..n} 1/((k - 1)!(n - k)!). - Paul Barry, Mar 26 2003
a(n+1) = Sum_{k=0..n} 4^k * A109466(n,k). - Philippe Deléham, Nov 13 2006
Row sums of A130300 starting (1, 4, 12, 32, ...). - Gary W. Adamson, May 20 2007
Equals row sums of triangle A134083. Equals A002064(n) + (2^n - 1). - Gary W. Adamson, Oct 07 2007
a(n) = 4*a(n-1) - 4*a(n-2), a(0)=0, a(1)=1. - Philippe Deléham, Nov 16 2008
Sum_{n>0} 1/a(n) = 2*log(2). - Jaume Oliver Lafont, Feb 10 2009
a(n) = A000788(A000225(n)) = A173921(A000225(n)). - Reinhard Zumkeller, Mar 04 2010
a(n) = n * A011782(n). - Omar E. Pol, Aug 28 2013
a(n-1) = Sum_{t_1+2*t_2+...+n*t_n=n} (t_1+t_2+...+t_n-1)*multinomial(t_1+t_2 +...+t_n,t_1,t_2,...,t_n). - Mircea Merca, Dec 06 2013
a(n+1) = Sum_{r=0..n} (2*r+1)*C(n,r). - J. M. Bergot, Apr 07 2014
a(n) = A007283(n)*n/6. - Enxhell Luzhnica, Apr 16 2016
a(n) = (A000225(n) + A000337(n))/2. - Anton Zakharov, Sep 17 2016
Sum_{n>0} (-1)^(n+1)/a(n) = 2*log(3/2) = 2*A016578. - Ilya Gutkovskiy, Sep 17 2016
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (i+1) * C(k,i). - Wesley Ivan Hurt, Sep 21 2017
a(n) = Sum_{i=1..n} Sum_{j=1..n} phi(i)*binomial(n, i*j). - Ridouane Oudra, Feb 17 2024

A002450 a(n) = (4^n - 1)/3.

Original entry on oeis.org

0, 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525, 1398101, 5592405, 22369621, 89478485, 357913941, 1431655765, 5726623061, 22906492245, 91625968981, 366503875925, 1466015503701, 5864062014805, 23456248059221, 93824992236885, 375299968947541
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

For n > 0, a(n) is the degree (n-1) "numbral" power of 5 (see A048888 for the definition of numbral arithmetic). Example: a(3) = 21, since the numbral square of 5 is 5(*)5 = 101(*)101(base 2) = 101 OR 10100 = 10101(base 2) = 21, where the OR is taken bitwise. - John W. Layman, Dec 18 2001
a(n) is composite for all n > 2 and has factors x, (3*x + 2*(-1)^n) where x belongs to A001045. In binary the terms greater than 0 are 1, 101, 10101, 1010101, etc. - John McNamara, Jan 16 2002
Number of n X 2 binary arrays with path of adjacent 1's from upper left corner to right column. - R. H. Hardin, Mar 16 2002
The Collatz-function iteration started at a(n), for n >= 1, will end at 1 after 2*n+1 steps. - Labos Elemer, Sep 30 2002 [corrected by Wolfdieter Lang, Aug 16 2021]
Second binomial transform of A001045. - Paul Barry, Mar 28 2003
All members of sequence are also generalized octagonal numbers (A001082). - Matthew Vandermast, Apr 10 2003
Also sum of squares of divisors of 2^(n-1): a(n) = A001157(A000079(n-1)), for n > 0. - Paul Barry, Apr 11 2003
Binomial transform of A000244 (with leading zero). - Paul Barry, Apr 11 2003
Number of walks of length 2n between two vertices at distance 2 in the cycle graph C_6. For n = 2 we have for example 5 walks of length 4 from vertex A to C: ABABC, ABCBC, ABCDC, AFABC and AFEDC. - Herbert Kociemba, May 31 2004
Also number of walks of length 2n + 1 between two vertices at distance 3 in the cycle graph C_12. - Herbert Kociemba, Jul 05 2004
a(n+1) is the number of steps that are made when generating all n-step random walks that begin in a given point P on a two-dimensional square lattice. To make one step means to mark one vertex on the lattice (compare A080674). - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Mar 13 2005
a(n+1) is the sum of square divisors of 4^n. - Paul Barry, Oct 13 2005
a(n+1) is the decimal number generated by the binary bits in the n-th generation of the Rule 250 elementary cellular automaton. - Eric W. Weisstein, Apr 08 2006
a(k) = [M^k]2,1, where M is the 3 X 3 matrix defined as follows: M = [1, 1, 1; 1, 3, 1; 1, 1, 1]. - _Simone Severini, Jun 11 2006
a(n-1) / a(n) = percentage of wasted storage if a single image is stored as a pyramid with a each subsequent higher resolution layer containing four times as many pixels as the previous layer. n is the number of layers. - Victor Brodsky (victorbrodsky(AT)gmail.com), Jun 15 2006
k is in the sequence if and only if C(4k + 1, k) (A052203) is odd. - Paul Barry, Mar 26 2007
This sequence also gives the number of distinct 3-colorings of the odd cycle C(2*n - 1). - Keith Briggs, Jun 19 2007
All numbers of the form m*4^m + (4^m-1)/3 have the property that they are sums of two squares and also their indices are the sum of two squares. This follows from the identity m*4^m + (4^m-1)/3 = 4(4(..4(4m + 1) + 1) + 1) + 1 ..) + 1. - Artur Jasinski, Nov 12 2007
For n > 0, terms are the numbers that, in base 4, are repunits: 1_4, 11_4, 111_4, 1111_4, etc. - Artur Jasinski, Sep 30 2008
Let A be the Hessenberg matrix of order n, defined by: A[1, j] = 1, A[i, i] := 5, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 1, a(n) = charpoly(A,1). - Milan Janjic, Jan 27 2010
This is the sequence A(0, 1; 3, 4; 2) = A(0, 1; 4, 0; 1) of the family of sequences [a, b : c, d : k] considered by G. Detlefs, and treated as A(a, b; c, d; k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
6*a(n) + 1 is every second Mersenne number greater than or equal to M3, hence all Mersenne primes greater than M2 must be a 6*a(n) + 1 of this sequence. - Roderick MacPhee, Nov 01 2010
Smallest number having alternating bit sum n. Cf. A065359.
For n = 1, 2, ..., the last digit of a(n) is 1, 5, 1, 5, ... . - Washington Bomfim, Jan 21 2011
Rule 50 elementary cellular automaton generates this sequence. This sequence also appears in the second column of array in A173588. - Paul Muljadi, Jan 27 2011
Sequence found by reading the line from 0, in the direction 0, 5, ... and the line from 1, in the direction 1, 21, ..., in the square spiral whose edges are the Jacobsthal numbers A001045 and whose vertices are the numbers A000975. These parallel lines are two semi-diagonals in the spiral. - Omar E. Pol, Sep 10 2011
a(n), n >= 1, is also the inverse of 3, denoted by 3^(-1), Modd(2^(2*n - 1)). For Modd n see a comment on A203571. E.g., a(2) = 5, 3 * 5 = 15 == 1 (Modd 8), because floor(15/8) = 1 is odd and -15 == 1 (mod 8). For n = 1 note that 3 * 1 = 3 == 1 (Modd 2) because floor(3/2) = 1 and -3 == 1 (mod 2). The inverse of 3 taken Modd 2^(2*n) coincides with 3^(-1) (mod 2^(2*n)) given in A007583(n), n >= 1. - Wolfdieter Lang, Mar 12 2012
If an AVL tree has a leaf at depth n, then the tree can contain no more than a(n+1) nodes total. - Mike Rosulek, Nov 20 2012
Also, this is the Lucas sequence V(5, 4). - Bruno Berselli, Jan 10 2013
Also, for n > 0, a(n) is an odd number whose Collatz trajectory contains no odd number other than n and 1. - Jayanta Basu, Mar 24 2013
Sum_{n >= 1} 1/a(n) converges to (3*(log(4/3) - QPolyGamma[0, 1, 1/4]))/log(4) = 1.263293058100271... = A321873. - K. G. Stier, Jun 23 2014
Consider n spheres in R^n: the i-th one (i=1, ..., n) has radius r(i) = 2^(1-i) and the coordinates of its center are (0, 0, ..., 0, r(i), 0, ..., 0) where r(i) is in position i. The coordinates of the intersection point in the positive orthant of these spheres are (2/a(n), 4/a(n), 8/a(n), 16/a(n), ...). For example in R^2, circles centered at (1, 0) and (0, 1/2), and with radii 1 and 1/2, meet at (2/5, 4/5). - Jean M. Morales, May 19 2015
From Peter Bala, Oct 11 2015: (Start)
a(n) gives the values of m such that binomial(4*m + 1,m) is odd. Cf. A003714, A048716, A263132.
2*a(n) = A020988(n) gives the values of m such that binomial(4*m + 2, m) is odd.
4*a(n) = A080674(n) gives the values of m such that binomial(4*m + 4, m) is odd. (End)
Collatz Conjecture Corollary: Except for powers of 2, the Collatz iteration of any positive integer must eventually reach a(n) and hence terminate at 1. - Gregory L. Simay, May 09 2016
Number of active (ON, black) cells at stage 2^n - 1 of the two-dimensional cellular automaton defined by "Rule 598", based on the 5-celled von Neumann neighborhood. - Robert Price, May 16 2016
From Luca Mariot and Enrico Formenti, Sep 26 2016: (Start)
a(n) is also the number of coprime pairs of polynomials (f, g) over GF(2) where both f and g have degree n + 1 and nonzero constant term.
a(n) is also the number of pairs of one-dimensional binary cellular automata with linear and bipermutive local rule of neighborhood size n+1 giving rise to orthogonal Latin squares of order 2^m, where m is a multiple of n. (End)
Except for 0, 1 and 5, all terms are Brazilian repunits numbers in base 4, and so belong to A125134. For n >= 3, all these terms are composite because a(n) = {(2^n-1) * (2^n + 1)}/3 and either (2^n - 1) or (2^n + 1) is a multiple of 3. - Bernard Schott, Apr 29 2017
Given the 3 X 3 matrix A = [2, 1, 1; 1, 2, 1; 1, 1, 2] and the 3 X 3 unit matrix I_3, A^n = a(n)(A - I_3) + I_3. - Nicolas Patrois, Jul 05 2017
The binary expansion of a(n) (n >= 1) consists of n 1's alternating with n - 1 0's. Example: a(4) = 85 = 1010101_2. - Emeric Deutsch, Aug 30 2017
a(n) (n >= 1) is the viabin number of the integer partition [n, n - 1, n - 2, ..., 2, 1] (for the definition of viabin number see comment in A290253). Example: a(4) = 85 = 1010101_2; consequently, the southeast border of the Ferrers board of the corresponding integer partition is ENENENEN, where E = (1, 0), N = (0, 1); this leads to the integer partition [4, 3, 2, 1]. - Emeric Deutsch, Aug 30 2017
Numbers whose binary and Gray-code representations are both palindromes (i.e., intersection of A006995 and A281379). - Amiram Eldar, May 17 2021
Starting with n = 1 the sequence satisfies {a(n) mod 6} = repeat{1, 5, 3}. - Wolfdieter Lang, Jan 14 2022
Terms >= 5 are those q for which the multiplicative order of 2 mod q is floor(log_2(q)) + 2 (and which is 1 more than the smallest possible order for any q). - Tim Seuré, Mar 09 2024
The order of 2 modulo a(n) is 2*n for n >= 2. - Joerg Arndt, Mar 09 2024

Examples

			Apply Collatz iteration to 9: 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5 and hence 16, 8, 4, 2, 1.
Apply Collatz iteration to 27: 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5 and hence 16, 8, 4, 2, 1. [Corrected by _Sean A. Irvine_ at the suggestion of Stephen Cornelius, Mar 04 2024]
a(5) = (4^5 - 1)/3 = 341 = 11111_4 = {(2^5 - 1) * (2^5 + 1)}/3 = 31 * 33/3 = 31 * 11. - _Bernard Schott_, Apr 29 2017

References

  • A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 112.
  • J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
  • 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).

Links

Crossrefs

Partial sums of powers of 4, A000302.
When converted to binary, this gives A094028.
Subsequence of A003714.
Primitive factors: A129735.
Cf. A000225, A002446, A006995, A007583, A018215, A020988, A024036, A047849, A048716, A080355, A080674, A112627, A113860, A139391, A155701, A156605, A160967, A163834, A163868, A178415, A263132, A281379, A347834.

Programs

  • GAP
    List([0..25], n -> (4^n-1)/3); # Muniru A Asiru, Feb 18 2018
  • Haskell
    a002450 = (`div` 3) . a024036
a002450_list = iterate ((+ 1) . (* 4)) 0
-- Reinhard Zumkeller, Oct 03 2012
  • Magma
    [ (4^n-1)/3: n in [0..25] ]; // Klaus Brockhaus, Oct 28 2008
  • Magma
    [n le 2 select n-1 else 5*Self(n-1)-4*Self(n-2): n in [1..70]]; // Vincenzo Librandi, Jun 13 2015
  • Maple
    [seq((4^n-1)/3,n=0..40)];
A002450:=1/(4*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation, dropping the initial zero
  • Mathematica
    Table[(4^n - 1)/3, {n, 0, 127}] (* Vladimir Joseph Stephan Orlovsky, Sep 29 2008 *)
LinearRecurrence[{5, -4}, {0, 1}, 30] (* Harvey P. Dale, Jun 23 2013 *)
  • Maxima
    makelist((4^n-1)/3, n, 0, 30); /* Martin Ettl, Nov 05 2012 */
  • PARI
    a(n) = (4^n-1)/3;
  • PARI
    my(z='z+O('z^40)); Vec(z/((1-z)*(1-4*z))) \\ Altug Alkan, Oct 11 2015
  • Python
    def A002450(n): return ((1<<(n<<1))-1)//3 # Chai Wah Wu, Jan 29 2023
  • Scala
    ((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)( * )).scanLeft(0: BigInt)( + ) // Alonso del Arte, Sep 17 2019

Formula

From Wolfdieter Lang, Apr 24 2001: (Start)
a(n+1) = Sum_{m = 0..n} A060921(n, m).
G.f.: x/((1-x)*(1-4*x)). (End)
a(n) = Sum_{k = 0..n-1} 4^k; a(n) = A001045(2*n). - Paul Barry, Mar 17 2003
E.g.f.: (exp(4*x) - exp(x))/3. - Paul Barry, Mar 28 2003
a(n) = (A007583(n) - 1)/2. - N. J. A. Sloane, May 16 2003
a(n) = A000975(2*n)/2. - N. J. A. Sloane, Sep 13 2003
a(n) = A084160(n)/2. - N. J. A. Sloane, Sep 13 2003
a(n+1) = 4*a(n) + 1, with a(0) = 0. - Philippe Deléham, Feb 25 2004
a(n) = Sum_{i = 0..n-1} C(2*n - 1 - i, i)*2^i. - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
a(n+1) = Sum_{k = 0..n} binomial(n+1, k+1)*3^k. - Paul Barry, Aug 20 2004
a(n) = center term in M^n * [1 0 0], where M is the 3 X 3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 0 0] = [A007583(n-1) a(n) A007583(n-1)]. E.g., a(4) = 85 since M^4 * [1 0 0] = [43 85 43] = [A007583(3) a(4) A007583(3)]. - Gary W. Adamson, Dec 18 2004
a(n) = Sum_{k = 0..n, j = 0..n} C(n, j)*C(j, k)*A001045(j - k). - Paul Barry, Feb 15 2005
a(n) = Sum_{k = 0..n} C(n, k)*A001045(n-k)*2^k = Sum_{k = 0..n} C(n, k)*A001045(k)*2^(n-k). - Paul Barry, Apr 22 2005
a(n) = A125118(n, 3) for n > 2. - Reinhard Zumkeller, Nov 21 2006
a(n) = Sum_{k = 0..n} 2^(n - k)*A128908(n, k), n >= 1. - Philippe Deléham, Oct 19 2008
a(n) = Sum_{k = 0..n} A106566(n, k)*A100335(k). - Philippe Deléham, Oct 30 2008
If we define f(m, j, x) = Sum_{k = j..m} binomial(m, k)*stirling2(k, j)*x^(m - k) then a(n-1) = f(2*n, 4, -2), n >= 2. - Milan Janjic, Apr 26 2009
a(n) = A014551(n) * A001045(n). - R. J. Mathar, Jul 08 2009
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3) = 5*a(n-1) - 4*a(n-2), a(0) = 0, a(1) = 1, a(2) = 5. - Wolfdieter Lang, Oct 18 2010
a(0) = 0, a(n+1) = a(n) + 2^(2*n). - Washington Bomfim, Jan 21 2011
A036555(a(n)) = 2*n. - Reinhard Zumkeller, Jan 28 2011
a(n) = Sum_{k = 1..floor((n+2)/3)} C(2*n + 1, n + 2 - 3*k). - Mircea Merca, Jun 25 2011
a(n) = Sum_{i = 1..n} binomial(2*n + 1, 2*i)/3. - Wesley Ivan Hurt, Mar 14 2015
a(n+1) = 2^(2*n) + a(n), a(0) = 0. - Ben Paul Thurston, Dec 27 2015
a(k*n)/a(n) = 1 + 4^n + ... + 4^((k-1)*n). - Gregory L. Simay, Jun 09 2016
Dirichlet g.f.: (PolyLog(s, 4) - zeta(s))/3. - Ilya Gutkovskiy, Jun 26 2016
A000120(a(n)) = n. - André Dalwigk, Mar 26 2018
a(m) divides a(m*n), in particular: a(2*n) == 0 (mod 5), a(3*n) == 0 (mod 3*7), a(5*n) == 0 (mod 11*31), etc. - M. F. Hasler, Oct 19 2018
a(n) = 4^(n-1) + a(n-1). - Bob Selcoe, Jan 01 2020
a(n) = A178415(1, n) = A347834(1, n-1), arrays, for n >= 1. - Wolfdieter Lang, Nov 29 2021
a(n) = A000225(2*n)/3. - John Keith, Jan 22 2022
a(n) = A080674(n) + 1 = A047849(n) - 1 = A163834(n) - 2 = A155701(n) - 3 = A163868(n) - 4 = A156605(n) - 7. - Ray Chandler, Jun 16 2023
From Peter Bala, Jul 23 2025: (Start)
The following are examples of telescoping products. Cf. A016153:
Product_{k = 1..2*n} 1 + 2^k/a(k+1) = a(n+1)/A007583(n) = (4^(n+1) - 1)/(2*4^n + 1).
Hence, Product_{k >= 1} 1 + 2^k/a(k+1) = 2.
Product_{k >= 1} 1 - 2^k/a(k+1) = 2/5, since 1 - 2^n/a(n+1) = b(n)/b(n-1), where b(n) = 2 - 3/(1 - 2^(n+1)).
Product_{k >= 1} 1 + (-2)^k/a(k+1) = 2/3, since 1 + (-2)^n/a(n+1) = c(n)/c(n-1), where c(n) = 2 - 1/(1 + (-2)^(n+1)).
Product_{k >= 1} 1 - (-2)^k/a(k+1) = 6/5, since 1 - (-2)^n/a(n+1) = d(n)/d(n-1), where d(n) = 2 - 1/(1 - (-2)^(n+1)). (End)

A118416 Triangle read by rows: T(n,k) = (2*k-1)*2^(n-1), 0 < k <= n.

Original entry on oeis.org

1, 2, 6, 4, 12, 20, 8, 24, 40, 56, 16, 48, 80, 112, 144, 32, 96, 160, 224, 288, 352, 64, 192, 320, 448, 576, 704, 832, 128, 384, 640, 896, 1152, 1408, 1664, 1920, 256, 768, 1280, 1792, 2304, 2816, 3328, 3840, 4352, 512, 1536, 2560, 3584, 4608, 5632, 6656, 7680
Offset: 1

Views

Author

Reinhard Zumkeller, Apr 27 2006

Keywords

Comments

Row sums give A014477: Sum_{k=1..n} T(n,k) = A014477(n-1);
central terms give A118415; T(2*k-1,k) = A058962(k-1);
T(n,1) = A000079(n-1);
T(n,2) = A007283(n-1) for n > 1;
T(n,3) = A020714(n-1) for n > 2;
T(n,4) = A005009(n-1) for n > 3;
T(n,5) = A005010(n-1) for n > 4;
T(n,n-1) = A118417(n-1) for n > 1;
T(n,n) = A014480(n-1) = A118413(n,n);
A001511(T(n,k)) = A002024(n,k);
A003602(T(n,k)) = A002260(n,k).
The alternating row sums, Sum_{k=1..n} (-1)^(k+1)*T(n,k), are: (a) in odd rows, the central term, T(n,(n+1)/2) = A058962((n-1)/2); (b) in even rows, the negation of the average of the two central terms, -(T(2n,n) + T(2n,+1))/2 = -A018215(m/2). The absolute values of the alternating row sums give the plain row means, Sum_{k=1..n} T(n,k)/n; the alternating sign row means are (-2)^(n-1). - Gregory Gerard Wojnar, Feb 10 2024

Examples

			Triangle begins:
   1;
   2,   6;
   4,  12,  20;
   8,  24,  40,  56;
  16,  48,  80, 112, 144;
  32,  96, 160, 224, 288, 352;
  64, 192, 320, 448, 576, 704, 832;

Links

Crossrefs

Cf. A118413, A117303, A054582.

Programs

  • Haskell
    a118416 n k = a118416_tabl !! (n-1) !! (k-1)
a118416_row 1 = [1]
a118416_row n = (map (* 2) $ a118416_row (n-1)) ++ [a014480 (n-1)]
a118416_tabl = map a118416_row [1..]
-- Reinhard Zumkeller, Jan 22 2012
  • Maple
    A118416 := proc(n,k) 2^(n-1)*(2*k-1) ; end proc: # R. J. Mathar, Sep 04 2011
  • Mathematica
    Flatten[Table[(2k-1)2^(n-1),{n,10},{k,n}]] (* Harvey P. Dale, Aug 26 2014 *)
  • Python
    from math import isqrt
def A118416(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(1-a)+(n<<1)-1<Chai Wah Wu, Jun 20 2025

Formula

T(n,k) = 2*T(n-1,k), 1 <= k < n; T(n,n) = A014480(n-1).

A123045 Number of frieze patterns of length n under a certain group (see Pisanski et al. for precise definition).

Original entry on oeis.org

0, 2, 6, 12, 39, 104, 366, 1172, 4179, 14572, 52740, 190652, 700274, 2581112, 9591666, 35791472, 134236179, 505290272, 1908947406, 7233629132, 27488079132, 104715393912, 399823554006, 1529755308212, 5864066561554, 22517998136936, 86607703209516
Offset: 0

Views

Author

N. J. A. Sloane, Nov 11 2006

Keywords

Links

Crossrefs

Cf. A054611, A018215.
The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

Programs

  • Maple
    with(numtheory):
V:=proc(n) local k, t1; t1:=0; for k in divisors(n) do t1 := t1+phi(k)*4^(n/k); od: t1; end; # A054611
H:=n-> if n mod 2 = 0 then (n/2)*4^(n/2); else 0; fi; # this is A018215 interleaved with 0's
A123045:=n-> `if`(n=0,0, (V(n)+H(n))/(2*n));
  • Mathematica
    V[n_] := Module[{t1 = 0}, Do[t1 = t1 + EulerPhi[k] 4^(n/k), {k, Divisors[n]}]; t1];
H[n_] := If[Mod[n, 2] == 0, (n/2) 4^(n/2), 0];
a[n_] := If[n == 0, 0, (V[n] + H[n])/(2n)];
a /@ Range[0, 26] (* Jean-François Alcover, Mar 20 2020, from Maple *)

Formula

See Maple program.

A060416 a(n) = n*4^n - 1, with a(0) = 1.

Original entry on oeis.org

1, 3, 31, 191, 1023, 5119, 24575, 114687, 524287, 2359295, 10485759, 46137343, 201326591, 872415231, 3758096383, 16106127359, 68719476735, 292057776127, 1236950581247, 5222680231935, 21990232555519, 92358976733183, 387028092977151, 1618481116086271, 6755399441055743
Offset: 0

Views

Author

Jason Earls, Apr 05 2001

Keywords

Links

Crossrefs

Cf. A003261, A018215.

Programs

  • Mathematica
    CoefficientList[Series[(1 - 6 x + 28 x^2 - 32 x^3)/((1 - x) (1 - 4 x)^2), {x, 0, 21}], x] (* Michael De Vlieger, Jan 04 2020 *)
  • PARI
    a(n) = { abs(n*4^n - 1) } \\ Harry J. Smith, Jul 04 2009

Formula

G.f.: (1 - 6*x + 28*x^2 - 32*x^3)/((1 - x)*(1 - 4*x)^2). - Colin Barker, Apr 22 2012
From Elmo R. Oliveira, May 05 2025: (Start)
E.g.f.: 2 + exp(x)*(4*x*exp(3*x) - 1).
a(n) = A018215(n) - 1 for n >= 1.
a(n) = 9*a(n-1) - 24*a(n-2) + 16*a(n-3) for n > 3. (End)

A083679 Decimal expansion of log(4/3).

Original entry on oeis.org

2, 8, 7, 6, 8, 2, 0, 7, 2, 4, 5, 1, 7, 8, 0, 9, 2, 7, 4, 3, 9, 2, 1, 9, 0, 0, 5, 9, 9, 3, 8, 2, 7, 4, 3, 1, 5, 0, 3, 5, 0, 9, 7, 1, 0, 8, 9, 7, 7, 6, 1, 0, 5, 6, 5, 0, 6, 6, 6, 5, 6, 8, 5, 3, 4, 9, 2, 9, 2, 9, 5, 0, 7, 2, 0, 7, 8, 0, 4, 6, 4, 3, 3, 8, 1, 1, 0, 8, 9, 9, 1, 7, 9, 1, 0, 5, 2, 8, 6, 2, 9, 6, 0, 3
Offset: 0

Views

Author

Benoit Cloitre, Jun 15 2003

Keywords

Examples

			log(4/3) = 0.2876820724517809274392190059938274315035097108977610565....

Links

Crossrefs

Cf. A016578, A018215, A036290.

Programs

Formula

Limit of a special sum: log(4/3) = Sum_{k>=1} (Sum_{i=1..k} 1/(i*2^i))/2^(k+1).
Asymptotically: log(4/3) = Sum_{k=1..n} (Sum_{i=1..k} 1/(i*2^i))/2^(k+1) + log(2)/2^(n+1) + o(1/2^n).
From Amiram Eldar, Aug 07 2020: (Start)
Equals 2 * arctanh(1/7).
Equals Sum_{n>=1} 1/(n * 4^n) = Sum_{n>=1} 1/A018215(n).
Equals Sum_{n>=1} (-1)^(n+1)/(n * 3^n) = Sum_{n>=1} (-1)^(n+1)/A036290(n).
Equals Integral_{x=0..oo} 1/(3*exp(x) + 1) dx. (End)
log(4/3) = 2*Sum_{n >= 1} 1/(n*P(n, 7)*P(n-1, 7)), where P(n, x) denotes the n-th Legendre polynomial. The first 10 terms of the series gives the approximation log(4/3) = 0.28768207245178092743921(31...), correct to 23 decimal places. - Peter Bala, Mar 18 2024
Equals Sum_{n >= 1} (-1)^(n+1) * 7/(n*binomial(2*n, n)*12^n). The n-th term of the series is O(7*sqrt(Pi/n)*1/48^n). - Peter Bala, Mar 04 2025
Equals Integral_{x=0..1} (x^(1/3) - 1)/log(x) dx. - Kritsada Moomuang, May 27 2025

A258997 A(n,k) = pi-based antiderivative of n^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 2, 0, 0, 0, 12, 12, 4, 0, 0, 0, 32, 54, 32, 3, 0, 0, 0, 80, 216, 192, 30, 7, 0, 0, 0, 192, 810, 1024, 225, 84, 4, 0, 0, 0, 448, 2916, 5120, 1500, 756, 56, 12, 0, 0, 0, 1024, 10206, 24576, 9375, 6048, 588, 192, 12, 0
Offset: 0

Views

Author

Alois P. Heinz, Jun 27 2015

Keywords

Examples

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0, ...
  0, 0,  0,   0,    0,     0,      0,       0, ...
  0, 1,  4,  12,   32,    80,    192,     448, ...
  0, 2, 12,  54,  216,   810,   2916,   10206, ...
  0, 4, 32, 192, 1024,  5120,  24576,  114688, ...
  0, 3, 30, 225, 1500,  9375,  56250,  328125, ...
  0, 7, 84, 756, 6048, 45360, 326592, 2286144, ...
  0, 4, 56, 588, 5488, 48020, 403368, 3294172, ...

Links

Crossrefs

Rows n=0+1,2,3,4,8 give: A000004, A001787, A212697, A018215, A230539.
Columns k=0,1 give: A000004, A258851.
Main diagonal gives A258846.
Cf. A000720.

Programs

  • Maple
    with(numtheory):
d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
A:= (n, k)-> `if`(k=0, 0, k*n^(k-1)*d(n)):
seq(seq(A(n, h-n), n=0..h), h=0..14);

Formula

A(n,k) = A258851(n^k) = k * n^(k-1) * A258851(n).

A171607 Expressible as A*B^A in a nontrivial way.

Original entry on oeis.org

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, 1215, 1250, 1352, 1458, 1536, 1568, 1682, 1800, 1922, 2048, 2178, 2187, 2312, 2450, 2500, 2592
Offset: 1

Views

Author

Robert Munafo, Dec 12 2009

Keywords

Examples

			8=2*2^2. 24=3*2^3. 375=3*5^3.

Links

Crossrefs

Cf. A171606. Union of the "KN^K" sequences A001105, A117642, A141046, ... or of the "NK^N" sequences A036289, A036290, A018215, A036291, ... but omitting the trivial initial terms.

Programs

  • PARI
    is(n)=if(n<8, return(0)); for(a=2,logint(n\2,2), if(n%a==0 && ispower(n/a,a), return(1))); 0 \\ Charles R Greathouse IV, Feb 19 2017
  • PARI
    list(lim)=my(v=List()); if(lim<8,return([])); for(a=2,logint(lim\2,2), for(b=2,sqrtnint(lim\a,a), listput(v,a*b^a))); Set(v) \\ Charles R Greathouse IV, Feb 19 2017

Formula

a(n) = 2n^2 - O(n^(5/3)). - Charles R Greathouse IV, Feb 19 2017

A158749 a(n) = n*9^n.

Original entry on oeis.org

0, 9, 162, 2187, 26244, 295245, 3188646, 33480783, 344373768, 3486784401, 34867844010, 345191655699, 3389154437772, 33044255768277, 320275094369454, 3088366981419735, 29648323021629456, 283512088894331673, 2701703435345984178, 25666182635786849691, 243153309181138576020
Offset: 0

Views

Author

Zerinvary Lajos, Mar 25 2009

Keywords

Links

Crossrefs

Cf. A001019, A018215, A036289, A036290, A036291, A036292, A036293, A036294.
Cf. A038299, A126431.

Programs

Formula

a(n) = n*9^n.
From R. J. Mathar, Mar 26 2009: (Start)
a(n) = 18*a(n-1) - 81*a(n-2) = A038299(n,1).
G.f.: 9*x/(1-9*x)^2. (End)
a(n) = A001019(n)*n. - Omar E. Pol, Mar 26 2009
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(9/8).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(10/9). (End)
E.g.f.: 9*x*exp(9*x). - Elmo R. Oliveira, Sep 09 2024

A230539 a(n) = 3*n*2^(3*n-1).

Original entry on oeis.org

0, 12, 192, 2304, 24576, 245760, 2359296, 22020096, 201326592, 1811939328, 16106127360, 141733920768, 1236950581248, 10720238370816, 92358976733184, 791648371998720, 6755399441055744, 57420895248973824, 486388759756013568, 4107282860161892352
Offset: 0

Views

Author

Bruno Berselli, Oct 23 2013

Keywords

Comments

Arithmetic derivative of 8^n: a(n) = A003415(8^n).
Sum of reciprocals of a(n), for n>0: (2/3)*log(8/7).

Links

Crossrefs

Cf. A001018, A003415.
Cf. arithmetic derivative of k^n: A001787 (k=2), A027471 (k=3), A018215 (k=4), A053464 (k=5), A212700 (k=6), A027473 (k=7), this sequence, A230540 (k=9), A085708 (k=10), A081127 (k=11).
Row n=8 of A258997.

Programs

  • Magma
    [3*n*2^(3*n-1): n in [0..20]];
  • Maple
    A230539:=n->3*n*2^(3*n-1): seq(A230539(n), n=0..30); # Wesley Ivan Hurt, May 03 2017
  • Mathematica
    Table[3 n 2^(3 n - 1), {n,0,20}]
LinearRecurrence[{16,-64},{0,12},20] (* Harvey P. Dale, Dec 25 2022 *)
  • PARI
    a(n) = 3*n*2^(3*n-1); \\ Michel Marcus, Oct 23 2013

Formula

G.f.: 12*x/(1-8*x)^2.
a(n) = 12*A053539(n).
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