A028403 -id:A028403 - cp's OEIS frontend

A004171 a(n) = 2^(2n+1).

2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832, 140737488355328, 562949953421312

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(2, 8), L(2, 8), P(2, 8), T(2, 8). See A008776 for definitions of Pisot sequences.
In the Chebyshev polynomial of degree 2n, a(n) is the coefficient of x^2n. - Benoit Cloitre, Mar 13 2002
1/2 - 1/8 + 1/32 - 1/128 + ... = 2/5. - Gary W. Adamson, Mar 03 2009
From Adi Dani, May 15 2011: (Start)
Number of ways of placing an even number of indistinguishable objects in n + 1 distinguishable boxes with at most 3 objects in box.
Number of compositions of even natural numbers into n + 1 parts less than or equal to 3 (0 is counted as part). (End)
Also the number of maximal cliques in the (n+1)-Sierpinski tetrahedron graph for n > 0. - Eric W. Weisstein, Dec 01 2017
Assuming the Collatz conjecture is true, any starting number eventually leads to a power of 2. A number in this sequence can never be the first power of 2 in a Collatz sequence except of course for the Collatz sequence starting with that number. For example, except for 8, 4, 2, 1, any Collatz sequence that includes 8 must also include 16 (e.g., 5, 16, 8, 4, 2, 1). - Alonso del Arte, Oct 01 2019
First differences of A020988, and thus the "wavelengths" of the local maxima in A020986. See the Brillhart and Morton link, pp. 855-856. - John Keith, Mar 04 2021

			G.f. = 2 + 8*x + 32*x^2 + 128*x^3 + 512*x^4 + 2048*x^5 + 8192*x^6 + 32768*x^7 + ...
From _Adi Dani_, May 15 2011: (Start)
a(1) = 8 because all compositions of even natural numbers into 2 parts less than or equal to 3 are:
  for 0: (0, 0)
  for 2: (0, 2), (2, 0), (1, 1)
  for 4: (1, 3), (3, 1), (2, 2)
  for 6: (3, 3).
a(2) = 32 because all compositions of even natural numbers into 3 parts less than or equal to 3 are:
  for 0: (0, 0, 0)
  for 2: (0, 0, 2), (0, 2, 0), (2, 0, 0), (0, 1, 1), (1, 0, 1) , (1, 1, 0)
  for 4: (0, 1, 3), (0, 3, 1), (1, 0, 3), (1, 3, 0), (3, 0, 1), (3, 1, 0), (0, 2, 2), (2, 0, 2), (2, 2, 0), (1, 1, 2), (1, 2, 1), (2, 1, 1)
  for 6: (0, 3, 3), (3, 0, 3), (3, 3, 0), (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1), (2, 2, 2)
  for 8: (2, 3, 3), (3, 2, 3), (3, 3, 2).
(End)

Adi Dani, Quasicompositions of natural numbers, Proceedings of III congress of mathematicians of Macedonia, Struga Macedonia 29 IX -2 X 2005 pages 225-238.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023).
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Tanya Khovanova, Recursive Sequences.
Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph.
Index entries for linear recurrences with constant coefficients, signature (4).
Index to divisibility sequences.

Cf. A013708-A013729.
Absolute value of A009117. Essentially the same as A081294.
Cf. A132020, A164632. Equals A000980(n) + 2*A181765(n). Cf. A013776.

List([0..30],n->2^(2*n+1)); # Muniru A Asiru, Mar 12 2019

a004171 = (* 2) . a000302
a004171_list = iterate (* 4) 2  -- Reinhard Zumkeller, Jan 09 2013

[2^(2*n+1): n in [0..30]]; // Vincenzo Librandi, May 16 2011

seq(2^(2*n+1),n=0..24); # Nathaniel Johnston, Jun 25 2011

Table[2^(2 n + 1), {n, 0, 24}]
2^(2 Range[20] - 1) (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{4}, {2}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[2/(1 - 4 x), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

a(n)=2<<(2*n) \\ Charles R Greathouse IV, Apr 07 2012

a(n) = 2^(2*n+1) \\ Michel Marcus, Aug 12 2014

[2**(2*n+1) for n in range(0,25)] # Stefano Spezia, Jul 23 2025

((List.fill(20)(4: BigInt)).scanLeft(1: BigInt)( * )).map(2 * ) // _Alonso del Arte, Sep 12 2019

a(n) = 2*4^n.
a(n) = 4*a(n-1).
1 = 1/2 + Sum_{n >= 1} 3/a(n) = 3/6 + 3/8 + 3/32 + 3/128 + 3/512 + 3/2048 + ...; with partial sums: 1/2, 31/32, 127/128, 511/512, 2047/2048, ... - Gary W. Adamson, Jun 16 2003
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 2*A000302(n).
G.f.: 2/(1-4*x). (End)
a(n) = A081294(n+1) = A028403(n+1) - A000079(n+1) for n >= 1. a(n-1) = A028403(n) - A000079(n). - Jaroslav Krizek, Jul 27 2009
E.g.f.: 2*exp(4*x). - Ilya Gutkovskiy, Nov 01 2016
a(n) = A002063(n)/3 - A000302(n). - Zhandos Mambetaliyev, Nov 19 2016
a(n) = Sum_{k = 0..2*n} (-1)^(k+n)*binomial(4*n + 2, 2*k + 1); a(2*n) = Sum_{k = 0..2*n} binomial(4*n + 2, 2*k + 1) = A013776(n). - Peter Bala, Nov 25 2016
Product_{n>=0} (1 - 1/a(n)) = A132020. - Amiram Eldar, May 08 2023

A081294 Expansion of (1-2x)/(1-4x).

Original entry on oeis.org

1, 2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912, 2147483648, 8589934592, 34359738368, 137438953472, 549755813888, 2199023255552, 8796093022208, 35184372088832

Offset: 0

Paul Barry, Mar 17 2003

Binomial transform of A046717. Second binomial transform of A000302 (with interpolated zeros). Partial sums are A007583.
Counts closed walks of length 2n at a vertex of the cyclic graph on 4 nodes C_4. With interpolated zeros, counts closed walks of length n at a vertex of the cyclic graph on 4 nodes C_4. - Paul Barry, Mar 10 2004
In general, Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j)*r^j has expansion (1 - (r+1)*x)/(1 - (r+3)*x - (r-1)*(r+3)*x^2 + (r-1)^3*x^3). - Paul Barry, Jun 04 2005 [corrected by Jason Yuen, Jan 20 2025]
a(n) is the number of binary strings of length 2n with an even number of 0's (and hence an even number of 1's). - Toby Gottfried, Mar 22 2010
Number of compositions of n where there are 2 sorts of part 1, 4 sorts of part 2, 8 sorts of part 3, ..., 2^k sorts of part k. - Joerg Arndt, Aug 04 2014
a(n) is also the number of permutations simultaneously avoiding 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
INVERT transform of powers of 2 (A000079). - Alois P. Heinz, Feb 11 2021
a(n) is the number of elements in an n-interval of the binomial poset of even-sized subsets of positive integers, cf. Stanley reference and second formula by Paul Barry.  Each multichain 0 = x_0 <= x_1 <= x_2 = 1 in such an n-interval corresponds to a closed walk described above by Paul Barry.  More generally, each multichain 0 = x_0 <= x_1 <= ... <= x_k = 1 corresponds to a closed walk of length 2n on the k-dimensional hypercube, cf. A054879, A092812, A121822. - Geoffrey Critzer, Apr 21 2023

			G.f. = 1 + 2*x + 8*x^2 + 32*x^3 + 128*x^4 + 512*x^5 + 2048*x^6 + 8192*x^7 + ...

Richard P. Stanley, Enumerative Combinatorics, Vol 1, second edition, Example 3.18.3-f, page 323.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
M. Paukner, L. Pepin, M. Riehl, and J. Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Index to divisibility sequences.
Index entries for linear recurrences with constant coefficients, signature (4).

Row sums of triangle A136158.

Cf. A000079, A081295, A009117, A016742, A054879, A092812, A121822. Essentially the same as A004171.

[(4^n+0^n)/2: n in [0..30]]; // Vincenzo Librandi, Jul 26 2011

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-2*x)/(1-4*x))); // Marius A. Burtea, Jan 20 2020

a:= n-> 2^max(0, (2*n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 20 2017

CoefficientList[Series[(1-2x)/(1-4x),{x,0,40}],x] (* or *)
Join[{1}, NestList[4 # &, 2, 40]] (* Harvey P. Dale, Apr 22 2011 *)

a(n)=1<Charles R Greathouse IV, Jul 25 2011

x='x+O('x^100); Vec((1-2*x)/(1-4*x)) \\ Altug Alkan, Dec 21 2015

G.f.: (1-2*x)/(1-4*x).
a(n) = 4*a(n-1) n > 1, with a(0)=1, a(1)=2.
a(n) = (4^n+0^n)/2 (i.e., 1 followed by 4^n/2, n > 0).
E.g.f.: exp(2*x)*cosh(2*x) = (exp(4*x)+exp(0))/2. - Paul Barry, May 10 2003
a(n) = Sum_{k=0..n} C(2*n, 2*k). - Paul Barry, May 20 2003
a(n) = A001045(2*n+1) - A001045(2*n-1) + 0^n/2. - Paul Barry, Mar 10 2004
a(n) = 2^n*A011782(n); a(n) = gcd(A011782(2n), A011782(2n+1)). - Paul Barry, Jan 12 2005
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j). - Paul Barry, Jun 04 2005
a(n) = Sum_{k=0..n} A038763(n,k). - Philippe Deléham, Sep 22 2006
a(n) = Integral_{x=0..4} p(n,x)^2/(Pi*sqrt(x(4-x))) dx, where p(n,x) is the sequence of orthogonal polynomials defined by C(2*n,n): p(n,x) = (2*x-4)*p(n-1,x) - 4*p(n-2,x), with p(0,x)=1, p(1,x)=-2+x. - Paul Barry, Mar 01 2007
a(n) = ((2+sqrt(4))^n + (2-sqrt(4))^n)/2. - Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008
a(n) = A000079(n) * A011782(n). - Philippe Deléham, Dec 01 2008
a(n) = A004171(n-1) = A028403(n) - A000079(n) for n >= 1. - Jaroslav Krizek, Jul 27 2009
a(n) = Sum_{k=0..n} A201730(n,k)*3^k. - Philippe Deléham, Dec 06 2011
a(n) = Sum_{k=0..n} A134309(n,k)*2^k = Sum_{k=0..n} A055372(n,k). - Philippe Deléham, Feb 04 2012
G.f.: Q(0), where Q(k) = 1 - 2*x/(1 - 2/(2 - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
E.g.f.: 1/2 + exp(4*x)/2 = (Q(0)+1)/2, where Q(k) = 1 + 4*x/(2*k+1 - 2*x*(2*k+1)/(2*x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
a(n) = ceiling( 2^(2n-1) ). - Wesley Ivan Hurt, Jun 30 2013
G.f.: 1 + 2*x/(1 + x)*( 1 + 5*x/(1 + 4*x)*( 1 + 8*x/(1 + 7*x)*( 1 + 11*x/(1 + 10*x)*( 1 + ... )))). - Peter Bala, May 27 2017
Sum_{n>=0} 1/a(n) = 5/3. - Amiram Eldar, Aug 18 2022
Sum_{n>=0} a(n)*x^n/A000680(n) = E(x)^2 where E(x) = Sum_{n>=0} x^n/A000680(n). - Geoffrey Critzer, Apr 21 2023

A007582 a(n) = 2^(n-1)*(1+2^n).

Original entry on oeis.org

1, 3, 10, 36, 136, 528, 2080, 8256, 32896, 131328, 524800, 2098176, 8390656, 33558528, 134225920, 536887296, 2147516416, 8590000128, 34359869440, 137439215616, 549756338176, 2199024304128, 8796095119360, 35184376283136, 140737496743936, 562949970198528

Offset: 0

Simon Plouffe

Let G_n be the elementary Abelian group G_n = (C_2)^n for n >= 1: A006516 is the number of times the number -1 appears in the character table of G_n and A007582 is the number of times the number 1. Together the two sequences cover all the values in the table, i.e., A006516(n) + A007582(n) = 2^(2n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 01 2001
Number of walks of length 2n+1 between two adjacent vertices in the cycle graph C_8. Example: a(1)=3 because in the cycle ABCDEFGH we have three walks of length 3 between A and B: ABAB, ABCB and AHAB. - Emeric Deutsch, Apr 01 2004
Smallest number containing in its binary representation two equal non-overlapping subwords of length n: A097295(a(n))=n and A097295(m)Reinhard Zumkeller, Aug 04 2004
a(n)^2 + (A006516(n))^2 = a(2n). E.g., a(3) = 36, A006516(3) = 28, a(6) = 2080. 36^2 + 28^2 = 2080. - Gary W. Adamson, Jun 17 2006
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either x equals y or x does not equal y. - Ross La Haye, Jan 02 2008
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A). This is just a simpler statement of my previous comment for this sequence. - Ross La Haye, Jan 10 2008
For n>0: A000120(a(n))=2, A023414(a(n))=2*(n-1), A087117(a(n))=n-1. - Reinhard Zumkeller, Jun 23 2009
a(n+1) written in base 2: 11, 1010, 100100, 10001000, 1000010000, ..., i.e., number 1, n times 0, number 1, n times 0 (A163449(n)). - Jaroslav Krizek, Jul 27 2009
a(n) for n >= 1 is a bisection of A001445(n+1). - Jaroslav Krizek, Aug 14 2009
Related to A102573: letting T(q,r) be the coefficient of n^(r+1) in the polynomial 2^(q-n)/n times sum_{k=0..n} binomial(n,k)*k^q, then A007582(x)= sum_{k=0..x-1} T(x,k)*2^k. - John M. Campbell, Nov 16 2011
a(n) gives the number of pairs (r, s) such that 0 <= r <= s <= (2^n)-1 that satisfy AND(r, s, XOR(r, s)) = 0. - Ramasamy Chandramouli, Aug 30 2012
a(n) = A000217(2^n) = 2^(2n-1) + 2^(n-1) is the nearest triangular number above 2^(2n-1); cf. A006516, A233327. - Antti Karttunen, Feb 26 2014
Consider the quantum spin-1/2 chain with even number of sites L (physics, condensed matter theory). The spectrum of the Hamiltonian can be classified according to symmetries. If the only symmetry of the spin Hamiltonian is Parity, i.e., reflection with respect to the middle of the chain (see e.g. the transverse-field Ising model with open boundary conditions), then the dimension of the p=+1 parity sector is given by a(n) with n=L/2. - Marin Bukov, Mar 11 2016
a(n) is also the total number of words of length n, over an alphabet of four letters, of which one of them appears an even number of times. See the Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter odd case), and the Balakrishnan reference there. For the 1- to 11-letter cases, see the crossrefs. - Wolfdieter Lang, Jul 17 2017
a(n) is the number of nonisomorphic spanning trees of the cyclic snake formed with n+1 copies of the cycle on 4 vertices. A cyclic snake is a connected graph whose block-cutpoint is a path and all its n blocks are isomorphic to the cycle C_m. - Christian Barrientos, Sep 05 2024
Also, with offset 1, the cogrowth sequence of the dihedral group with 16 elements, D8 = . - Sean A. Irvine, Nov 06 2024

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
S. Hong and J. H. Kwak, Regular fourfold coverings with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 168
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Mircea Merca, A Note on Cosine Power Sums, J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
Index entries for linear recurrences with constant coefficients, signature (6,-8)

Cf. A000217, A049773, A049775.
Cf. A006516.
Cf. A134308.
Cf. A000225, A000392, A032263, A028243, A000079.
Cf. A102573.
The number of words of length n with m letters, one of them appearing an even number of times is for m = 1..11: A000035, A011782,  A007051, A007582, A081186, A081187, A081188, A081189, A081190, A060531, A081192. - Wolfdieter Lang, Jul 17 2017

[Binomial(2^n + 1, 2) : n in [0..30]]; // Wesley Ivan Hurt, Jul 03 2020

seq(binomial(-2^n, 2), n=0..23); # Zerinvary Lajos, Feb 22 2008

Table[ Binomial[2^n + 1, 2], {n, 0, 23}] (* Robert G. Wilson v, Jul 30 2004 *)
LinearRecurrence[{6,-8},{1,3},30] (* Harvey P. Dale, Apr 08 2013 *)

A007582(n):=2^(n-1)*(1+2^n)$ makelist(A007582(n),n,0,30); /* Martin Ettl, Nov 15 2012 */

PARI
```
a(n)=if(n<0,0,2^(n-1)*(1+2^n))
```

a(n)=sum(k=-n\4,n\4,binomial(2*n+1,n+1+4*k))

G.f.: (1-3*x)/((1-2*x)*(1-4*x)). C(1+2^n, 2) where C(n, 2) is n-th triangular number A000217.
Binomial transform of A007051. Inverse binomial transform of A081186. - Paul Barry, Apr 07 2003
E.g.f.: exp(3*x)*cosh(x). - Paul Barry, Apr 07 2003
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*3^(n-2*k). - Paul Barry, May 08 2003
a(n+1) = 4*a(n) - 2^n; see also A049775. a(n) = 2^(n-1)*A000051(n). - Philippe Deléham, Feb 20 2004
a(n) = 6*a(n-1) - 8*a(n-2). - Emeric Deutsch, Apr 01 2004
Row sums of triangle A134308. - Gary W. Adamson, Oct 19 2007
a(n) = StirlingS2(2^n + 1,2^n) = 1 + 2*StirlingS2(n+1,2) + 3*StirlingS2(n+1,3) + 3*StirlingS2(n+1,4) = StirlingS2(n+2,2) + 3(StirlingS2(n+1,3) + StirlingS2(n+1,4)). - Ross La Haye, Mar 01 2008
a(n) = StirlingS2(2^n + 1,2^n) = 1 + 2*StirlingS2(n+1,2) + 3*StirlingS2(n+1,3) + 3*StirlingS2(n+1,4) = StirlingS2(n+2,2) + 3(StirlingS2(n+1,3) + StirlingS2(n+1,4)). - Ross La Haye, Apr 02 2008
a(n) = A000079(n) + A006516(n). - Yosu Yurramendi, Aug 06 2008
a(n) = A028403(n+1) / 4. - Jaroslav Krizek, Jul 27 2009
a(n) = Sum_{k=-floor(n/4)..floor(n/4)} binomial(2*n,n+4*k)/2. - Mircea Merca, Jan 28 2012
G.f.: Q(0)/2 where Q(k) = 1 + 2^k/(1 - 2*x/(2*x + 2^k/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 10 2013
a(n) = Sum_{k=1..2^n} k. - Joerg Arndt, Sep 01 2013
a(n) = (1/3) * Sum_{k=2^n..2^(n+1)} k. - J. M. Bergot, Jan 26 2015
a(n+1) = 2*a(n) + 4^n. - Yuchun Ji, Mar 10 2017

A085601 a(n) = 2 * (4^n + 2^n) + 1.

Original entry on oeis.org

5, 13, 41, 145, 545, 2113, 8321, 33025, 131585, 525313, 2099201, 8392705, 33562625, 134234113, 536903681, 2147549185, 8590065665, 34360000513, 137439477761, 549756862465, 2199025352705, 8796097216513, 35184380477441

Offset: 0

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jul 07 2003

Begin with a square tile.
Place square tiles on each edge to form a diamond shape.
Count the tiles: a(0) = 5.
Add tiles to fill the enclosing square.
Go to step 2.

Iain Fox, Table of n, a(n) for n = 0..1660
Index entries for linear recurrences with constant coefficients, signature (7,-14,8)

Cf. A028403, A046142, A056640, A064412.

Cf. A343175 (essentially the same).

Table[2(4^n+2^n)+1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{5,13,41},30] (* Harvey P. Dale, Dec 30 2017 *)

first(n) = Vec((5 - 22*x + 20*x^2)/(1 - 7*x + 14*x^2 - 8*x^3) + O(x^n)) \\ Iain Fox, Dec 30 2017

From R. J. Mathar, Apr 20 2009: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3).
G.f.: -(5 - 22*x + 20*x^2)/((x - 1)*(2*x - 1)*(4*x - 1)).
(End)
E.g.f.: e^x + 2*(e^(2*x) + e^(4*x)). - Iain Fox, Dec 30 2017

Edited by Franklin T. Adams-Watters and Don Reble, Aug 15 2006

A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 68, 72, 80, 96, 128, 129, 130, 132, 136, 144, 160, 192, 256, 257, 258, 260, 264, 272, 288, 320, 384, 512, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048

Offset: 0

Reinhard Zumkeller, Feb 28 2010

Essentially the same as A048645. - T. D. Noe, Mar 28 2011

			Triangle begins as:
     2;
     3,    4;
     5,    6,    8;
     9,   10,   12,   16;
    17,   18,   20,   24,   32;
    33,   34,   36,   40,   48,   64;
    65,   66,   68,   72,   80,   96,  128;
   129,  130,  132,  136,  144,  160,  192,  256;
   257,  258,  260,  264,  272,  288,  320,  384,  512;
   513,  514,  516,  520,  528,  544,  576,  640,  768, 1024;
  1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048;

T. D. Noe, Rows n = 0..100 of triangle, flattened
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025).

Cf. A048645, A118413, A118416.

Cf. also A087112, A370121.

[2^n + 2^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 07 2021

Flatten[Table[2^n + 2^m, {n,0,10}, {m, 0, n}]] (* T. D. Noe, Jun 18 2013 *)

A173786(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); 1 << c + 1 << (n - binomial(c + 1, 2)); }; \\ Antti Karttunen, Feb 29 2024, after David A. Corneth's PARI-program in A048645

from math import isqrt, comb
def A173786(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    return (1<Chai Wah Wu, Jun 20 2025

flatten([[2^n + 2^k for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 07 2021

1 <= A000120(T(n,k)) <= 2.
For n>0, 0<=kA048645(n+1,k+2) and T(n,n) = A048645(n+2,1).
Row sums give A006589(n).
Central terms give A161168(n).
T(2*n+1,n) = A007582(n+1).
T(2*n+1,n+1) = A028403(n+1).
T(n,k) = A140513(n,k) - A173787(n,k), 0<=k<=n.
T(n,k) = A059268(n+1,k+1) + A173787(n,k), 0T(n,k) * A173787(n,k) = A173787(2*n,2*k), 0<=k<=n.
T(n,0) = A000051(n).
T(n,1) = A052548(n) for n>0.
T(n,2) = A140504(n) for n>1.
T(n,3) = A175161(n-3) for n>2.
T(n,4) = A175162(n-4) for n>3.
T(n,5) = A175163(n-5) for n>4.
T(n,n-4) = A110287(n-4) for n>3.
T(n,n-3) = A005010(n-3) for n>2.
T(n,n-2) = A020714(n-2) for n>1.
T(n,n-1) = A007283(n-1) for n>0.
T(n,n) = 2*A000079(n).

Typo in first comment line fixed by Reinhard Zumkeller, Mar 07 2010

A225910 Square array read by antidiagonals: a(m,n) is the number of binary pattern classes in the (m,n)-rectangular grid, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 7, 6, 1, 1, 10, 24, 24, 10, 1, 1, 20, 76, 168, 76, 20, 1, 1, 36, 288, 1120, 1120, 288, 36, 1, 1, 72, 1072, 8640, 16576, 8640, 1072, 72, 1, 1, 136, 4224, 66816, 263680, 263680, 66816, 4224, 136, 1, 1, 272, 16576, 529920, 4197376, 8407040, 4197376, 529920, 16576, 272, 1

Offset: 0

Yosu Yurramendi, May 20 2013

In the square table A000012, A005418, and A225826 to A225834 are the first 11 rows (see example).
In the square table, m odd (see formula). The order of the recurrence equations is 4. Let it be (a1(m),a2(m),a3(m),a4(m)) the characterizing 4-plet of a(m). The sequence a1(m) belongs to A028403 (2^m+2^((m+1)/2)), -a2(m) to A147538 (2^m*(2^((m+1)/2)-1)) and a4(m) to A013824 (2^(2m)*2^((m+1)/2)). -a3(m) sequence formula is 2^m*(2^m+2^((m+1)/2)).
All the coefficients of x in generating functions from A225826 to A225834 belong to A113979.

			Array begins:
  1   1      1         1            1               1                  1 ...
  1   2      3         6           10              20                 36 ...
  1   3      7        24           76             288               1072 ...
  1   6     24       168         1120            8640              66816 ...
  1  10     76      1120        16576          263680            4197376 ...
  1  20    288      8640       263680         8407040          268517376 ...
  1  36   1072     66816      4197376       268517376        17180065792 ...
  1  72   4224    529920     67133440      8590786560      1099516870656 ...
  1 136  16576   4212736   1073790976    274882625536     70368756760576 ...
  1 272  66048  33632256  17180262400   8796137062400   4503599962914816 ...
  1 528 262912 268713984 274878693376 281475261923328 288230376957018112 ...
  ...

Alois P. Heinz, Antidiagonals n = 0..65, flattened
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.

Cf. A005418, A225826-A225834.

m even and n even:
a(m,n) = 2^(m*n/2-2)*(2^(m*n/2) + 3);
m even and n odd:
a(m,n) = 2^(m*n/2-1)*(2^(m*n/2-1) + 2^(m/2-1) + 1);
m odd and n even:
a(m,n) = 2^(m*n/2-1)*(2^(m*n/2-1) + 2^(n/2-1) + 1);
m odd and n odd:
a(m,n) = 2^((m*n-1)/2-1)*(2^((m*n-1)/2) + 2^((m-1)/2) + 2^((n-1)/2) + 1).
m even:
a(m,n) = 2^m*a(m,n-1) + 2^m*a(m,n-2) - (2^m)^2*a(m,n-3) with n>2, a(m,0)=1, a(m,1)=a(1,m), a(m,2)=a(2,m).
m odd:
a(m,n) = 2^m*a(m,n-1) + 2^m*a(m,n-2) - (2^m)^2*a(m,n-3) - 2^(((m+1)/2)*n-3)*(2^((m-1)/2)-1) with n>2, a(m,0)=1, a(m,1)=a(1,m), a(m,2)=a(2,m).
Only a(1,n) and a(2,n) (A005418 and A225826) sequences are needed to define the others.

A092431 Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.

Original entry on oeis.org

2, 9, 35, 135, 527, 2079, 8255, 32895, 131327, 524799, 2098175, 8390655, 33558527, 134225919, 536887295, 2147516415, 8590000127, 34359869439, 137439215615, 549756338175, 2199024304127, 8796095119359, 35184376283135, 140737496743935, 562949970198527

Offset: 1

Reinhard Zumkeller, Mar 23 2004

Smallest numbers having in binary representation n 0's and n 1's: a(n) = Min{m: A023416(m)=A000120(m)=n}.

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Binary
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A007088, A001700.
Subsequence of A031443.
Cf. A000120, A023416.
Cf. A007582, A056326, A005367, A063376, A032125, A056309, A028403, A001576, A028400, A049775.

LinearRecurrence[{7, -14, 8}, {2, 9, 35}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
Table[FromDigits[Join[PadRight[{1},n,0],PadRight[{},n-2,1]],2],{n,2,30}]//Sort (* or *) Rest[CoefficientList[Series[x (-2+5x)/((x-1)(2x-1)(4x-1)),{x,0,30}],x]] (* Harvey P. Dale, Jul 30 2021 *)

a(n+1) = 2*a(n) + 4^n + 1.
a(n) = 2^(2*n-1) + 2^(n-1) - 1.
a(n) = A007582(n)-1 = A056326(2n+1) = A005367(n-1)/2 = A063376(n)/2-1 = A032125(n+1)/3-1 = A056309(2n+1)/2 = A028403(n+1)/4-1 = (A001576(n)-3)/2 = (A028400(n+1)-9)/8 = Sum_{k=2..n+1} A049775(k). - Ralf Stephan, Mar 24 2004
G.f.: x*(-2+5*x) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Jun 01 2011
E.g.f.: exp(x)*(exp(3*x) + exp(x) - 2)/2. - Stefano Spezia, Sep 27 2023

A161168 a(n) = 2^n + 4^n.

Original entry on oeis.org

2, 6, 20, 72, 272, 1056, 4160, 16512, 65792, 262656, 1049600, 4196352, 16781312, 67117056, 268451840, 1073774592, 4295032832, 17180000256, 68719738880, 274878431232, 1099512676352, 4398048608256, 17592190238720, 70368752566272

Offset: 0

Zerinvary Lajos, Jun 04 2009

Essentially a duplicate of A063376 and A028402.
a(n) written in base 2: a(0) = 10, a(n) for n >= 1: 110, 10100, 1001000, 100010000, ..., i.e., number 1, (n-1) times 0, number 1, n times 0 (see A163664). a(n) is a bisection of A005418. - Jaroslav Krizek, Aug 14 2009
Central terms of the triangle in A173786. - Reinhard Zumkeller, Feb 28 2010
For n > 0 let 2^(n+1) be the length of the even leg of a primitive Pythagorean triangle (PPT); then the odd leg is constrained to have a length of 4^n-1 and the hypotenuse to have a length of 4^n+1. The resulting triangle has a semiperimeter of 4^n + 2^n. - Frank M Jackson, Dec 28 2017
a(n) is also the number of distinct planar embeddings of the (2n+7)-triangular snake graph. - Eric W. Weisstein, May 21 2024

Iain Fox, Table of n, a(n) for n = 0..1660
Eric Weisstein's World of Mathematics, Planar Embedding.
Eric Weisstein's World of Mathematics, Triangular Snake Graph.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A005418, A007582, A020522, A028403, A063376, A163664, A173786.

Magma
```
[ 2^n+4^n: n in [0..25] ];
```

A161168:=n->2^n+4^n: seq(A161168(n), n=0..40); # Wesley Ivan Hurt, Jul 24 2017

a[n_]:=4^n+2^n; Array[a,24] (* Frank M Jackson, Dec 28 2017 *)

a(n)=2^n+4^n \\ Charles R Greathouse IV, Oct 07 2015

first(n) = Vec(2*(1 - 3*x)/((1 - 2*x)*(1 - 4*x)) + O(x^n)) \\ Iain Fox, Dec 28 2017

Sage
```
[2^n + 4^n for n in range(0,25)]
```
Sage
```
[sigma(4,n)-1for n in range(0,25)]
```

a(n) = 6*a(n-1) - 8*a(n-2); a(0)=2, a(1)=6. - Vincenzo Librandi, Dec 27 2010
G.f.: -2*(3*x-1) / ((2*x-1)*(4*x-1)). - Colin Barker, Mar 19 2013
E.g.f.: e^(2*x) + e^(4*x). - Iain Fox, Dec 28 2017
a(n) = 2*A007582(n). - R. J. Mathar, Feb 26 2018

A257273 a(n) = 2^(n-1)*(2^n+3).

Original entry on oeis.org

2, 5, 14, 44, 152, 560, 2144, 8384, 33152, 131840, 525824, 2100224, 8394752, 33566720, 134242304, 536920064, 2147581952, 8590131200, 34360131584, 137439739904, 549757386752, 2199026401280, 8796099313664, 35184384671744, 140737513521152, 562950003752960, 2251799914348544, 9007199456067584

Offset: 0

M. F. Hasler, Apr 27 2015

a(n) is in A125246 <=> n is in A057732 <=> A062709(n) is in A057733.

These are also the row sum of the triangle A146769: For n>=1, a(n-1) is the sum of row n of A146769.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A000079, A007582, A028403, A256873, A256871, A257272.

Cf. A062709, A057732, A057733.

[2^(n-1)*(2^n+3): n in [0..35]]; // Vincenzo Librandi, Apr 27 2015

Table[2^(n - 1) (2^n + 3), {n, 0, 30}] (* Bruno Berselli, Apr 27 2015 *)
CoefficientList[Series[(2 - 7 x)/((1 - 4 x) (1 - 2 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 27 2015 *)
LinearRecurrence[{6,-8},{2,5},30] (* Harvey P. Dale, Dec 21 2024 *)

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

Vec((2-7*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 27 2015

G.f.: (2-7*x)/((1-4*x)*(1-2*x)). - Vincenzo Librandi, Apr 27 2015

a(n) = 6*a(n-1)-8*a(n-2). - Colin Barker, Apr 27 2015

Showing 1-10 of 11 results. Next

cp's OEIS Frontend

A163450 A028403 written in base 2.

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A004171 a(n) = 2^(2n+1).

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A081294 Expansion of (1-2*x)/(1-4*x).

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A007582 a(n) = 2^(n-1)*(1+2^n).

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A085601 a(n) = 2 * (4^n + 2^n) + 1.

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A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.

A081294 Expansion of (1-2x)/(1-4x).