A020707 -id:A020707 - cp's OEIS frontend

A151821 Powers of 2, omitting 2 itself.

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

Offset: 1

N. J. A. Sloane, Jul 08 2009

Different from A046055.
An elephant sequence, see A175655. For the central square just one A[5] vector, with decimal value 170, leads to this sequence. For the corner squares this vector leads to the companion sequence A095121. - Johannes W. Meijer, Aug 15 2010
This is a subsequence of A055744, numbers n such that n and phi(n) have same prime factors. - Michel Marcus, Mar 20 2015
INVERTi transform of A007483: (1, 5, 17, 61, 217, 773, ...). - Gary W. Adamson, Aug 06 2016
Nonprimes that are also powers of 2. Intersection of A000079 and A018252. - Omar E. Pol, Jan 27 2017
Also the chromatic number of the n-Keller graph. - Eric W. Weisstein, Nov 17 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, Keller Graph
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000079, A007483, A018252, A020707, A046055, A055744, A063759, A095121, A175655.

Partial sums are given by 2*A000225(n)-1, which is not the same as A000918.

a151821 n = a151821_list !! (n-1)
a151821_list = x : xs where (x : _ : xs) = a000079_list
-- Reinhard Zumkeller, Dec 16 2013

[1] cat [2^n: n in [2..35]]; // Vincenzo Librandi, Jul 21 2013

CoefficientList[Series[(1 + 2 x)/(1 - 2 x), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
DeleteCases[2^Range[0, 33], p_ /; PrimeQ @ p] (* Michael De Vlieger, Aug 06 2016 *)
Join[{1}, 2^Range[2, 20]] (* Eric W. Weisstein, Nov 17 2017 *)

a(n)=if(n>1,2^n,1) \\ Charles R Greathouse IV, Dec 08 2015

Vec(x*(1+2*x)/(1-2*x) + O(x^100)) \\ Altug Alkan, Dec 09 2015

def A151821(n): return 1<1 else 1 # Chai Wah Wu, Jun 10 2025

G.f.: x*(1+2*x)/(1-2*x). - Philippe Deléham, Sep 17 2009
a(1) = 1 and a(n) = 3 + Sum_{k=1..n-1} a(k) for n>=2. - Joerg Arndt, Aug 15 2012
E.g.f.: exp(2*x) - x - 1. - Stefano Spezia, Jan 31 2023

A010709 Constant sequence: the all 4's sequence.

Original entry on oeis.org

4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, May 25 2010: (Start)
Continued fraction expansion of 2+sqrt(5).
Decimal expansion of 4/9.
Inverse binomial transform of A020707. (End)

Tom Edgar, Proof without words: sum of powers of 4/9, Math. Mag. 89 no. 3 (2016) 191.
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1012
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (1).

From Klaus Brockhaus, May 25 2010: (Start)
Equals 4*A000012, 2*A007395, A010731/2, A010855/4, A010871/8.
Cf. A098317 (decimal expansion of 2+sqrt(5)), A020707 (2^(n+2)). (End)

makelist(4, n, 0, 30); /* Martin Ettl, Nov 09 2012 */

a(n) = 4 \\ Charles R Greathouse IV, Apr 07 2012

def A010709(n): return 4 # Chai Wah Wu, Mar 22 2023

From Klaus Brockhaus, May 25 2010: (Start)
a(n) = 4.
G.f.: 4/(1-x). (End)
E.g.f.: 4*e^x. - Vincenzo Librandi, Jan 29 2012

A300454 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial 2(x + 1)^(n + 1) + x^3 + 2x^2 - x - 2.

Original entry on oeis.org

0, 1, 2, 1, 0, 3, 4, 1, 0, 5, 8, 3, 0, 7, 14, 9, 2, 0, 9, 22, 21, 10, 2, 0, 11, 32, 41, 30, 12, 2, 0, 13, 44, 71, 70, 42, 14, 2, 0, 15, 58, 113, 140, 112, 56, 16, 2, 0, 17, 74, 169, 252, 252, 168, 72, 18, 2, 0, 19, 92, 241, 420, 504, 420, 240, 90, 20, 2, 0

Offset: 0

Franck Maminirina Ramaharo, Mar 06 2018

Row sums of column 1,2 and 3 yields {4, 8, 16, 30, 52, ...}, in A046127.
Almost twice Pascal's triangle A028326 (up to horizontal shift), except column 0 to 3.
The polynomial P(n;x) = 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2 is a simplified version of the bracket polynomial associated with a twist knot of n half twists that is only concerned with the enumeration of the state diagrams. The simplification arises when the twist knot is thought of as a planar diagram with no crossing information at each double point. In this case, P(n;x) = x*(A,B,x), where (A,B,d) denotes the bracket polynomial for the n-twist knot (see links for the definition of the bracket polynomial). For example, the bracket polynomial for the trefoil (n = 2) is A^3*d^1 + 3*BA^2*d^0 + 3*AB^2*d^1 + B^3*d^2, where A and B are the "splitting variables". Then setting A = B = 1 and d = x, we obtain 3 + 4*x + x^2 (also see A299989, row 1).

			The triangle T(n,k) begins
n\k  0   1    2    3     4     5     6     7     8     9    10   11   12  13 14
0:   0   1    2    1
1:   0   3    4    1
2:   0   5    8    3
3:   0   7   14    9     2
4:   0   9   22   21    10     2
5:   0  11   32   41    30    12     2
6:   0  13   44   71    70    42    14     2
7:   0  15   58  113   140   112    56    16     2
8:   0  17   74  169   252   252   168    72    18     2
9:   0  19   92  241   420   504   420   240    90    20     2
10:  0  21  112  331   660   924   924   660   330   110    22    2
11:  0  23  134  441   990  1584  1848  1584   990   440   132   24    2
12:  0  25  158  573  1430  2574  3432  3432  2574  1430   572  156   26   2
13:  0  27  184  729  2002  4004  6006  6864  6006  4004  2002  728  182  28  2

Inga Johnson and Allison K. Henrich, An Interactive Introduction to Knot Theory, Dover Publications, Inc., 2017.

Agnijo Banerjee, Knot theory.
Răzvan Gelca and Fumikazu Nagasato,Some results about the kauffman bracket skein module of the twist knot exterior, J. Knot Theory Ramifications 15 (2006), 1095-1106.
L. H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Kelsey Lafferty, The three-variable bracket polynomial for reduced, alternating links, Rose-Hulman Undergraduate Mathematics Journal, Vol. 14: Iss. 2, Article 7 (2013).
Franck Ramaharo, Enumerating the states of the twist knot, arXiv preprint arXiv:1712.06543 [math.CO], 2017.
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Alexander Stoimenow, Generating functions, Fibonacci numbers and rational knots, Journal of Algebra, Volume 310, Issue 2 (2007), 491-525.
Eric Weisstein's World of Mathematics, Bracket Polynomial.
Wikipedia, Twist knot.

Row sums: A020707(Pisot sequences).
Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300453 ((2,n)-torus knot).
Cf. A002061, A005408, A007318, A014206, A028326, A028326, A046127, A046127, A046127, A064999, A155753, A299989, A300454, A300454.

P(n, x) := 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2$
T : []$
for i:0 thru 20 do
  T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(3, i + 1)))$
T;

row(n) = Vecrev(2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Mar 12 2018

T(n,1) = A005408(n).
T(n,2) = A014206(n).
T(n,3) = A064999(n+1).
T(n,1) + T(n,2) = A002061(n+2).
T(n,1) + T(n,3) = A046127(n+1).
T(n,2) + T(n,3) = A155753(n+1).
T(n,1) + T(n,2) + T(n,3) = A046127(n+2).
T(n,k) = A028326(n,k-1), k >= 4 and n >= k - 1.
T(n,k) = A300454(n,k-1) + 2*A300454(n,k) + A007318(n,k-1), with T(n,0) = 0.
G.f: (2*x + 2)/(1 - y*(x + 1)) + (x^3 + 2*x^2 - x - 2)/(1 - y).

A051916 The Greek sequence: 2^a * 3^b * 5^c where a = 0,1,2,3,..., b,c in {0,1}, excluding the terms 1,2; that is: (a,b,c) != (0,0,0), (1,0,0).

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 128, 160, 192, 240, 256, 320, 384, 480, 512, 640, 768, 960, 1024, 1280, 1536, 1920, 2048, 2560, 3072, 3840, 4096, 5120, 6144, 7680, 8192, 10240, 12288, 15360, 16384, 20480

Offset: 1

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Dec 17 1999

From Reinhard Zumkeller, Mar 19 2010: (Start)
Union of A007283, A020707, A020714, and A110286.
Intersection of A051037 and A003401 apart from terms 1 and 2. (End)

George E. Martin, Geometric Constructions, New York: Springer, 1997, p. 140.

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

Cf. A003401, A007283, A020707, A020714, A051037, A110286.

CoefficientList[Series[x(3x^7+2x^6+2x^5+2x^4+6x^3+5x^2+4x+3)/(1-2x^4),{x,0,60}],x] (* Harvey P. Dale, Dec 23 2012 *)

Vec(x*(3*x^7+2*x^6+2*x^5+2*x^4+6*x^3+5*x^2+4*x+3)/(1-2*x^4)+O(x^99)) \\ Charles R Greathouse IV, Oct 12 2012

def A051916(n): return n+2 if n<5 else (15,1,5,3)[m:=n&3]<<(n>>2)+(-2,2,0,1)[m] # Chai Wah Wu, Apr 02 2025

G.f.: x*(3*x^7 + 2*x^6 + 2*x^5 + 2*x^4 + 6*x^3 + 5*x^2 + 4*x + 3)/(1 - 2*x^4).
a(n+4) = 2*a(n) for n > 8. - Reinhard Zumkeller, Mar 19 2010
Sum_{n>=1} 1/a(n) = 17/10. - Amiram Eldar, Jan 18 2023

More terms from James Sellers, Dec 18 1999

Offset corrected by Reinhard Zumkeller, Mar 10 2010

A198633 Total number of round trips, each of length 2*n on the graph P_3 (o-o-o).

Original entry on oeis.org

3, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 0

Wolfdieter Lang, Nov 02 2011

See the array and triangle A198632 for the general case for the graph P_N (there N is n and the length is l=2*k).

			With the graph P_3 as 1-2-3:
n=0: 3, from the length 0 walks starting at 1, 2 and 3.
n=2: 8, from the walks of length 4, namely 12121, 12321, 21212, 23232, 21232, 23212, 32323 and 32123.

Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A198632, 2*A005248, A198635.

Essentially the same as A000079, A020707, A077552 etc.

Join[{3},NestList[2#&,4,30]] (* Harvey P. Dale, Nov 07 2020 *)

a(n)=if(n,2<Charles R Greathouse IV, Jan 02 2012

a(n) = w(3,2*n), n>=0, with w(3,l) the total number of closed walks on the graph P_3 (the simple path with 3 points (vertices) and 2 lines (or edges)).
O.g.f. for w(3,l) (with zeros for odd l): y*(d/dy)S(3,y)/S(3,y) with y=1/x and Chebyshev S-polynomials (coefficients A049310). See A198632, also for a rewritten form.
Empirical g.f.: (3-2*x)/(1-2*x). - Colin Barker, Jan 02 2012
This g.f. follows from the Chebyshev o.g.f. given above with x -> sqrt(x). Therefore a(0) = 3 and a(n) = 2^(n+1), n >= 1. - Wolfdieter Lang, Feb 18 2013.

A166956 a(n) = 2^n +(-1)^n - 2.

Original entry on oeis.org

0, -1, 3, 5, 15, 29, 63, 125, 255, 509, 1023, 2045, 4095, 8189, 16383, 32765, 65535, 131069, 262143, 524285, 1048575, 2097149, 4194303, 8388605, 16777215, 33554429, 67108863, 134217725, 268435455, 536870909, 1073741823, 2147483645, 4294967295, 8589934589

Offset: 0

Paul Curtz, Oct 25 2009

The inverse binomial transform yields 0,-1,5,-7,17,-31,..., a sign alternating variant of A014551.
In a table of a(n) and higher-order differences in successive rows, the main diagonal contains 0, 4, 8, 16, ... (zero followed by A020707).
Similar to the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero, which begins with 1,3,5,15,29,63,125. - Robert Price, Aug 08 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..240
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,1,-2)

Cf. A166920, A166956, A290660, A290661, A290662.

[2^n-2+(-1)^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

LinearRecurrence[{2,1,-2},{0,-1,3},20] (* G. C. Greubel, May 29 2016 *)

a(n) = A000079(n) - A010684(n).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
G.f.: x*(5*x -1)/((1-x)*(1-2*x)*(1+x)).
E.g.f.: exp(2*x) - 2*exp(x) + exp(-x). - G. C. Greubel, May 29 2016

Edited and extended by R. J. Mathar, Mar 02 2010

A176040 Periodic sequence: Repeat 3, 1.

Original entry on oeis.org

3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3

Offset: 0

Klaus Brockhaus, Apr 07 2010

Interleaving of A010701 and A000012.
Also continued fraction expansion of (3+sqrt(21))/2.
Also decimal expansion of 31/99.
Essentially first differences of A014601.
Inverse binomial transform of 3 followed by A020707.
Second inverse binomial transform of A052919 without initial term 2.
Third inverse binomial transform of A007582 without initial term 1.
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + ... is the o.g.f. for A008619. - Peter Bala, Mar 13 2015

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A153284, A010701 (all 3's sequence), A000012 (all 1's sequence), A090458 (decimal expansion of (3+sqrt(21))/2), A010684 (repeat 1, 3), A014601 (congruent to 0 or 3 mod 4), A020707 (2^(n+2)), A052919, A007582 (2^(n-1)*(1+2^n)), A008619.

&cat[ [3, 1]: n in [0..52] ];
[ 2+(-1)^n: n in [0..104] ];

PadRight[{},120,{3,1}] (* or *) LinearRecurrence[{0,1},{3,1},120] (* Harvey P. Dale, Mar 11 2015 *)

a(n) = 2+(-1)^n.
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = 1.
a(n) = -a(n-1)+4 for n > 0; a(0) = 3.
a(n) = 3*((n+1) mod 2)+(n mod 2).
a(n) = A010684(n+1).
G.f.: (3+x)/((1-x)*(1+x)).
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2^e) = 3, and a(p^e) = 1 for p >= 3.
Dirichlet g.f.: zeta(s)*(1+2^(1-s)). (End)

A293228 a(n) is the sum of proper divisors of n that are squarefree.

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 3, 4, 8, 1, 12, 1, 10, 9, 3, 1, 12, 1, 18, 11, 14, 1, 12, 6, 16, 4, 24, 1, 42, 1, 3, 15, 20, 13, 12, 1, 22, 17, 18, 1, 54, 1, 36, 24, 26, 1, 12, 8, 18, 21, 42, 1, 12, 17, 24, 23, 32, 1, 72, 1, 34, 32, 3, 19, 78, 1, 54, 27, 74, 1, 12, 1, 40, 24, 60, 19, 90, 1, 18, 4, 44, 1, 96, 23, 46, 33, 36, 1, 72, 21, 72, 35, 50, 25, 12, 1, 24

Offset: 1

Antti Karttunen, Oct 08 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005117, A008966, A048250, A293227, A293235.

with(numtheory): seq(coeff(series(add(mobius(k)^2*k*x^(2*k)/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 120); # Muniru A Asiru, Oct 28 2018

a[n_] := Times @@ (1 + (f = FactorInteger[n])[[;; , 1]]) - If[AllTrue[f[[;;, 2]], # == 1 &], n, 0]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Oct 09 2022 *)
Table[Total[Select[Most[Divisors[n]],SquareFreeQ]],{n,100}] (* Harvey P. Dale, Apr 20 2025 *)

PARI
```
A293228(n) = sumdiv(n, d, (d
				
```

a(n) = Sum_{d|n, dA008966(d)*d.
a(n) = A048250(n) - (A008966(n)*n).
G.f.: Sum_{k>=1} mu(k)^2*k*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Oct 28 2018
From Amiram Eldar, Oct 09 2022: (Start)
a(n) = 1 iff n is a prime.
a(n) = 3 iff n is a power of 2 greater than 2 (A020707).
Sum_{k=1..n} a(k) ~ (1/2 - 3/Pi^2) * n^2. (End)

A018921 Define the generalized Pisot sequence T(a(0),a(1)) by: a(n+2) is the greatest integer such that a(n+2)/a(n+1) < a(n+1)/a(n). This is T(4,8).

Original entry on oeis.org

4, 8, 15, 28, 52, 96, 177, 326, 600, 1104, 2031, 3736, 6872, 12640, 23249, 42762, 78652, 144664, 266079, 489396, 900140, 1655616, 3045153, 5600910, 10301680, 18947744, 34850335, 64099760, 117897840, 216847936, 398845537, 733591314, 1349284788, 2481721640

Offset: 0

R. K. Guy

Not to be confused with the Pisot T(4,8) sequence, which is A020707. - R. J. Mathar, Feb 13 2016

Colin Barker, Table of n, a(n) for n = 0..1000
D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).
Index entries for Pisot sequences

Cf. A008937.

Tiv:=[4,8]; [n le 2 select Tiv[n] else Ceiling(Self(n-1)^2/Self(n-2))-1: n in [1..40]]; // Bruno Berselli, Feb 17 2016

RecurrenceTable[{a[1] == 4, a[2] == 8, a[n] == Ceiling[a[n-1]^2/a[n-2]] - 1}, a, {n, 40}] (* Bruno Berselli, Feb 17 2016 *)
LinearRecurrence[{2,0,0,-1},{4,8,15,28},40] (* Harvey P. Dale, Mar 05 2019 *)

Vec((4-x^2-2*x^3)/((1-x)*(1-x-x^2-x^3)) + O(x^40)) \\ Colin Barker, Feb 13 2016

T(a0, a1, maxn) = a=vector(maxn); a[1]=a0; a[2]=a1; for(n=3, maxn, a[n]=ceil(a[n-1]^2/a[n-2])-1); a
T(4, 8, 30) \\ Colin Barker, Feb 14 2016

a(n) = 2*a(n-1) - a(n-4).
G.f.: (4-x^2-2*x^3) / ((1-x)*(1-x-x^2-x^3)). - Colin Barker, Feb 08 2012
a(n) = A008937(n+3) = A027084(n+3)+1. [first index correct by R. J. Mathar, Jun 24 2020]
a(n) = 2*a(n-1) - A008937(n). - Vincenzo Librandi, Feb 12 2016

Comments moved to formula, and typo in data fixed by Colin Barker, Feb 13 2016

A146541 Binomial transform of A010688.

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Oct 31 2008

Hankel transform is := 1,-48,0,0,0,0,0,0,0,...

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

Cf. A000007, A011782, A000079, A003945, A046055, A146523, A082505, A135092.

Essentially the same as A132479, A077552, A020707.

Join[{1},2^Range[3,40]] (* Harvey P. Dale, Feb 28 2016 *)

Vec((1+6*x)/(1-2*x) + O(x^50)) \\ Colin Barker, Mar 17 2016

a(0)=1, a(n) = 2^(n+2) for n>0.
a(n) = Sum_{k, 0..n} A109466(n,k)*A146534(k).
a(n) = A132479(n), n>1. - R. J. Mathar, Nov 02 2008
G.f.: (1+6*x) / (1-2*x). - Colin Barker, Mar 17 2016

Corrected and extended by Harvey P. Dale, Feb 28 2016

cp's OEIS Frontend

A151821 Powers of 2, omitting 2 itself.

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A010709 Constant sequence: the all 4's sequence.

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A300454 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2.

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A051916 The Greek sequence: 2^a * 3^b * 5^c where a = 0,1,2,3,..., b,c in {0,1}, excluding the terms 1,2; that is: (a,b,c) != (0,0,0), (1,0,0).

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A198633 Total number of round trips, each of length 2*n on the graph P_3 (o-o-o).

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A166956 a(n) = 2^n +(-1)^n - 2.

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A300454 Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial 2(x + 1)^(n + 1) + x^3 + 2x^2 - x - 2.