A016116
a(n) = 2^floor(n/2).
Original entry on oeis.org
1, 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
Offset: 0
For n=5 the a(5)=4 symmetric subsets of [4] are {1,4}, {2,3}, {1,2,3,4} and the empty set. - _Dennis P. Walsh_, Oct 27 2009
For n=5 the a(5)=4 length-6 binary sequences are <1,1,0,0,0,0>, <1,1,0,0,1,1>, <1,1,1,1,0,0> and <1,1,1,1,1,1>. - _Dennis P. Walsh_, Sep 06 2018
- Vincenzo Librandi, Table of n, a(n) for n = 0..5000
- Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Volume 332, Oct 06 2014, Pages 45-54.
- Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, arXiv:1407.2197 [math.CO], 2014.
- Arvind Ayyer, Amritanshu Prasad, and Steven Spallone, Odd partitions in Young's lattice, arXiv:1601.01776 [math.CO], 2016. See Theorem 6 p. 12.
- Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.
- Francesco Battistoni and Giuseppe Molteni, An elementary proof for a generalization of a Pohst's inequality, arXiv:2101.06163 [math.NT], 2021.
- Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
- Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
- Emeric Deutsch, Problem 1633, Math. Mag., 74 #5 (2001), p. 403.
- James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017.
- A. Goupil, H. Cloutier, and F. Nouboud, Enumeration of polyominoes inscribed in a rectangle Discrete Applied Mathematics 158(2010), pp. 2014-2023.
- S. Heubach and T. Mansour, Counting rises, levels and drops in compositions, arXiv:math/0310197 [math.CO], 2003.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1067
- D. Levin, L. Pudwell, M. Riehl, and A. Sandberg, Pattern Avoidance on k-ary Heaps, Slides of Talk, 2014.
- D. Merlini, F. Uncini and M. C. Verri, A unified approach to the study of general and palindromic compositions, Integers 4 (2004), A23, 26 pp.
- Agustín Moreno Cañadas, Hernán Giraldo, and Robinson Julian Serna Vanegas, Some integer partitions induced by orbits of Dynkin type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 12 (2017), pp. 2745-2766.
- Laurent Noé, Spaced seed design on profile HMMs for precise HTS read-mapping efficient sliding window product on the matrix semi-group, in Rapide Bilan 2012-2013, Laurent, LIFL, Université Lille 1 - INRIA Journées au vert 11 et 12 juin 2013, Laurent, Année 2012-2013.
- Valentin Ovsienko, Villes paires et impaires (Oddtown and Eventown) I, Images des Mathématiques, CNRS, 2013 (in French).
- Dennis Walsh, Notes on symmetric subsets of {1, 2, ..., n}
- A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
- Index to divisibility sequences
- Index entries for linear recurrences with constant coefficients, signature (0,2).
See
A052955 for partial sums (without the initial term).
A000079 gives the odd-indexed terms of a(n).
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
-
List([0..45],n->2^Int(n/2)); # Muniru A Asiru, Apr 03 2018
-
[2^Floor(n/2): n in [0..50]]; // Vincenzo Librandi, Aug 16 2011
-
A016116:= proc(n): 2^floor(n/2) end: seq(A016116(n), n=0..42); # Dennis P. Walsh, Oct 27 2009
-
Table[ 2^Floor[n/2], {n, 0, 42}] (* Robert G. Wilson v, Jun 05 2004 *)
With[{c=2^Range[0,30]},Riffle[c,c]] (* Harvey P. Dale, Jan 23 2015 *)
CoefficientList[Series[(1+x)/(1-2*x^2), {x, 0, 50}], x] (* Stefano Spezia, Sep 07 2018 *)
-
makelist(2^floor(n/2), n, 0, 50); /* Martin Ettl, Oct 17 2012 */
-
a(n)=if(n<0,0,2^(n\2))
-
def A016116(n): return 1 << n//2 # Chai Wah Wu, Jun 07 2022
-
def A016116():
x, y = -1, 0
while True:
yield -x
x, y = x + y, x - y
a = A016116(); [next(a) for i in range(40)] # Peter Luschny, Jul 11 2013
A027383
a(2*n) = 3*2^n - 2; a(2*n+1) = 2^(n+2) - 2.
Original entry on oeis.org
1, 2, 4, 6, 10, 14, 22, 30, 46, 62, 94, 126, 190, 254, 382, 510, 766, 1022, 1534, 2046, 3070, 4094, 6142, 8190, 12286, 16382, 24574, 32766, 49150, 65534, 98302, 131070, 196606, 262142, 393214, 524286, 786430, 1048574, 1572862, 2097150, 3145726, 4194302, 6291454
Offset: 0
After 3 folds one sees 4 fold lines.
Example: a(3) = 6 because the strings 001, 010, 100, 011, 101, 110 have the property.
Binary: 1, 10, 100, 110, 1010, 1110, 10110, 11110, 101110, 111110, 1011110, 1111110, 10111110, 11111110, 101111110, 111111110, 1011111110, 1111111110, 10111111110, ... - _Jason Kimberley_, Nov 02 2011
Example: Partial sums of powers of 2 repeated 2 times:
a(3) = 1+1+2 = 4;
a(4) = 1+1+2+2 = 6;
a(5) = 1+1+2+2+4 = 10.
_Yuchun Ji_, Nov 16 2018
- John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010. [John P. McSorley, Sep 28 2010]
- Vincenzo Librandi, Table of n, a(n) for n = 0..5000
- J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. - From _N. J. A. Sloane_, Feb 03 2013
- Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, Under misprinted head B3, Amer Math. Monthly, 104(1997) 753-754.
- Laurent Noé, Spaced seed design on profile HMMs for precise HTS read-mapping efficient sliding window product on the matrix semi-group, in Rapide Bilan 2012-2013, LIFL, Université Lille 1 - INRIA Journées au vert 11 et 12 juin 2013.
- Eric Weisstein's World of Mathematics, Cage Graph
- Index entries for sequences obtained by enumerating foldings
- Index entries for linear recurrences with constant coefficients, signature (1,2,-2).
Moore lower bound on the order of a (k,g) cage:
A198300 (square); rows:
A000027 (k=2), this sequence (k=3),
A062318 (k=4),
A061547 (k=5),
A198306 (k=6),
A198307 (k=7),
A198308 (k=8),
A198309 (k=9),
A198310 (k=10),
A094626 (k=11); columns:
A020725 (g=3),
A005843 (g=4),
A002522 (g=5),
A051890 (g=6),
A188377 (g=7). -
Jason Kimberley, Oct 30 2011
Cf.
A000066 (actual order of a (3,g)-cage).
a(n) =
A305540(n+2,2), the second column of the triangle.
Numbers whose binary expansion is a balanced word are
A330029.
The complementary compositions are counted by
A274230(n-1) + 1, with bisections
A060867 (even) and
A134057 (odd).
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
-
import Data.List (transpose)
a027383 n = a027383_list !! n
a027383_list = concat $ transpose [a033484_list, drop 2 a000918_list]
-- Reinhard Zumkeller, Jun 17 2015
-
[2^Floor((n+2)/2)+2^Floor((n+1)/2)-2: n in [0..50]]; // Vincenzo Librandi, Aug 16 2011
-
a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=2*a[n-2]+2 od: seq(a[n], n=1..41); # Zerinvary Lajos, Mar 16 2008
-
a[n_?EvenQ] := 3*2^(n/2)-2; a[n_?OddQ] := 2^(2+(n-1)/2)-2; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 21 2011, after Quim Castellsaguer *)
LinearRecurrence[{1, 2, -2}, {1, 2, 4}, 41] (* Robert G. Wilson v, Oct 06 2014 *)
Table[Length[Select[Tuples[{0,1},n],And[Max@@Length/@Split[#]<=2,!MatchQ[Length/@Split[#],{_,2,ins:1..,2,_}/;OddQ[Plus[ins]]]]&]],{n,0,15}] (* Gus Wiseman, Nov 28 2019 *)
-
a(n)=2^(n\2+1)+2^((n+1)\2)-2 \\ Charles R Greathouse IV, Oct 21 2011
-
def a(n): return 2**((n+2)//2) + 2**((n+1)//2) - 2
print([a(n) for n in range(43)]) # Michael S. Branicky, Feb 19 2022
More terms from Larry Reeves (larryr(AT)acm.org), Mar 24 2000
A029744
Numbers of the form 2^n or 3*2^n.
Original entry on oeis.org
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728, 4194304
Offset: 1
- Vincenzo Librandi, Table of n, a(n) for n = 1..2000
- Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 11, 22.
- Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, The Binary Two-Up Sequence, arXiv:2209.04108 [math.CO], Sep 11 2022.
- David Eppstein, Making Change in 2048, arXiv:1804.07396 [cs.DM], 2018.
- Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
- John P. McSorley and Alan H. Schoen, Rhombic tilings of (n,k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics, Discrete Math., 313 (2013), 129-154. - From _N. J. A. Sloane_, Nov 26 2012
- Index entries for linear recurrences with constant coefficients, signature (0,2).
- Index entries for sequences related to necklaces
First differences are in
A016116(n-1).
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. There may be minor differences from (s(n)) at the start, and a shift of indices.
A029744 (s(n));
A052955 (s(n)-1),
A027383 (s(n)-2),
A354788 (s(n)-3),
A060482 (s(n)-3);
A136252 (s(n)-3);
A347789 (s(n)-4),
A209721 (s(n)+1),
A209722 (s(n)+2),
A343177 (s(n)+3),
A209723 (s(n)+4);
A354785 (3*s(n)),
A061776 (3*s(n)-6);
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
-
a029744 n = a029744_list !! (n-1)
a029744_list = 1 : iterate
(\x -> if x `mod` 3 == 0 then 4 * x `div` 3 else 3 * x `div` 2) 2
-- Reinhard Zumkeller, Mar 18 2012
-
1,seq(op([2^i,3*2^(i-1)]),i=1..100); # Robert Israel, Sep 23 2014
-
CoefficientList[Series[(-x^2 - 2*x - 1)/(2*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
Function[w, DeleteCases[Union@ Flatten@ w, k_ /; k > Max@ First@ w]]@ TensorProduct[{1, 3}, 2^Range[0, 22]] (* Michael De Vlieger, Nov 24 2016 *)
LinearRecurrence[{0,2},{1,2,3},50] (* Harvey P. Dale, Jul 04 2017 *)
-
a(n)=if(n%2,3/2,2)<<((n-1)\2)\1
-
def A029744(n):
if n == 1: return 1
elif n % 2 == 0: return 2**(n//2)
else: return 3 * 2**((n-3)//2) # Karl-Heinz Hofmann, Sep 08 2022
-
(define (A029744 n) (cond ((<= n 1) n) ((even? n) (expt 2 (/ n 2))) (else (* 3 (expt 2 (/ (- n 3) 2)))))) ;; Antti Karttunen, Sep 23 2014
Corrected and extended by Joe Keane (jgk(AT)jgk.org), Feb 20 2000
A052955
a(2n) = 2*2^n - 1, a(2n+1) = 3*2^n - 1.
Original entry on oeis.org
1, 2, 3, 5, 7, 11, 15, 23, 31, 47, 63, 95, 127, 191, 255, 383, 511, 767, 1023, 1535, 2047, 3071, 4095, 6143, 8191, 12287, 16383, 24575, 32767, 49151, 65535, 98303, 131071, 196607, 262143, 393215, 524287, 786431, 1048575, 1572863, 2097151, 3145727
Offset: 0
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 7*x^4 + 11*x^5 + 15*x^6 + 23*x^7 + ... - _Michael Somos_, Jun 24 2018
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- Andrei Asinowski, Cyril Banderier, and Benjamin Hackl, On extremal cases of pop-stack sorting, Permutation Patterns (Zürich, Switzerland, 2019).
- Andrei Asinowski, Cyril Banderier, and Benjamin Hackl, Flip-sort and combinatorial aspects of pop-stack sorting, arXiv:2003.04912 [math.CO], 2020.
- J.-L. Baril, T. Mansour, and A. Petrossian, Equivalence classes of permutations modulo excedances, preprint, 2014.
- J.-L. Baril, T. Mansour, and A. Petrossian, Equivalence classes of permutations modulo excedances, Journal of Combinatorics 5 (2014), 453-469.
- David Blackman and Sebastiano Vigna, Scrambled Linear Pseudorandom Number Generators, ACM Transactions on Mathematical Software, Vol. 47, No. 4, p. 1-32, 2021; arXiv preprint, arXiv:1805.01407 [cs.DS], 2018.
- James East, Jitender Kumar, James D. Mitchell, and Wilf A. Wilson, Maximal subsemigroups of finite transformation and partition monoids, arXiv:1706.04967 [math.GR], 2017. [_Wilf A. Wilson_, Jul 21 2017]
- Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 9.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1026
- Mohammed A. Raouf, Fazirulhisyam Hashim, Jiun Terng Liew, and Kamal Ali Alezabi, Pseudorandom sequence contention algorithm for IEEE 802.11ah based internet of things network, PLoS ONE (2020) Vol. 15, No. 8, e0237386.
- Mark Shattuck, Further Results for the Capacity Statistic Distribution on Compositions of 1's and 2's, arXiv:2501.09931 [math.CO], 2025. See p. 3.
- Index to sequences related to the complexity of n
- Index entries for linear recurrences with constant coefficients, signature (1,2,-2).
See
A016116 for the first differences.
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
-
List([0..45], n-> ((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4)-1); # G. C. Greubel, Oct 22 2019
-
a052955 n = a052955_list !! n
a052955_list = 1 : 2 : map ((+ 1) . (* 2)) a052955_list
-- Reinhard Zumkeller, Feb 22 2012
-
[((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4)-1: n in [0..45]]; // G. C. Greubel, Oct 22 2019
-
spec := [S,{S=Prod(Sequence(Prod(Union(Z,Z),Z)),Union(Sequence(Z),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=2*a[n-2]+2 od: seq(a[n]/2, n=2..43); # Zerinvary Lajos, Mar 16 2008
-
a[n_]:= If[EvenQ[n], 2^(n/2+1) -1, 3*2^((n-1)/2) -1]; Table[a[n], {n, 0, 41}] (* Robert G. Wilson v, Jun 05 2004 *)
a[0]=1; a[1]=2; a[n_]:= a[n]= 2 a[n-2] +1; Array[a, 42, 0]
a[n_]:= (2 + Mod[n, 2]) 2^Quotient[n, 2] - 1; (* Michael Somos, Jun 24 2018 *)
-
a(n)=(2+n%2)<<(n\2)-1 \\ Charles R Greathouse IV, Jun 19 2011
-
{a(n) = (n%2 + 2) * 2^(n\2) - 1}; /* Michael Somos, Jun 24 2018 */
-
# command line argument tells how high to take n
# Beyond a(38) = 786431 you may need a special code to handle large integers
$lim = shift;
sub show{};
$n = $incr = $P = 1;
show($n, $incr, $P);
$incr = 1;
for $n (2..$lim) {
$P += $incr;
show($n, $P, $incr, $P);
$incr *=2 if ($n % 2); # double the increment after an odd n
}
sub show {
my($n, $P) = @_;
printf("%4d\t%16g\n", $n, $P);
}
# Mark A. Mandel (thnidu aT g ma(il) doT c0m), Dec 29 2010
-
def A052955(n): return ((2|n&1)<<(n>>1))-1 # Chai Wah Wu, Jul 13 2023
-
[((5-(-1)^n)/2)*2^((2*n-1+(-1)^n)/4)-1 for n in (0..45)] # G. C. Greubel, Oct 22 2019
A005578
a(2*n) = 2*a(2*n-1), a(2*n+1) = 2*a(2*n)-1.
Original entry on oeis.org
1, 1, 2, 3, 6, 11, 22, 43, 86, 171, 342, 683, 1366, 2731, 5462, 10923, 21846, 43691, 87382, 174763, 349526, 699051, 1398102, 2796203, 5592406, 11184811, 22369622, 44739243, 89478486, 178956971, 357913942, 715827883, 1431655766, 2863311531, 5726623062, 11453246123
Offset: 0
- R. K. Guy, Graphs and the strong law of small numbers. Graph theory, combinatorics and applications. Vol. 2 (Kalamazoo, MI, 1988), 597-614, Wiley-Intersci. Publ., Wiley, New York, 1991.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Sujith Uthsara Kalansuriya Arachchi, Hung Viet Chu, Jiasen Liu, Qitong Luan, Rukshan Marasinghe, and Steven J. Miller, On a Pair of Diophantine Equations, arXiv:2309.04488 [math.NT], 2023.
- Joerg Arndt, Matters Computational (The Fxtbook), p. 315.
- Ji Young Choi, Ternary Modified Collatz Sequences And Jacobsthal Numbers, Journal of Integer Sequences, Vol. 19 (2016), #16.7.5.
- Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
- Fan Chung and Shlomo Sternberg, Mathematics and the Buckyball
- Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
- Johann Cigler, Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials, arXiv:2212.02118 [math.NT], 2022.
- 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 pp. 6, 17.
- R. K. Guy, Letter to N. J. A. Sloane, Apr 08 1988 (annotated scanned copy, included with permission)
- Clemens Heuberger and Helmut Prodinger, On minimal expansions in redundant number systems: Algorithms and quantitative analysis, Computing 66(2001), 377-393.
- Andreas M. Hinz, The Lichtenberg sequence, Fib. Quart., 55 (2017), 2-12.
- Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
- Gurmeet Singh Manku and Joe Sawada, A loopless Gray code for minimal signed-binary representations, 13th Annual European Symposium on Algorithms (ESA), LNCS 3669 (2005), 438-447.
- Thor Martinsen, Non-Fisherian generalized Fibonacci numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 2, 370-389. See pp. 385, 387.
- John P. McSorley, Counting structures in the Moebius ladder, Discrete Math., 184 (1998), 137-164.
- Hans Olofsen, Blending functions based on trigonometric and polynomial approximations of the Fabius function, The Arctic University of Norway (Narvik, 2019).
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
- Chloe E. Shiff and Noah A. Rosenberg, Enumeration of rooted binary perfect phylogenies, arXiv:2410.14915 [q-bio.PE], 2024. See p. 16.
- Andrew V. Sills and Hua Wang, On the maximal Wiener index and related questions, Discrete Applied Mathematics, Volume 160, Issues 10-11, July 2012, Pages 1615-1623.
- Eric Weisstein's World of Mathematics, Walsh Function
- Index entries for linear recurrences with constant coefficients, signature (2,1,-2).
-
List([0..40],n->(2^(n+1)+3+(-1)^n)/6); # Muniru A Asiru, Dec 22 2018
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[(2^(n+1)+3+(-1)^n)/6: n in [0..40]]; // Vincenzo Librandi, Aug 14 2011
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A005578:=-(-1+z+z^2)/((z-1)*(2*z-1)*(z+1)); # Simon Plouffe in his 1992 dissertation
with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), a):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL3), b=ZL3], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, {Q}, unlabelled], size=n), n=2..34); # Zerinvary Lajos, Mar 08 2008
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a=0; Table[a=2^n-a;(a/2+1)/2,{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Nov 22 2009 *)
LinearRecurrence[{2,1,-2}, {1,1,2}, 40] (* G. C. Greubel, Aug 26 2019 *)
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a(n)=(2^(n+1)+3+(-1)^n)/6 \\ Charles R Greathouse IV, Mar 22 2016
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print([1+2**n//3 for n in range(40)]) # Gennady Eremin, Feb 05 2022
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[(2^(n+1)+3+(-1)^n)/6 for n in (0..40)] # G. C. Greubel, Aug 26 2019
A060546
a(n) = 2^ceiling(n/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: 0
André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001
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
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[2^Ceiling(n/2): n in [0..50]]; // G. C. Greubel, Nov 07 2018
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for n from 0 to 100 do printf(`%d,`,2^ceil(n/2)) od:
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2^Ceiling[Range[0,50]/2] (* or *) Riffle[2^Range[0, 25], 2^Range[25]] (* Harvey P. Dale, Mar 05 2013 *)
LinearRecurrence[{0, 2}, {1, 2}, 40] (* Robert A. Russell, Nov 07 2018 *)
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a(n) = { 2^ceil(n/2) } \\ Harry J. Smith, Jul 06 2009
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
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 + ...
Binomial transform is
A078057, second binomial transform is
A007070, third binomial transform is
A102285, fourth binomial transform is
A163350, fifth binomial transform is
A163346.
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
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[ n le 2 select n else 2*Self(n-2): n in [1..43] ];
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LinearRecurrence[{0, 2}, {1, 2}, 50] (* Paolo Xausa, Feb 02 2024 *)
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{a(n) = if( n<1, 0, 2^(n\2))} /* Michael Somos, Mar 20 2011 */
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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
A060482
New record highs reached in A060030.
Original entry on oeis.org
1, 2, 3, 5, 9, 13, 21, 29, 45, 61, 93, 125, 189, 253, 381, 509, 765, 1021, 1533, 2045, 3069, 4093, 6141, 8189, 12285, 16381, 24573, 32765, 49149, 65533, 98301, 131069, 196605, 262141, 393213, 524285, 786429, 1048573, 1572861, 2097149, 3145725, 4194301, 6291453
Offset: 1
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
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LinearRecurrence[{1,2,-2},{1,2,3,5,9},50] (* Harvey P. Dale, Sep 11 2016 *)
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{ for (n=1, 1000, if (n%2==0, m=n/2; a=2^(m + 1) - 3, m=(n - 1)/2; a=3*2^m - 3); if (n<3, a=n); write("b060482.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 05 2009
A136252
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).
Original entry on oeis.org
1, 3, 5, 9, 13, 21, 29, 45, 61, 93, 125, 189, 253, 381, 509, 765, 1021, 1533, 2045, 3069, 4093, 6141, 8189, 12285, 16381, 24573, 32765, 49149, 65533, 98301, 131069, 196605, 262141, 393213, 524285, 786429, 1048573, 1572861, 2097149, 3145725, 4194301, 6291453, 8388605
Offset: 0
Cf.
A007664 (Optimal 4-peg Tower of Hanoi).
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
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a:=proc(n) options operator,arrow: 2^((1/2)*n-1)*(4+4*(-1)^n+3*sqrt(2)*(1-(-1)^n))-3 end proc: seq(a(n),n=0..40); # Emeric Deutsch, Mar 31 2008
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LinearRecurrence[{1, 2, -2}, {1, 3, 5}, 100] (* G. C. Greubel, Feb 18 2017 *)
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x='x+O('x^50); Vec((1+2*x)/((1-x)*(1-2*x^2))) \\ G. C. Greubel, Feb 18 2017
A152166
a(2*n) = 2^n; a(2*n+1) = -(2^(n+1)).
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
Offset: 0
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
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LinearRecurrence[{0, 2}, {1, -2}, 50] (* Paolo Xausa, Jul 19 2024 *)
Showing 1-10 of 23 results.
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