A061776 -id:A061776 - cp's OEIS frontend

A029744 Numbers of the form 2^n or 3*2^n.

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

N. J. A. Sloane

This entry is a list, and so has offset 1. WARNING: However, in this entry several comments, formulas and programs seem to refer to the original version of this sequence which had offset 0. - M. F. Hasler, Oct 06 2014
Number of necklaces with n-1 beads and two colors that are the same when turned over and hence have reflection symmetry. [edited by Herbert Kociemba, Nov 24 2016]
The subset {a(1),...,a(2k)} contains all proper divisors of 3*2^k. - Ralf Stephan, Jun 02 2003
Let k = any nonnegative integer and j = 0 or 1. Then n+1 = 2k + 3j and a(n) = 2^k*3^j. - Andras Erszegi (erszegi.andras(AT)chello.hu), Jul 30 2005
Smallest number having no fewer prime factors than any predecessor, a(0)=1; A110654(n) = A001222(a(n)); complement of A116451. - Reinhard Zumkeller, Feb 16 2006
A093873(a(n)) = 1. - Reinhard Zumkeller, Oct 13 2006
a(n) = a(n-1) + a(n-2) - gcd(a(n-1), a(n-2)), n >= 3, a(1)=2, a(2)=3. - Ctibor O. Zizka, Jun 06 2009
Where records occur in A048985: A193652(n) = A048985(a(n)) and A193652(n) < A048985(m) for m < a(n). - Reinhard Zumkeller, Aug 08 2011
A002348(a(n)) = A000079(n-3) for n > 2. - Reinhard Zumkeller, Mar 18 2012
Without initial 1, third row in array A228405. - Richard R. Forberg, Sep 06 2013
Also positions of records in A048673. A246360 gives the record values. - Antti Karttunen, Sep 23 2014
Known in numerical mathematics as "Bulirsch sequence", used in various extrapolation methods for step size control. - Peter Luschny, Oct 30 2019
For n > 1, squares of the terms can be expressed as the sum of two powers of two: 2^x + 2^y. - Karl-Heinz Hofmann, Sep 08 2022

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

Cf. A056493, A038754, A063759. Union of A000079 and A007283.
First differences are in A016116(n-1).
Cf. A082125, A094958, A048739, A048985, A193652, A048673, A064216, A246360, A354785.
Row sums of the triangle in sequence A119963. - John P. McSorley, Aug 31 2010
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 *)

PARI
```
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

a(n) = 2*A000029(n) - A000031(n).
For n > 2, a(n) = 2*a(n - 2); for n > 3, a(n) = a(n - 1)*a(n - 2)/a(n - 3). G.f.: (1 + x)^2/(1 - 2*x^2). - Henry Bottomley, Jul 15 2001, corrected May 04 2007
a(0)=1, a(1)=1 and a(n) = a(n-2) * ( floor(a(n-1)/a(n-2)) + 1 ). - Benoit Cloitre, Aug 13 2002
(3/4 + sqrt(1/2))*sqrt(2)^n + (3/4 - sqrt(1/2))*(-sqrt(2))^n. a(0)=1, a(2n) = a(n-1)*a(n), a(2n+1) = a(n) + 2^floor((n-1)/2). - Ralf Stephan, Apr 16 2003 [Seems to refer to the original version with offset=0. - M. F. Hasler, Oct 06 2014]
Binomial transform is A048739. - Paul Barry, Apr 23 2004
E.g.f.: (cosh(x/sqrt(2)) + sqrt(2)sinh(x/sqrt(2)))^2.
a(1) = 1; a(n+1) = a(n) + A000010(a(n)). - Stefan Steinerberger, Dec 20 2007
u(2)=1, v(2)=1, u(n)=2*v(n-1), v(n)=u(n-1), a(n)=u(n)+v(n). - Jaume Oliver Lafont, May 21 2008
For n => 3, a(n) = sqrt(2*a(n-1)^2 + (-2)^(n-3)). - Richard R. Forberg, Aug 20 2013
a(n) = A064216(A246360(n)). - Antti Karttunen, Sep 23 2014
a(n) = sqrt((17 - (-1)^n)*2^(n-4)) for n >= 2. - Anton Zakharov, Jul 24 2016
Sum_{n>=1} 1/a(n) = 8/3. - Amiram Eldar, Nov 12 2020
a(n) = 2^(n/2) if n is even. a(n) = 3 * 2^((n-3)/2) if n is odd and for n>1. - Karl-Heinz Hofmann, Sep 08 2022

Corrected and extended by Joe Keane (jgk(AT)jgk.org), Feb 20 2000

A061777 Start with a single triangle; at n-th generation add a triangle at each vertex, allowing triangles to overlap; sequence gives total population of triangles at n-th generation.

Original entry on oeis.org

1, 4, 10, 22, 40, 70, 112, 178, 268, 406, 592, 874, 1252, 1822, 2584, 3730, 5260, 7558, 10624, 15226, 21364, 30574, 42856, 61282, 85852, 122710, 171856, 245578, 343876, 491326, 687928, 982834, 1376044, 1965862, 2752288, 3931930, 5504788

Offset: 0

N. J. A. Sloane, R. K. Guy, Jun 23 2001

From the definition, assign label value "1" to an origin triangle; at n-th generation add a triangle at each vertex. Each non-overlapping triangle will have the same label value as that of the predecessor triangle to which it is connected; for the overlapping ones, the label value will be the sum of the label values of predecessors. a(n) is the sum of all label values at the n-th generation. The triangle count is A005448. See illustration. For n >= 1, (a(n) - a(n-1))/3 is A027383. - Kival Ngaokrajang, Sep 05 2014

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6.

Kival Ngaokrajang, Illustration of initial terms
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,2).

Partial sums of A061776.

Cf. A005448, A027383.

seq(`if`(n::even, 21*2^(n/2) - 6*n-20, 30*2^((n-1)/2)-6*n-20),n=0..100); # Robert Israel, Sep 14 2014

Table[If[EvenQ[n],21 2^(n/2)-6n-20,30 2^((n-1)/2)-6(n-1)-26],{n,0,40}] (* Harvey P. Dale, Nov 06 2011 *)

a(n)=if(n%2, 30, 21)<<(n\2) - 6*n - 20 \\ Charles R Greathouse IV, Sep 19 2014

From Colin Barker, May 08 2012: (Start)
a(n) = 21*2^(n/2) - 6*n - 20 if n is even.
a(n) = 30*2^((n-1)/2) - 6*(n - 1) - 26 if n is odd.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + 2*a(n-4).
G.f.: (1 + 2*x)*(1 + x^2)/((1 - x)^2*(1 - 2*x^2)). (End)
From Robert Israel, Sep 14 2014: (Start)
a(n) = -20 - 6*n + (21 + 15*sqrt(2))*sqrt(2)^(n-2) + (21 - 15*sqrt(2))*(-sqrt(2))^(n-2).
a(n) = 2*a(n-2) + ((3*n-2)/(3*n-5))*(a(n-1)-2*a(n-3)). (End)
E.g.f.: 21*cosh(sqrt(2)*x) + 15*sqrt(2)*sinh(sqrt(2)*x) - 2*exp(x)*(10 + 3*x). - Stefano Spezia, Aug 13 2022

Corrected by T. D. Noe, Nov 08 2006

A099942 Start with 1, then alternately double or add 2.

Original entry on oeis.org

1, 2, 4, 8, 10, 20, 22, 44, 46, 92, 94, 188, 190, 380, 382, 764, 766, 1532, 1534, 3068, 3070, 6140, 6142, 12284, 12286, 24572, 24574, 49148, 49150, 98300, 98302, 196604, 196606, 393212, 393214, 786428, 786430, 1572860, 1572862, 3145724, 3145726

Offset: 0

N. J. A. Sloane, Nov 12 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A033484, A061776, A075427, A083416.

[3*2^Ceiling(n/2) + (-1)^n - 3: n in [0..50]]; // Vincenzo Librandi, Aug 17 2011

LinearRecurrence[{0,3,0,-2},{1,2,4,8},50] (* Harvey P. Dale, May 03 2016 *)

print1(a=1,",");for(n=1,20,print1(a=2*a,",",a=a+2,","))

a(0)=1; for n > 0, a(n) = a(n-1)*(1 + n mod 2) + 2*((n+1) mod 2).
G.f.: (2*x^3 + x^2 + 2*x + 1)/(2*x^4 - 3*x^2 + 1).
3*2^ceiling(n/2) + (-1)^n - 3. - Ralf Stephan, Dec 04 2004
a(2*n) = A033484(n).
a(n-1) + a(n) = A061776(n) for n > 0.
E.g.f.: -2*cosh(x) + 3*cosh(sqrt(2)*x) - 4*sinh(x) + 3*sqrt(2)*sinh(sqrt(2)*x). - Franck Maminirina Ramaharo, Nov 08 2018

Edited and extended by Klaus Brockhaus, Nov 13 2004

cp's OEIS Frontend

A029744 Numbers of the form 2^n or 3*2^n.

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A061777 Start with a single triangle; at n-th generation add a triangle at each vertex, allowing triangles to overlap; sequence gives total population of triangles at n-th generation.

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A099942 Start with 1, then alternately double or add 2.

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