A081294 -id:A081294 - cp's OEIS frontend

A360617 Half the number of prime factors of n (counted with multiplicity, A001222), rounded up.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 08 2023

nonn

			The prime indices of 378 are {1,2,2,2,4}, so a(378) = ceiling(5/2) = 3.

Positions of 0's and 1's are 1 and A037143.
Positions of first appearances are A081294.
Rounding down instead of up gives A360616.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A000040, A000302, A001248, A026424, A168645, A359912, A360006, A360457.

Mathematica
```
Table[Ceiling[PrimeOmega[n]/2],{n,100}]
```

A081654 a(n) = 2*4^n - 0^n.

Original entry on oeis.org

1, 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 26 2003

Binomial transform of A081632. Inverse binomial transform of A081655.

			a(0) = 2*4^0 - 0^0 = 2 - 1 = 1 (use 0^0 = 1).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000244 (3^n), A187093.

Essentially the same as A004171.

[2*4^n-0^n: n in [0..30]]; // Vincenzo Librandi, Aug 10 2013

CoefficientList[Series[(1 + 4 x) / (1 - 4 x), {x, 0, 40}], x] (* Vincenzo Librandi, Aug 10 2013 *)

a(n)=2*4^n-0^n \\ Charles R Greathouse IV, Apr 09 2012

x='x+O('x^100); Vec((1+4*x)/(1-4*x)) \\ Altug Alkan, Dec 14 2015

a(0)=1, a(n) = 2*4^n, n>0
G.f.: (1+4*x)/(1-4*x).
E.g.f. 2*exp(4*x)-1.
With interpolated zeros, this is 2^n - 0^n + (-2)^n. - Paul Barry, Sep 06 2003
a(n) = A081294(n+1), n>0. - R. J. Mathar, Sep 17 2008
For n>0, a(n) = 2 * (1 + 3^(n-1) + Sum{x=1..n-2}Sum{k=0..x-1}(binomial(x-1,k)*(3^(k+1) + 3^(n-x+k)))). - J. Conrad, Dec 10 2015

A092812 Number of closed walks of length 2*n on the 4-cube.

Original entry on oeis.org

1, 4, 40, 544, 8320, 131584, 2099200, 33562624, 536903680, 8590065664, 137439477760, 2199025352704, 35184380477440, 562949986975744, 9007199388958720, 144115188612726784, 2305843011361177600

Offset: 0

Paul Barry, Mar 11 2004

With interpolated zeros this has a(n) = (6*0^n + 4^n + (-4)^n + 4*2^n + 4*(-2)^n)/16 and counts closed walks of length n at a vertex of the 4-cube. [Typo corrected by Alexander R. Povolotsky, May 26 2008]

Also, cogrowth sequence of the 16-element group C2^4. - Sean A. Irvine, Nov 10 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..200
G. R. Franssens, On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, How big is BCI fragment of BCK logic, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. [From _N. J. A. Sloane_, Feb 21 2012]
L. Reyzin, Mathoverflow, Number of closed walks on an n-cube.
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Essentially the same as A075878.

Cf. A026244, A081294, A054879, A121822.

[3*0^n/8+16^n/8+4^n/2: n in [0..30]]; // Vincenzo Librandi, May 31 2011

CoefficientList[Series[(1-16x+24x^2)/((1-4x)(1-16x)),{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{20,-64},{4,40},30]] (* Harvey P. Dale, Aug 23 2011 *)

G.f.: (1-16*x+24*x^2)/((1-4*x)*(1-16*x)).
a(n) = 3*0^n/8 + 16^n/8 + 4^n/2.
From Peter Bala, Nov 13 2006: (Start)
E.g.f.: cosh^4(x).
O.g.f.: 1/(1-4*1*x/(1-3*2*x/(1-2*3*x/(1-1*4*x)))) (continued fraction). (End)
(-1)^n*a(n) = Sum_{k=0..n} A086872(n,k)*(-5)^(n-k). - Philippe Deléham, Aug 17 2007
a(n) = 20*a(n-1) - 64*a(n-2); a(0) = 1, a(1) = 4, a(2) = 40. - Harvey P. Dale, Aug 23 2011
a(n) = 4*A026244(n-1), n > 0. - R. J. Mathar, Oct 24 2014
a(n) = (1/2^4)*Sum_{j = 0..4} binomial(4, j)*(4 - 2*j)^(2*n). See Reyzin link. - Peter Bala, Jun 03 2019

Title improved by Sean A. Irvine at the suggestion of Peter Bala, Jun 04 2019

A147538 Numbers whose binary representation is the concatenation of n 1's and 2n-1 digits 0.

Original entry on oeis.org

2, 24, 224, 1920, 15872, 129024, 1040384, 8355840, 66977792, 536346624, 4292870144, 34351349760, 274844352512, 2198889037824, 17591649173504, 140735340871680, 1125891316908032, 9007164895002624, 72057456598974464, 576460202547609600

Offset: 1

Omar E. Pol, Nov 06 2008

a(n) is the number whose binary representation is A138119(n).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (12,-32).

Cf. A138119.

Cf. A016152. - Omar E. Pol, Nov 13 2008

List([1..20], n-> 2^(2*n-1)*(2^n -1)); # G. C. Greubel, Jan 12 2020

[2^(2*n-1)*(2^n -1): n in [1..20]]; // G. C. Greubel, Jan 12 2020

seq(2^(2*n-1)*(2^n -1), n=1..20); # G. C. Greubel, Jan 12 2020

Table[FromDigits[Join[Table[1, {n}], Table[0, {2n - 1}]], 2], {n, 1, 20}] (* Stefan Steinerberger, Nov 11 2008 *)

vector(20, n, 2^(2*n-1)*(2^n -1)) \\ G. C. Greubel, Jan 12 2020

def a(n): return ((1 << n) - 1) << (2*n-1)
print([a(n) for n in range(1, 22)]) # Michael S. Branicky, Feb 24 2021

[2^(2*n-1)*(2^n -1) for n in (1..20)] # G. C. Greubel, Jan 12 2020

a(n) = 2^(2*n-1)*(2^n -1) = A081294(n)*A000225(n). - R. J. Mathar, Nov 09 2008
a(n) = 2*A016152(n). - Omar E. Pol, Nov 13 2008
From Colin Barker, Nov 04 2012: (Start)
a(n) = 12*a(n-1) - 32*a(n-2).
G.f.: 2*x/((1-4*x)*(1-8*x)). (End)

Extended by R. J. Mathar and Stefan Steinerberger, Nov 09 2008

A183119 Magnetic Tower of Hanoi, total number of moves generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [RED ; NEUTRAL ; BLUE] pre-colored puzzle.

Original entry on oeis.org

0, 1, 4, 11, 32, 93, 276, 823, 2464, 7385, 22148, 66435, 199296, 597877, 1793620, 5380847, 16142528, 48427569, 145282692, 435848059, 1307544160, 3922632461, 11767897364, 35303692071, 105911076192, 317733228553, 953199685636, 2859599056883, 8578797170624, 25736391511845

Offset: 0

Uri Levy, Jan 03 2011

The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [RED ; NEUTRAL ; BLUE], given in [Source ; Intermediate ; Destination] order. The solution algorithm producing the presented sequence is NOT optimal. The particular "75" algorithm solving the puzzle at hand is presented in a paper referenced by link 1 listed below. For the optimal solution of the Magnetic Tower of Hanoi puzzle with the given pre-coloring configuration see A183113 and A183114. Optimal solutions are discussed and their optimality is proved in link 2 listed below.

Large N limit of the sequence is 0.5*(3/4)*3^N = 0.5*0.75*3^N. Series designation: S75(n).

Uri Levy, The Magnetic Tower of Hanoi, Journal of Recreational Mathematics, Volume 35 Number 3 (2006), 2010, pp 173.

Muniru A Asiru, Table of n, a(n) for n = 0..2070
Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 [math.CO], 2010.
Uri Levy, to play The Magnetic Tower of Hanoi, web applet [Broken link]
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

A122983 - "Binomial transform of aeration of A081294" is an "original" sequence (also) describing the number of moves of disk number k, solving the pre-colored puzzle at hand when executing the "75" algorithm mentioned above and presented in the paper referenced by link 1 above. The integer sequence listed above is the partial sums of the A122983 original sequence.
A003462 "Partial sums of A000244" is the sequence (also) describing the total number of moves solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle.
Cf. A183111-A183125.

[3^(n+1)/8+(n-1)/2+(-1)^n/8: n in [0..30]]; // Vincenzo Librandi, Dec 04 2018

seq(coeff(series(x*(3*x^2-1)/((1+x)*(3*x-1)*(x-1)^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Dec 04 2018

LinearRecurrence[{4, -2, -4, 3}, {0, 1, 4, 11}, 30] (* Jean-François Alcover, Dec 04 2018 *)
Table[3^(n + 1) / 8 + (n - 1) / 2 + (-1)^n / 8, {n, 0, 30}] (* Vincenzo Librandi, Dec 04 2018 *)

a(n) = 3^(n+1)/8 + (n-1)/2 + (-1)^n/8 \\ Charles R Greathouse IV, Jun 11 2015

G.f.: x*(-1+3*x^2) / ( (1+x)*(3*x-1)*(x-1)^2 ).
(a(n)=S75(n) in referenced paper):
a(n) = 3*a(n-1) - n + 3; n even; n >= 2
a(n) = 3*a(n-1) - n + 2; n odd; n >= 1
a(n) = a(n-2) + 3^(n-1) + 1; n >= 2
a(n) = 3^(n+1)/8 + (n-1)/2 +(-1)^n/8.

More terms from Jean-François Alcover, Dec 04 2018

A033469 Denominator of Bernoulli(2n,1/2).

Original entry on oeis.org

1, 12, 240, 1344, 3840, 33792, 5591040, 49152, 16711680, 104595456, 173015040, 289406976, 22900899840, 201326592, 116769423360, 7689065201664, 1095216660480, 51539607552, 65942866278481920, 824633720832, 7438196161904640, 3971435999526912

Offset: 0

N. J. A. Sloane

nonn

From the von Staudt-Clausen theorem it follows that a(n) can be computed without using Bernoulli polynomials or the 'denominator'-function (see the Sage implementation). - Peter Luschny, Mar 24 2014

J. R. Philip, The symmetrical Euler-Maclaurin summation formula, Math. Sci., 6, 1981, pp. 35-41.

Vincenzo Librandi, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem.
Index entries for sequences related to Bernoulli numbers.

Cf. A001896.

with(numtheory); seq(denom(bernoulli(2*n, 1/2)), n=0..20);

Table[ BernoulliB[2*n, 1/2] // Denominator, {n, 0, 18}] (* Jean-François Alcover, Apr 15 2013 *)
a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ x/2 / Sinh[x/2], {x, 0, 2 n}] // Denominator]; (* Michael Somos, Sep 21 2016 *)

a(n)=denominator(subst(bernpol(2*n,x),x,1/2)); \\ Joerg Arndt, Apr 17 2013

def A033469(n):
    if n == 0: return 1
    M = map(lambda i: i+1, divisors(2*n))
    return 2^(2*n-1)*mul(filter(lambda s: is_prime(s), M))
[A033469(n) for n in (0..21)] # Peter Luschny, Mar 24 2014

a(n) = denominator(2*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(-1)). - Peter Luschny, Jun 29 2012
a(n) = A081294(n) * A002445(n) for n > 0. - Paul Curtz, Apr 17 2013
Apparently, denominators of the fractions with e.g.f. (x/2) / sinh(x/2). - Tom Copeland, Sep 17 2016

More terms from Joerg Arndt, Apr 17 2013

A099140 a(n) = 4^n * T(n,3/2) where T is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 6, 56, 576, 6016, 62976, 659456, 6905856, 72318976, 757334016, 7930904576, 83053510656, 869747654656, 9108115685376, 95381425750016, 998847258034176, 10460064284409856, 109539215284371456, 1147109554861899776

Offset: 0

Paul Barry, Sep 30 2004

In general, r^n * T(n,(r+2)/r) has g.f. (1-(r+2)*x)/(1-2*(r+2)*x + r^2*x^2), e.g.f. exp((r+2)*x)*cosh(2*sqrt(r+1)*x), a(n) = Sum_{k=0..n} (r+1)^k*binomial(2n,2k) and a(n) = (1+sqrt(r+1))^(2n)/2 + (1-sqrt(r+1))^(2n)/2.

Harvey P. Dale, Table of n, a(n) for n = 0..900
P. J. Szablowski, On moments of Cantor and related distributions, arXiv preprint arXiv:1403.0386 [math.PR], 2014.
Index entries for linear recurrences with constant coefficients, signature (12,-16).

Cf. A001541, A081294, A083884, A090965, A099141, A099142.

LinearRecurrence[{12,-16},{1,6},30] (* Harvey P. Dale, Oct 23 2012 *)

a(n) = 4^n*polchebyshev(n, 1, 3/2); \\ Michel Marcus, Sep 08 2019

G.f.: (1-6*x)/(1-12*x+16*x^2);
E.g.f.: exp(6*x)*cosh(2*sqrt(5)*x);
a(n) = 4^n * T(n, 6/4) where T is the Chebyshev polynomial of the first kind;
a(n) = Sum_{k=0..n} 5^k*binomial(2n, 2k);
a(n) = (1+sqrt(5))^(2n)/2 + (1-sqrt(5))^(2n)/2.
a(n) = a(0)=1, a(1)=6, 12*a(n-1) - 16*a(n-2) for n > 1. - Philippe Deléham, Sep 08 2009

A099141 a(n) = 5^n * T(n,7/5) where T is the Chebyshev polynomial of the first kind.

Original entry on oeis.org

1, 7, 73, 847, 10033, 119287, 1419193, 16886527, 200931553, 2390878567, 28449011113, 338514191407, 4027973401873, 47928772841047, 570303484727833, 6786029465163487, 80746825394092993, 960804818888214727

Offset: 0

Paul Barry, Sep 30 2004

In general, r^n * T(n,(r+2)/r) has g.f. (1-(r+2)*x)/(1-2*(r+2)*x + r^2*x^2), e.g.f. exp((r+2)*x)*cosh(2*sqrt(r+1)*x), a(n) = Sum_{k=0..n} (r+1)^k*binomial(2*n,2*k) and a(n) = (1+sqrt(r+1))^(2*n)/2 + (1-sqrt(r+1))^(2*n)/2.

Seiichi Manyama, Table of n, a(n) for n = 0..500
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (14,-25).

Column k=6 of A333988.

Cf. A081294, A001541, A090965, A083884, A099140, A099142.

LinearRecurrence[{14,-25},{1,7},30] (* Harvey P. Dale, Dec 26 2014 *)

G.f.: (1-7*x)/(1-14*x+25*x^2);
e.g.f.: exp(7*x)*cosh(2*sqrt(6)*x);
a(n) = 5^n * T(n, 7/5) where T is the Chebyshev polynomial of the first kind;
a(n) = Sum_{k=0..n} 6^k * binomial(2n, 2k);
a(n) = (1+sqrt(6))^(2n)/2 + (1-sqrt(6))^(2n)/2.
a(0)=1, a(1)=7, a(n) = 14*a(n-1) - 25*a(n-2) for n > 1. - Philippe Deléham, Sep 08 2009

A121822 Number of closed walks of length 2*n on the 5-cube.

Original entry on oeis.org

1, 5, 65, 1205, 26465, 628805, 15424865, 382964405, 9550195265, 238539648005, 5961554097665, 149021418519605, 3725378557692065, 93133051794619205, 2328313585536338465, 58207725254446186805, 1455192101905494196865

Offset: 0

Philippe Deléham, Aug 27 2006

G. C. Greubel, Table of n, a(n) for n = 0..700
G. R. Franssens, On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
L. Reyzin, Mathoverflow, Number of closed walks on an n-cube
Index entries for linear recurrences with constant coefficients, signature (35,-259,225).

Cf. A054879, A081294, A092812.

List([0..20], n-> (25^n +5*9^n +10)/16); # G. C. Greubel, Jun 07 2019

[(25^n +5*9^n +10)/16: n in [0..20]]; // G. C. Greubel, Jun 07 2019

Table[(25^n +5*9^n +10)/16, {n,0,20}] (* G. C. Greubel, Jun 07 2019 *)

a(n)=(25^n+5*9^n+10)>>4 \\ Charles R Greathouse IV, Jan 17 2012

[(25^n +5*9^n +10)/16 for n in (0..20)] # G. C. Greubel, Jun 07 2019

a(n) = (25^n + 5*9^n + 10)/16.
G.f.: (1 - 30*x + 149*x^2)/(1 - 35*x + 259*x^2 - 225*x^3).
From Peter Bala, Nov 13 2006: (Start)
E.g.f.: cosh^5(x).
O.g.f.: 1/(1-5*1x/(1-4*2x/(1-3*3x/(1-2*4x/(1-1*5x))))) (continued fraction). (End)
(-1)^n*a(n) = Sum_{k=0..n} A086872(n,k)*(-6)^(n-k). - Philippe Deléham, Aug 17 2007
a(n) = (1/2^5)*Sum_{j = 0..5} binomial(5,j)*(5 - 2*j)^(2*n). See Reyzin link. - Peter Bala, Jun 03 2019

Corrected by T. D. Noe, Nov 07 2006

A009117 Expansion of e.g.f.: 1/2 + exp(-4*x)/2.

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, 140737488355328, -562949953421312

Offset: 0

R. H. Hardin

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, How big is BCI fragment of BCK logic, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. - _N. J. A. Sloane_, Feb 21 2012
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 171
Index entries for linear recurrences with constant coefficients, signature (-4).

a(n) = (-1)^n * A004171(n-1).

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2*x+1)/(1+4*x))); // G. C. Greubel, Jul 26 2018

A009117:=n->`if`(n=0, 1, (-4)^n/2); seq(A009117(n), n=0..30); # Wesley Ivan Hurt, Mar 10 2014

With[{nn=30},CoefficientList[Series[1/2+Exp[-4x]/2,{x,0,nn}],x] Range[ 0,nn]!] (* or *) LinearRecurrence[{-4},{1,-2},30] (* Harvey P. Dale, Apr 09 2015 *)

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

1 followed by (-4)^n /2.
E.g.f.: cos(x)^2 (even powers).
a(n) = Sum_{k, 0<=k<=n} A086872(n,k)*(-3)^(n-k). - Philippe Deléham, Aug 17 2007
G.f. (2*x+1)/(1+4*x). - R. J. Mathar, Mar 08 2011
E.g.f.: 1/2 + exp(-4*x)/2 = (G(0)+1)/2 ; G(k) = 1 - 4*x/(2*k+1 - 2*x*(2*k+1)/(2*x - (k+1)/G(k+1))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 20 2011
a(n) = (-1)^n * A081294(n). - Philippe Deléham, Mar 09 2014

Signs added and formula corrected by Olivier Gérard, Mar 15 1997
More terms from Olaf Voß, Feb 13 2008
Definition corrected by Joerg Arndt, May 16 2011

cp's OEIS Frontend

A360617 Half the number of prime factors of n (counted with multiplicity, A001222), rounded up.

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A081654 a(n) = 2*4^n - 0^n.

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A092812 Number of closed walks of length 2*n on the 4-cube.

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A147538 Numbers whose binary representation is the concatenation of n 1's and 2n-1 digits 0.

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A183119 Magnetic Tower of Hanoi, total number of moves generated by a certain algorithm, yielding a "forward moving" non-optimal solution of the [RED ; NEUTRAL ; BLUE] pre-colored puzzle.

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A033469 Denominator of Bernoulli(2n,1/2).

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