A162436 -id:A162436 - cp's OEIS frontend

A164123 Partial sums of A162436.

1, 4, 7, 16, 25, 52, 79, 160, 241, 484, 727, 1456, 2185, 4372, 6559, 13120, 19681, 39364, 59047, 118096, 177145, 354292, 531439, 1062880, 1594321, 3188644, 4782967, 9565936, 14348905, 28697812, 43046719, 86093440, 129140161, 258280324, 387420487, 774840976, 1162261465

Offset: 1

Klaus Brockhaus, Aug 10 2009

Interleaving of A058481 and A100774 without initial term 0.
Apparently a(n) = A062318(n+2) - 1.
The terms beginning with a(2) are the row numbers in Pascal's Triangle where every 3rd element in those rows is divisible by 3 and none of the other elements in those rows are divisible by 3. - Thomas M. Green, Apr 03 2013

			For n = 3, a(3) = 7. The binomial coefficients of the 7th row of Pascal's Triangle are 1 7 21 35 35 21 7 1 and every 3rd element is a multiple of 3. - _Thomas M. Green_, Apr 03 2013

Thomas M. Green, Prime Patterns in Pascal's Triangle, paper in review process, 2013.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A162436, A058481 (3^n-2), A100774 (2*(3^n - 1)), A062318, A038754, A038754.

T:=[ n le 2 select 2*n-1 else 3*Self(n-2): n in [1..33] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

Accumulate[Transpose[NestList[{Last[#],3*First[#]}&,{1,3},40]][[1]]] (* Harvey P. Dale, Feb 17 2012 *)

a(n) = (2+n%2)*3^(n\2)-2 \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A038754(n+1) - 2.
a(n) = 3*a(n-2) + 4 for n > 2; a(1) = 1, a(2) = 4.
a(n) = (5 - (-1)^n)*3^(1/4*(2*n - 1 + (-1)^n))/2 - 2.
G.f.: x*(1 + 3*x)/((1 - x)*(1 - 3*x^2)).
E.g.f.: 2*(cosh(sqrt(3)*x) - cosh(x)) + sqrt(3)*sinh(sqrt(3)*x) - 2*sinh(x). - Stefano Spezia, Dec 31 2022

Incorrect formula removed by Stefano Spezia, Dec 31 2022

A108411 a(n) = 3^floor(n/2). Powers of 3 repeated.

Original entry on oeis.org

1, 1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467

Offset: 0

Ralf Stephan, Jun 05 2005

a(n) is the Parker sequence for the automorphism group of the limit of the class of oriented graphs; a(n) counts the finite circulant structures in that class. - N-E. Fahssi, Feb 18 2008
Complete sequence: every positive integer is the sum of members of this sequence. - Charles R Greathouse IV, Jul 19 2012
Conjecture: a(n+1) is the number of distinct subsets S of {0,1,2,...,n} such that the sumset S+S does not contain n. - Michael Chu, Oct 05 2021. Andrew Howroyd, Nov 20 2021: The conjecture is true: If there are m pairs of numbers that add to n then inclusion/exclusion gives sum(k=0, m, binomial(m,k)*(-1)^k*2^(2*m-2*k)) as the number of sets that don't contain any of those pairs which equals 3^m. For even n , n/2 cannot be included in any set.
Also, number of walks of length n in the graph K_{1,3} (the graph with edges {1,2}, {1,3}, {1,4}) starting at one of the degree 1 vertices. - Sean A. Irvine, May 30 2025

			a(6) = 27; 3^floor(6/2) = 3^floor(3) = 3^3 = 27.

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
D. A. Gewurz and F. Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., 6 (2003), 03.1.6.
Index entries for linear recurrences with constant coefficients, signature (0,3).

Essentially the same as A056449 and A162436.

Cf. A000244, A004526, A016116, A152815, A158020.

a108411 = (3 ^) . flip div 2  -- Reinhard Zumkeller, May 01 2014

[3^Floor(n/2): n in [0..50]]; // Vincenzo Librandi, Aug 17 2011

A108411:=n->3^floor(n/2); seq(A108411(k), k=0..100); # Wesley Ivan Hurt, Nov 01 2013

Table[3^Floor[n/2], {n,0,100}] (* Wesley Ivan Hurt, Nov 01 2013 *)

PARI
```
a(n)=3^floor(n/2);
```

def A108411(n): return 3**(n>>1) # Chai Wah Wu, Oct 28 2024

O.g.f.: (1+x)/(1-3*x^2). - R. J. Mathar, Apr 01 2008
a(n) = 3^(n/2)*((1+(-1)^n)/2+(1-(-1)^n)/(2*sqrt(3))). - Paul Barry, Nov 12 2009
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = (-1)^n*sum(A158020(n,k)*2^k, 0<=k<=n). - Philippe Deléham, Dec 01 2011
a(n) = sum(A152815(n,k)*2^k, 0<=k<=n). - Philippe Deléham, Apr 22 2013
a(n) = 3^A004526(n). - Michel Marcus, Aug 30 2014
E.g.f.: cosh(sqrt(3)*x) + sinh(sqrt(3)*x)/sqrt(3). - Stefano Spezia, Dec 31 2022

Incorrect formula removed by Michel Marcus, Oct 06 2021

A056449 a(n) = 3^floor((n+1)/2).

Original entry on oeis.org

1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467

Offset: 0

Marks R. Nester

One followed by powers of 3 with positive exponent, repeated. - Omar E. Pol, Jul 27 2009

Number of achiral rows of n colors using up to three colors. E.g., for a(3) = 9, the rows are AAA, ABA, ACA, BAB, BBB, BCB, CAC, CBC, and CCC. - Robert A. Russell, Nov 07 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0,3).

Column k=3 of A321391.
Cf. A000007, A000244, A016116, A056391, A056453, A056454, A057427.
Essentially the same as A108411 and A162436.
Cf. A000244 (oriented), A032120 (unoriented), A032086(n>1) (chiral).

[3^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

Riffle[3^Range[0, 20], 3^Range[20]] (* Harvey P. Dale, Jan 21 2015 *)
Table[3^Ceiling[n/2], {n,0,40}] (* or *)
LinearRecurrence[{0, 3}, {1, 3}, 40] (* Robert A. Russell, Nov 07 2018 *)

a(n)=3^floor((n+1)/2); \\ Joerg Arndt, Apr 23 2013

def A056449(n): return 3**(n+1>>1) # Chai Wah Wu, Oct 28 2024

G.f.: (1 + 3*x) / (1 - 3*x^2). - R. J. Mathar, Jul 06 2011 [Adapted to offset 0 by Robert A. Russell, Nov 07 2018]
a(n) = k^ceiling(n/2), where k = 3 is the number of possible colors. - Robert A. Russell, Nov 07 2018
a(n) = C(3,0)*A000007(n) + C(3,1)*A057427(n) + C(3,2)*A056453(n) + C(3,3)*A056454(n). - Robert A. Russell, Nov 08 2018
E.g.f.: cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x). - Stefano Spezia, Dec 31 2022

Edited by N. J. A. Sloane at the suggestion of Klaus Brockhaus, Jul 03 2009

a(0)=1 prepended by Robert A. Russell, Nov 07 2018

A162766 a(n) = 3*a(n-2) for n > 2; a(1) = 4, a(2) = 3.

Original entry on oeis.org

4, 3, 12, 9, 36, 27, 108, 81, 324, 243, 972, 729, 2916, 2187, 8748, 6561, 26244, 19683, 78732, 59049, 236196, 177147, 708588, 531441, 2125764, 1594323, 6377292, 4782969, 19131876, 14348907, 57395628, 43046721, 172186884, 129140163

Offset: 1

Klaus Brockhaus, Jul 13 2009

nonn

Binomial transform is A162559. Second binomial transform is A077236.

Index entries for linear recurrences with constant coefficients, signature (0, 3).

Cf. A074324, A166552, A162559, A077236, A162436.

[ n le 2 select 5-n else 3*Self(n-2): n in [1..34] ];

a(n)=3^(n\2)*4^(n%2) \\ M. F. Hasler, Dec 03 2014

a(n) = (5-3*(-1)^n)*3^(1/4*(2*n-1+(-1)^n))/2.
G.f.: x*(4+3*x)/(1-3*x^2).
a(n) = A074324(n+1) = A166552(n+1) = 3^floor(n/2)*4^(n%2), where n%2 = 0 for n even, 1 for n odd. - M. F. Hasler, Dec 03 2014

G.f. corrected by Klaus Brockhaus, Sep 18 2009

A161728 a(n) = ((3+sqrt(3))(4+sqrt(3))^n-(3-sqrt(3))(4-sqrt(3))^n)/sqrt(12).

Original entry on oeis.org

1, 7, 43, 253, 1465, 8431, 48403, 277621, 1591729, 9124759, 52305595, 299822893, 1718610409, 9851185663, 56467549987, 323674986277, 1855321740385, 10634799101479, 60959210186827, 349421293175389, 2002900612974361, 11480728092514831, 65808116771451955, 377215468968922837

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009

Fourth binomial transform of A162436, inverse binomial transform of A162272.

The inverse binomial transform yields A030192. The binomial transform yields A162272. - R. J. Mathar, Jul 07 2009

Index entries for linear recurrences with constant coefficients, signature (8,-13).

Cf. A162436, A162272, A030192, A161729.

F=nfinit(x^2-3); for(n=0, 20, print1(nfeltdiv(F, ((3+x)*(4+x)^n-(3-x)*(4-x)^n), (2*x))[1], ",")) \\ Klaus Brockhaus, Jun 19 2009

Vec((1-x)/(1-8*x+13*x^2)+O(x^25)) \\ M. F. Hasler, Dec 03 2014

a(n) = 8*a(n-1) - 13(n-2) for n > 1; a(0) = 1, a(1) = 7.
G.f.: (1 - x)/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Jun 19 2009
E.g.f.: exp(4*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Dec 31 2022

Extended beyond a(5) by Klaus Brockhaus, Jun 19 2009

Edited by Klaus Brockhaus, Jul 05 2009; M. F. Hasler, Dec 03 2014

A162813 a(n) = 3*a(n-2) for n > 2; a(1) = 5, a(2) = 3.

Original entry on oeis.org

5, 3, 15, 9, 45, 27, 135, 81, 405, 243, 1215, 729, 3645, 2187, 10935, 6561, 32805, 19683, 98415, 59049, 295245, 177147, 885735, 531441, 2657205, 1594323, 7971615, 4782969, 23914845, 14348907, 71744535, 43046721, 215233605, 129140163, 645700815, 387420489

Offset: 1

Klaus Brockhaus, Jul 20 2009

Binomial transform is A162562.

Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A162562, A162436, A162766.

[ n le 2 select 7-2*n else 3*Self(n-2): n in [1..34] ];

nxt[{a_,b_}]:={b,3a}; NestList[nxt,{5,3},40][[All,1]] (* or *) LinearRecurrence[ {0,3},{5,3},40] (* Harvey P. Dale, May 29 2021 *)

a(n) = (3-2*(-1)^n)*3^(1/4*(2*n-1+(-1)^n)).

G.f.: x*(5+3*x)/(1-3*x^2)

a(35)-a(36) from Yifan Xie, Jul 20 2022

A162272 a(n) = ((1+sqrt(3))(5+sqrt(3))^n + (1-sqrt(3))(5-sqrt(3))^n)/2.

Original entry on oeis.org

1, 8, 58, 404, 2764, 18752, 126712, 854576, 5758096, 38780288, 261124768, 1758081344, 11836068544, 79682895872, 536435450752, 3611330798336, 24311728066816, 163668003104768, 1101822013577728, 7417524067472384, 49935156376013824, 336166034275745792, 2263086902485153792

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009

Fifth binomial transform of A162436, binomial transform of A161728.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-22).

Cf. A162436, A161728.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((1+r)*(5+r)^n+(1-r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 02 2009

seq(expand(((1+sqrt(3))*(5+sqrt(3))^n+(1-sqrt(3))*(5-sqrt(3))^n)*1/2), n = 0 .. 20); # Emeric Deutsch, Jul 05 2009

LinearRecurrence[{10, -22}, {1, 8}, 40] (* Vincenzo Librandi, Feb 03 2018 *)

From Emeric Deutsch, Jul 05 2009: (Start)
G.f.: (1 - 2*x)/(1 - 10*x + 22*x^2).
a(n) = 10*a(n-1) - 22*a(n-2) for n >= 2; a(0)=1, a(1)=8. (End)
E.g.f.: exp(5*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Dec 31 2022

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 05 2009

Extended by Emeric Deutsch, Jul 05 2009

A162852 a(n) = 3*a(n-2) for n > 2; a(1) = 3, a(2) = -1.

Original entry on oeis.org

3, -1, 9, -3, 27, -9, 81, -27, 243, -81, 729, -243, 2187, -729, 6561, -2187, 19683, -6561, 59049, -19683, 177147, -59049, 531441, -177147, 1594323, -531441, 4782969, -1594323, 14348907, -4782969, 43046721, -14348907, 129140163

Offset: 1

Klaus Brockhaus, Jul 14 2009

sign

Third binomial transform is A162560.

Equivalently, 3^n followed by -3^(n-1), n > 0. - Muniru A Asiru, Oct 25 2018

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3).

Cf. A162560, A162436, A162766, A162813.

a:=[3,-1];; for n in [3..25] do a[n]:=3*a[n-2]; od; a; # Muniru A Asiru, Oct 25 2018

[ n le 2 select 7-4*n else 3*Self(n-2): n in [1..34] ];

seq(op([3^n,-3^(n-1)]),n=1..18); # Muniru A Asiru, Oct 25 2018

Rest[CoefficientList[Series[x*(3-x)/(1-3*x^2), {x, 0, 40}], x]] (* or *) LinearRecurrence[{0,3}, {3,-1}, 40] (* G. C. Greubel, Oct 24 2018 *)

x='x+O('x^40); Vec(x*(3-x)/(1-3*x^2)) \\ G. C. Greubel, Oct 24 2018

a(n) = ((4-5*(-1)^n)*3^(1/4*(2*n-1+(-1)^n)))/3.
G.f.: x*(3-x)/(1-3*x^2). [corrected by Klaus Brockhaus, Sep 18 2009]
E.g.f.: (1 - cosh(sqrt(3)*x) + 3*sqrt(3)*sinh(sqrt(3)*x))/3. - G. C. Greubel, Oct 24 2018

A164560 Partial sums of A164532.

Original entry on oeis.org

1, 5, 11, 35, 71, 215, 431, 1295, 2591, 7775, 15551, 46655, 93311, 279935, 559871, 1679615, 3359231, 10077695, 20155391, 60466175, 120932351, 362797055, 725594111, 2176782335, 4353564671, 13060694015, 26121388031, 78364164095

Offset: 1

Klaus Brockhaus, Aug 16 2009

Interleaving of A164559 and A024062 without initial term 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..900
Index entries for linear recurrences with constant coefficients, signature (1,6,-6).

Cf. A164532, A164123 (partial sums of A162436), A164559 (6^n/3-1), A024062 (6^n-1), A026549.

T:=[ n le 2 select 3*n-2 else 6*Self(n-2): n in [1..28] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

a(n) = 6*a(n-2)+5 for n > 2; a(1) = 1, a(2) = 5.
a(n) = (3-(-1)^n)*6^(1/4*(2*n-1+(-1)^n))/2-1.
G.f.: x*(1+4*x)/((1-x)*(1-6*x^2)).
a(n) = A026549(n) - 1.

A167710 a(n) = 102^n - 3A083658(n+2).

Original entry on oeis.org

1, 5, 13, 35, 79, 185, 397, 875, 1831, 3905, 8053, 16835, 34399, 70985, 144157, 294875, 596311, 1212305, 2444293, 4947635, 9954319, 20085785, 40348717, 81228875, 162989191, 327572705, 656739733, 1318262435, 2641307839, 5296964585, 10608278077, 21259602875

Offset: 0

Paul Curtz, Nov 10 2009

The sequence can be defined as the row sums of the triangle T(n,k)
.1;
.3,.2;
.3,.6,.4;
.9,.6,12,.8;
.9,18,12,24,16;
27,18,36,24,48,32;
with left column A162436, diagonal the powers of 2, and the recurrence T(n+2,k) = 3*T(n,k).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-6).

LinearRecurrence[{2,3,-6},{1,5,13},40] (* Harvey P. Dale, Oct 03 2014 *)

a(n+1) - 2*a(n) = A162436(n+2).
a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3).
G.f.: (1+3*x)/((2*x-1) * (3*x^2-1)). - R. J. Mathar, Feb 27 2010

Replaced cross-references by link to the index - R. J. Mathar, Feb 27 2010

cp's OEIS Frontend

A164123 Partial sums of A162436.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A108411 a(n) = 3^floor(n/2). Powers of 3 repeated.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A056449 a(n) = 3^floor((n+1)/2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

Extensions

A162766 a(n) = 3*a(n-2) for n > 2; a(1) = 4, a(2) = 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

PARI

Formula

Extensions

A161728 a(n) = ((3+sqrt(3))*(4+sqrt(3))^n-(3-sqrt(3))*(4-sqrt(3))^n)/sqrt(12).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A162813 a(n) = 3*a(n-2) for n > 2; a(1) = 5, a(2) = 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

A161728 a(n) = ((3+sqrt(3))(4+sqrt(3))^n-(3-sqrt(3))(4-sqrt(3))^n)/sqrt(12).

A162272 a(n) = ((1+sqrt(3))(5+sqrt(3))^n + (1-sqrt(3))(5-sqrt(3))^n)/2.

A167710 a(n) = 102^n - 3A083658(n+2).