A158020 -id:A158020 - cp's OEIS frontend

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

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

A094014 Expansion of (1-2x)/(1-8x^2).

Original entry on oeis.org

1, -2, 8, -16, 64, -128, 512, -1024, 4096, -8192, 32768, -65536, 262144, -524288, 2097152, -4194304, 16777216, -33554432, 134217728, -268435456, 1073741824, -2147483648, 8589934592, -17179869184, 68719476736, -137438953472

Offset: 0

Paul Barry, Apr 21 2004

Second inverse binomial transform of A094013. Third inverse binomial transform of A000129(2n-1).

The unsigned sequence has g.f. (1+2*x)/(1-8*x^2) and abs(a(n)) = 2^(3*n/2)*(1/2 + sqrt(2)/4 + (1/2 - sqrt(2)/4)*(-1)^n).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,8).

Cf. A000129, A016116, A094013, A113836, A158020.

[n le 2 select (-2)^(n-1) else 8*Self(n-2): n in [1..41]]; // G. C. Greubel, Dec 04 2021

LinearRecurrence[{0,8}, {1,-2}, 40] (* G. C. Greubel, Dec 04 2021 *)

[(-2)^n*2^(n//2) for n in (0..40)] # G. C. Greubel, Dec 04 2021

a(n) = (2*sqrt(2))^n*(1/2 - sqrt(2)/4) + (-2*sqrt(2))^n*(1/2 + sqrt(2)/4).
a(n) = (-2)^n * A016116(n). - R. J. Mathar, Apr 28 2008
Abs(a(n)) = A113836(n+1) - A113836(n) for n > 0. - Reinhard Zumkeller, Feb 22 2010
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = Sum_{k=0..n} A158020(n,k)*3^k. - Philippe Deléham, Dec 01 2011
E.g.f.: cosh(2*sqrt(2)*x) - (1/sqrt(2))*sinh(2*sqrt(2)*x). - G. C. Greubel, Dec 04 2021

A157785 Triangle of coefficients of the polynomials defined by q^binomial(n, 2)*QPochhammer(x, 1/q, n), where q = -2.

Original entry on oeis.org

1, 1, -1, -2, 1, 1, -8, 6, 3, -1, 64, -40, -30, 5, 1, 1024, -704, -440, 110, 11, -1, -32768, 21504, 14784, -3080, -462, 21, 1, -2097152, 1409024, 924672, -211904, -26488, 1806, 43, -1, 268435456, -178257920, -119767040, 26199040, 3602368, -204680, -7310, 85, 1

Offset: 0

Roger L. Bagula, Mar 06 2009

Triangle T(n,k), 0 <= k <= n, read by rows given by [1, q-1, q^2, q^3-q, q^4, q^5-q^2, q^6, q^7-q^3, q^8, ...] DELTA [-1, 0, -q, 0, -q^2, 0, -q^3, 0, -q^4, 0, ...] (for q=-2) = [1, -3, 4, -6, 16, -36, 64,...] DELTA [ -1, 0, 2, 0, -4, 0, 8, 0, -16, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2009

			Triangle begins as:
          1;
          1,         -1;
         -2,          1,          1;
         -8,          6,          3,       -1;
         64,        -40,        -30,        5,       1;
       1024,       -704,       -440,      110,      11,      -1;
     -32768,      21504,      14784,    -3080,    -462,      21,     1;
   -2097152,    1409024,     924672,  -211904,  -26488,    1806,    43, -1;
  268435456, -178257920, -119767040, 26199040, 3602368, -204680, -7310, 85, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. this sequence (q=-2), A158020 (q=-1), A007318 (q=1), A157963 (q=2).
Cf. A135950 (q=2; alternative).
Cf. A022166, A135950.

p[x_, n_, q_]:= q^Binomial[n, 2]*QPochhammer[x, 1/q, n];
Table[CoefficientList[Series[p[x, n, -2], {x,0,n}], x], {n,0,10}]//Flatten (* G. C. Greubel, Nov 29 2021 *)

Sum_{k=0..n} T(n, k) = 0^n.
From G. C. Greubel, Nov 29 2021: (Start)
T(n, k) = [x^k] coefficients of the polynomials defined by q^binomial(n, 2)*QPochhammer(x, 1/q, n), where q = -2.
T(n, k) = [x^k] Product_{j=0..n-1} (q^j - x). (End)

Edited by G. C. Greubel, Nov 29 2021

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

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A094014 Expansion of (1-2*x)/(1-8*x^2).

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A157785 Triangle of coefficients of the polynomials defined by q^binomial(n, 2)*QPochhammer(x, 1/q, n), where q = -2.

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Mathematica

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A094014 Expansion of (1-2x)/(1-8x^2).