A152815 -id:A152815 - 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

A152198 Triangle read by rows, A007318 rows repeated.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1

Offset: 0

Gary W. Adamson, Nov 28 2008

Eigensequence of the triangle = A051163: (1, 2, 5, 12, 30, 76,...)
Another version of A152815. - Philippe Deléham, Dec 13 2008
Row sums : A016116(n); Diagonal sums: A000931(n+5). - Philippe Deléham, Dec 13 2008
Triangle, with zeros omitted, given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 16 2012
Sums along rising diagonals are A134816. - John Molokach, Jul 09 2013

			The triangle starts
1;
1;
1, 1;
1, 1;
1, 2, 1;
1, 2, 1;
1, 3, 3, 1;
1, 3, 3, 1;
1, 4, 6, 4, 1;
1, 4, 6, 4, 1;
1, 5, 10, 10, 5, 1;
1, 5, 10, 10, 5, 1;
...
Triangle (1,0,-1,0,0,...) DELTA (0,1,-1,0,0,...) begins:
1
1, 0
1, 1, 0
1, 1, 0, 0
1, 2, 1, 0, 0
1, 2, 1, 0, 0, 0
1, 3, 3, 1, 0, 0, 0
1, 3, 3, 1, 0, 0, 0, 0
1, 4, 6, 4, 1, 0, 0, 0, 0
1, 4, 6, 4, 1, 0, 0, 0, 0, 0
1, 5, 10, 10, 5, 1, 0, 0, 0, 0, 0...

Cf. A007318, A133156, A051163.

t[n_, k_] := Binomial[ Floor[n/2], k]; Table[t[n, k], {n, 0, 17}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Sep 13 2012 *)

Triangle read by rows, Pascal's triangle rows repeated.
Equals inverse binomial transform of A133156 unsigned.
G.f. : (1+x)/(1-(1+y)*x^2). - Philippe Deléham, Jan 16 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057077(n), A019590(n+1), A000012(n), A016116(n), A108411(n), A074872(n+1) for x = -2, -1, 0, 1, 2, 4 respectively. - Philippe Deléham, Jan 16 2012
T(n,k) = A065941(n-k, n-2*k) = abs(A108299(n-k, n-2*k)). - Johannes W. Meijer, Sep 05 2013

More terms from Philippe Deléham, Dec 14 2008

A152828 Triangle read by rows, A007318 rows repeated three times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 7, 21

Offset: 0

Philippe Deléham, Dec 13 2008

Diagonal sums : A079398 . Lengths of row are : 1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,... (A008620) .

			Triangle begins : 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; 1,3,3,1 ; 1,3,3,1 ; 1,4,6,4,1 ; ...

Cf. A007318, A152198, A152815

{#,#,#}&/@Table[Binomial[n,k],{n,0,11},{k,0,n}]//Flatten (* Harvey P. Dale, Jul 22 2024 *)

A152830 Triangle read by rows, A007318 rows repeated four times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 6, 15, 20, 15, 6, 1, 1

Offset: 0

Philippe Deléham, Dec 13 2008

Diagonal sums : A103372 .

			Triangle begins : 1 ; 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A152198, A152815, A152828.

Flatten[Table[#,{4}]&/@Table[Binomial[n,k],{n,0,6},{k,0,n}]] (* Harvey P. Dale, Sep 23 2015 *)

A213173 a(n) = 4^floor(n/2), Powers of 4 repeated.

Original entry on oeis.org

1, 1, 4, 4, 16, 16, 64, 64, 256, 256, 1024, 1024, 4096, 4096, 16384, 16384, 65536, 65536, 262144, 262144, 1048576, 1048576, 4194304, 4194304, 16777216, 16777216, 67108864, 67108864, 268435456, 268435456, 1073741824, 1073741824, 4294967296, 4294967296

Offset: 0

Philippe Deléham, Apr 14 2013

1, followed by A056450. - Joerg Arndt, Sep 17 2013
Binomial transform of A084567.
Unsigned version of A164111.
Also, number of walks of length n on the star S_4 = K_1,4 starting at a specific vertex of degree 1. - Sean A. Irvine, Jun 03 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sean A. Irvine, Walks on Graphs.
Wikipedia, Star (graph theory)
Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A000302, A056450, A164111.

LinearRecurrence[{0, 4}, {1, 1}, 40] (* T. D. Noe, Apr 17 2013 *)
CoefficientList[Series[(1 + x)/(1 - 4*x^2), {x,0,50}], x] (* G. C. Greubel, Apr 30 2017 *)
With[{p4=4^Range[0,30]},Riffle[p4,p4]] (* Harvey P. Dale, Mar 17 2022 *)

a(n)=4^(n\2) \\ Charles R Greathouse IV, Oct 03 2016

O.g.f.: (1+x)/(1-4*x^2).
a(n) = A016116(n)^2.
a(n) = 4*a(n-2) with a(0) = a(1) = 1.
a(n) = a(n-1)*a(n-2)/a(n-3) = 4^A004526(n).
a(n) = sum(A152815(n,k)*3^k, 0<=k<=n). - Philippe Deléham, Apr 22 2013

A152831 Triangle read by rows, A007318 repeated five times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1, 5

Offset: 0

Philippe Deléham, Dec 13 2008

Diagonal sums : A103373 .

			Triangle begins : 1 ; 1 ; 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A152198, A152815, A152828, A152830.

Flatten[Table[#, {5}] & /@ Table[Binomial[n, k], {n,0,10}, {k, 0, n}]] (* G. C. Greubel, May 03 2017 *)

A152844 Triangle read by rows, A007318 rows repeated six times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 5, 10, 10, 5, 1, 1

Offset: 0

Philippe Deléham, Dec 14 2008

Diagonal sums : A103374 .

			Triangle begins : 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A152198, A152815, A152828, A152830, A152831.

Flatten[Table[#, {6}] & /@ Table[Binomial[n, k], {n,0,10}, {k, 0, n}]] (* G. C. Greubel, May 03 2017 *)

A152845 Triangle read by rows, A007318 rows repeated seven times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1

Offset: 0

Philippe Deléham, Dec 14 2008

Diagonal sums : A103375 .

			Triangle begins : 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A007318, A152198, A152815, A152828, A152830, A152831, A152844

Flatten[Table[#, {7}] & /@ Table[Binomial[n, k], {n, 0, 10}, {k, 0, n}]] (* G. C. Greubel, May 03 2017 *)

A152842 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 1, 7, 15, 9, 1, 8, 22, 24, 9, 1, 11, 46, 90, 81, 27, 1, 12, 57, 136, 171, 108, 27, 1, 15, 93, 307, 579, 621, 351, 81, 1, 16, 108, 400, 886, 1200, 972, 432, 81, 1, 19, 156, 724, 2086, 3858, 4572, 3348, 1377, 243, 1, 20, 175, 880, 2810, 5944, 8430, 7920

Offset: 0

Philippe Deléham, Dec 14 2008

			The triangle T(n,k) begins:
n\k  0   1    2     3     4      5      6      7      8      9     10    11   12
0:   1
1:   1   3
2:   1   4    3
3:   1   7   15     9
4:   1   8   22    24     9
5:   1  11   46    90    81     27
6:   1  12   57   136   171    108     27
7:   1  15   93   307   579    621    351     81
8:   1  16  108   400   886   1200    972    432     81
9:   1  19  156   724  2086   3858   4572   3348   1377    243
10:  1  20  175   880  2810   5944   8430   7920   4725   1620    243
11:  1  23  235  1405  5450  14374  26262  33210  28485  15795   5103   729
12:  1  24  258  1640  6855  19824  40636  59472  61695  44280  20898  5832  729
... reformatted and extended. - _Franck Maminirina Ramaharo_, Feb 28 2018

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Cf. A152815, A007318, A064861.

a152842 n k = a152842_tabl !! n !! k
a152842_row n = a152842_tabl !! n
a152842_tabl = map fst $ iterate f ([1], 3) where
   f (xs, z) = (zipWith (+) ([0] ++ map (* z) xs) (xs ++ [0]), 4 - z)
-- Reinhard Zumkeller, May 01 2014

T(n,k) = T(n-1,k) + (2-(-1)^n)*T(n-1,k-1).
Sum_{k=0..n} T(n,k) = A094015(n).
T(n,n) = A108411(n+1).
T(2n,n) = A069835(n).
G.f.: (1+x+x*y)/(1-x^2-4*x^2*y-3*x^2*y^2). - Philippe Deléham , Nov 09 2013
T(n,k) = T(n-2,k) + 4*T(n-2,k-1) + 3*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = 3, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 09 2013

A152846 Triangle read by rows, A007318 rows repeated eight times .

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1, 1, 4, 6, 4, 1

Offset: 0

Philippe Deléham, Dec 14 2008

Diagonal sums : A103376 .

			Triangle begins : 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,2,1 ; 1,3,3,1 ; ...

Cf. A007318, A152198, A152815, A152828, A152830, A152831, A152844, A152845.

Flatten[Table[#, {8}] & /@ Table[Binomial[n, k], {n, 0, 10}, {k, 0, n}]] (* G. C. Greubel, May 03 2017 *)

cp's OEIS Frontend

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

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A152198 Triangle read by rows, A007318 rows repeated.

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A152828 Triangle read by rows, A007318 rows repeated three times .

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A152830 Triangle read by rows, A007318 rows repeated four times .

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A213173 a(n) = 4^floor(n/2), Powers of 4 repeated.

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A152831 Triangle read by rows, A007318 repeated five times .

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A152844 Triangle read by rows, A007318 rows repeated six times .

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A152845 Triangle read by rows, A007318 rows repeated seven times .