A002001 -id:A002001 - cp's OEIS frontend

A055841 Number of compositions of n into 3*j-1 kinds of j's for all j >= 1.

1, 2, 9, 36, 144, 576, 2304, 9216, 36864, 147456, 589824, 2359296, 9437184, 37748736, 150994944, 603979776, 2415919104, 9663676416, 38654705664, 154618822656, 618475290624, 2473901162496, 9895604649984, 39582418599936, 158329674399744, 633318697598976, 2533274790395904, 10133099161583616, 40532396646334464, 162129586585337856, 648518346341351424

Offset: 0

Barry E. Williams, May 30 2000

First differences of A002001.
For n >= 2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 19 2007
Convolved with [1, 2, 3, ...] = powers of 4: [1, 4, 16, 64, ...]. - Gary W. Adamson, Jun 04 2009
a(n) is the number of generalized compositions of n when there are 3 *i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Daniel Birmajer, Juan B. Gil, Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000302 and A002001.

Essentially the same as A002063.

Join[{1, 2}, 9*4^Range[0, 30]] (* Jean-François Alcover, Jul 21 2018 *)

a(n) = 9*4^(n-2), a(0)=1, a(1)=2.
a(0)=1, a(1)=2, a(3)=9, a(n+1)=4*a(n) for n >= 3.
G.f.: (1-x)^2/(1-4*x).
G.f.: 1/(1 - Sum_{j>=1} (3*j-1)*x^j). - Joerg Arndt, Jul 06 2011
a(n) = 4*a(n-1) + (-1)^n*C(2,2-n).
a(n) = Sum_{k=0..n} A201780(n,k)*2^k. - Philippe Deléham, Dec 05 2011

New name from Joerg Arndt, Jul 06 2011

A094554 Number of closed walks of length n at a base vertex of a truncated tetrahedron (triangular prism).

Original entry on oeis.org

1, 0, 3, 2, 19, 30, 143, 322, 1179, 3110, 10183, 28842, 89939, 262990, 802623, 2380562, 7196299, 21479670, 64657463, 193535482, 581480259, 1742693150, 5231574703, 15687733602, 47077181819, 141203583430, 423666674343

Offset: 0

Paul Barry, May 11 2004

For n > 0, 6*a(n) is the number of 3-colorings of the prism of size 2 X n (i.e., C_2 X C_n).More generally, the number of k-colorings of the prism of size 2 X n is given by (k^2 - 3*k + 3)^n + (k - 1) * ((3 - k)^n + (1 - k)^n) + k^2 - 3*k + 1 (chromatic polynomial). - Sela Fried, Oct 07 2023

Andrew Howroyd, Table of n, a(n) for n = 0..1000
N. L. Biggs, R. M. Damerell, and D. A. Sands, Recursive families of graphs, Journal of Combinatorial Theory Series B Volume 12 (1972), 123-131.
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 3.
Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

Cf. A002001, A025192, A094555, A094556, A328778.

LinearRecurrence[{2, 5, -6}, {1, 0, 3, 2}, 30] (* Greg Dresden, Jun 19 2021 *)

a(n) = if(n==0, 1, (1 + 3^n + 2*(-2)^n)/6) \\ Andrew Howroyd, Jun 14 2021

G.f.: (1 - 2*x - 2*x^2 + 2*x^3)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
a(n) = 1/6 + 3^n/6 + (-2)^n/3 for n > 0.
a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
E.g.f.: exp(-2*x)*(1 + exp(2*x))*(2 + exp(3*x))/6. - Stefano Spezia, Sep 26 2023

A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

Original entry on oeis.org

1, 3, 5, 7, 5, 11, 17, 15, 5, 11, 17, 23, 29, 35, 41, 31, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 63, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 127, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137

Offset: 1

Omar E. Pol, Apr 04 2015

Row sums give A002001.
The sum of all terms of first n rows gives A000302(n-1).
The rows of triangle A256263 converge to this sequence.
Rows converge to A016969.
First 11 terms agree with A151548.

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
3;
5,7;
5,11,17,15;
5,11,17,23,29,35,41,31;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,63;
5,11,17,23,29,35,41,47,53,59,65,71,77,83,89,95,101,107,113,119,125,131,137,143,149,155,161,167,173,179,185,127;
...
Illustration of initial terms in the fourth quadrant of the square grid:
------------------------------------------------------------------------
n   a(n)             Compact diagram
------------------------------------------------------------------------
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
1    1      |_| | | |_ _  | |_ _ _ _ _ _  | |
2    3      |_ _| | |_  | | |_ _ _ _ _  | | |
3    5      |_ _ _| | | | | |_ _ _ _  | | | |
4    7      |_ _ _ _| | | | |_ _ _  | | | | |
5    5      | | |_ _ _| | | |_ _  | | | | | |
6   11      | |_ _ _ _ _| | |_  | | | | | | |
7   17      |_ _ _ _ _ _ _| | | | | | | | | |
8   15      |_ _ _ _ _ _ _ _| | | | | | | | |
9    5      | | | | | | |_ _ _| | | | | | | |
10  11      | | | | | |_ _ _ _ _| | | | | | |
11  17      | | | | |_ _ _ _ _ _ _| | | | | |
12  23      | | | |_ _ _ _ _ _ _ _ _| | | | |
13  29      | | |_ _ _ _ _ _ _ _ _ _ _| | | |
14  35      | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
15  41      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
16  31      |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
a(n) is also the number of cells in the n-th region of the diagram.
It appears that A241717 can be represented by a similar diagram.

Ivan Neretin, Table of n, a(n) for n = 1..8192

Cf. A000225, A000302, A002001, A011782, A016969, A141548, A241717, A256260, A256261, A256263, A256264.

Nest[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 7] (* Ivan Neretin, Feb 14 2017 *)

A134683 Expansion of 1+x(2+3x)/(1-4*x^2).

Original entry on oeis.org

1, 2, 3, 8, 12, 32, 48, 128, 192, 512, 768, 2048, 3072, 8192, 12288, 32768, 49152, 131072, 196608, 524288, 786432, 2097152, 3145728, 8388608, 12582912, 33554432, 50331648, 134217728, 201326592, 536870912, 805306368, 2147483648, 3221225472

Offset: 0

Paul Curtz, Jan 26 2008, Feb 09 2008

A002001 interleaved with A081294. Gary W. Adamson, Jul 08 2012

Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A001045, A045623, A131577, A002001, A081294, A000302.

A134683 := proc(n)
    if n =0 then
        1 ;
    else
        -(-1)^n*A131577(n-1)+2*procname(n-1) ;
    end if;
end proc: # R. J. Mathar, Jul 22 2012

CoefficientList[Series[1 + x*(2 + 3*x)/(1 - 4*x^2), {x, 0, 32}], x] (* Amiram Eldar, Aug 18 2022 *)

a(n) = 2*a(n-1)-(-1)^n*A131577(n-1), n>0.
a(n) = 4*a(n-2), n>2. Gary W. Adamson, Jul 08 2012
a(n) = 2^(n-3)*(7-(-1)^n), n>0. - R. J. Mathar, Jul 22 2012
Sum_{n>=0} 1/a(n) = 19/9. - Amiram Eldar, Aug 18 2022

Edited by N. J. A. Sloane, Feb 20 2008

A193723 Mirror of the fusion triangle A193722.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 18, 21, 8, 1, 54, 81, 45, 11, 1, 162, 297, 216, 78, 14, 1, 486, 1053, 945, 450, 120, 17, 1, 1458, 3645, 3888, 2295, 810, 171, 20, 1, 4374, 12393, 15309, 10773, 4725, 1323, 231, 23, 1, 13122, 41553, 58320, 47628, 24948, 8694, 2016, 300, 26, 1

Offset: 0

Clark Kimberling, Aug 04 2011

A193723 is obtained by reversing the rows of the triangle A193722.
Triangle T(n,k), read by rows, given by [2,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011
From Philippe Deléham, Nov 14 2011: (Start)
Riordan array ((1-x)/(1-3x), x/(1-3x)).
Product A200139*A007318 as infinite lower triangular arrays. (End)

			First six rows:
    1;
    2,   1;
    6,   5,   1;
   18,  21,   8,   1;
   54,  81,  45,  11,   1;
  162, 297, 216,  78,  14,   1;

Cf. A084938, A193722, A052924 (antidiagonal sums), Diagonals: A000012, A016789, A081266, Columns: A025192, A081038.

z = 9; a = 1; b = 1; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193722 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193723 *)

Write w(n,k) for the triangle at A193722.  The triangle at A193723 is then given by w(n,n-k).
T(n,k) = T(n-1,k-1) + 3*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
From Philippe Deléham, Nov 14 2011: (Start)
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A025192(n), A002001(n), A005054(n), A052934(n), A055272(n), A055274(n), A055275(n), A052268(n), A055276(n), A196731(n) for x=-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.
T(n,k) = Sum_{j>=0} T(n-1-j,k-1)*3^j.
G.f.: (1-x)/(1-(3+y)*x). (End)

A196663 Expansion of g.f. (1-4x)/(1-13x).

Original entry on oeis.org

1, 9, 117, 1521, 19773, 257049, 3341637, 43441281, 564736653, 7341576489, 95440494357, 1240726426641, 16129443546333, 209682766102329, 2725875959330277, 35436387471293601, 460673037126816813, 5988749482648618569, 77853743274432041397, 1012098662567616538161

Offset: 0

Philippe Deléham, Oct 05 2011

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

Cf. A002001, A193722, A196661, A196662.

a(0) = 1, a(n) = 9*13^(n-1) for n>0.
a(n) = Sum_{k=0..n} A193722(n,k)*4^k.
From Elmo R. Oliveira, Mar 18 2025: (Start)
E.g.f.: (9*exp(13*x) + 4)/13.
a(n) = 13*a(n-1). (End)

A085388 First differences of n^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 18, 8, 0, 1, 5, 20, 48, 54, 16, 0, 1, 6, 30, 100, 192, 162, 32, 0, 1, 7, 42, 180, 500, 768, 486, 64, 0, 1, 8, 56, 294, 1080, 2500, 3072, 1458, 128, 0, 1, 9, 72, 448, 2058, 6480, 12500, 12288, 4374, 256, 0, 1, 10, 90, 648

Offset: 1

Paul Barry, Jun 30 2003

T(n,k) is the number of k-digit numbers in base n; n,k >= 2. - Mohammed Yaseen, Nov 11 2022

			Rows begin
  1,   0,   0,   0,   0, ...
  1,   1,   2,   4,   8, ...
  1,   2,   6,  18,  54, ...
  1,   3,  12,  48, 192, ...
  1,   4,  20, 100, 500, ...

Rows include A011782, A025192, A002001, A005054, A052934, A055272, A055274, A055275, A052268, A055276.
Diagonals include A053506, A085389, A085390.
Row-wise binomial transform is A083064.

T(n,k) = (n-1)*n^(k-1) + 0^k/n. - Corrected by Mohammed Yaseen, Nov 11 2022

T(n,0) = 1; T(n,k) = n^k - n^(k-1) for k >= 1. - Mohammed Yaseen, Nov 11 2022

Offset corrected by Mohammed Yaseen, Nov 11 2022

A098646 Trace sequence of 3 X 3 Krawtchouk matrix.

Original entry on oeis.org

3, 2, 12, 8, 48, 32, 192, 128, 768, 512, 3072, 2048, 12288, 8192, 49152, 32768, 196608, 131072, 786432, 524288, 3145728, 2097152, 12582912, 8388608, 50331648, 33554432, 201326592, 134217728, 805306368, 536870912, 3221225472, 2147483648

Offset: 0

Paul Barry, Sep 18 2004

Let A=[1,1,1;2,0,-2;1,-1,1], the 3 X 3 Krawtchouk matrix. Then a(n)=trace(A^n).

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

Cf. A074535, A098647.

[(-2)^n+2*2^n: n in [0..45]]; // Vincenzo Librandi, Jun 11 2011

G.f.: (3+2*x)/((1+2*x)*(1-2*x)).
a(n) = (-2)^n+2*2^n.
Recurrence: a(n) = 4a(n-2), a(0)=3, a(1)=2. - Ralf Stephan, Jul 17 2013
a(2n+1)=A081294(n+1). a(2n)=A002001(n+1). - R. J. Mathar, Nov 11 2013

A330941 a(n) is the greatest value whose binary representation can be obtained by interleaving (or shuffling) two copies of the binary representation of n.

Original entry on oeis.org

0, 3, 12, 15, 48, 53, 60, 63, 192, 201, 212, 219, 240, 245, 252, 255, 768, 785, 804, 819, 848, 853, 876, 887, 960, 969, 980, 987, 1008, 1013, 1020, 1023, 3072, 3105, 3140, 3171, 3216, 3237, 3276, 3303, 3392, 3401, 3412, 3435, 3504, 3509, 3548, 3567, 3840, 3857

Offset: 0

Rémy Sigrist, Jan 04 2020

The binary representation of all positive terms are square binary words (see A191755).

			The first terms, alongside the binary representations of n and of a(n), are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ----------
   0     0       0           0
   1     3       1          11
   2    12      10        1100
   3    15      11        1111
   4    48     100      110000
   5    53     101      110101
   6    60     110      111100
   7    63     111      111111
   8   192    1000    11000000
   9   201    1001    11001001
  10   212    1010    11010100
  11   219    1011    11011011
  12   240    1100    11110000

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Logarithmic scatterplot of the first difference of the first 2^13 terms
Rémy Sigrist, PARI program for A330941
Index entries for sequences related to binary expansion of n

See A330940 for the minimum variant.

Cf. A002001, A024036, A191755, A193020.

PARI
```
See Links section.
```

a(2^k) = 3*4^k = A002001(k+1) for any k >= 0.
a(2^k-1) = 4^k-1 = A024036(k) for any k >= 0.
a(n) >= A330940(n).

A163913 Number of integers i in range [A000302(n-1)..A024036(n)] of permutation A163355/A163356 with A163915(i)=i, but not A163355(i)=i.

Original entry on oeis.org

0, 0, 6, 3, 30, 27, 162, 171, 885, 987, 4839, 5502, 26436, 30216

Offset: 0

Antti Karttunen, Sep 19 2009

nonn

a(n) = 3*A163914(n). See also A163903.

cp's OEIS Frontend

A055841 Number of compositions of n into 3*j-1 kinds of j's for all j >= 1.

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A094554 Number of closed walks of length n at a base vertex of a truncated tetrahedron (triangular prism).

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A256258 Triangle read by rows in which the row lengths are the terms of A011782 and row n lists the terms of A016969 except for the right border which gives the positive terms of A000225.

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A134683 Expansion of 1+x*(2+3*x)/(1-4*x^2).

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A193723 Mirror of the fusion triangle A193722.

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A196663 Expansion of g.f. (1-4*x)/(1-13*x).

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A085388 First differences of n^k.

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A098646 Trace sequence of 3 X 3 Krawtchouk matrix.

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A134683 Expansion of 1+x(2+3x)/(1-4*x^2).

A196663 Expansion of g.f. (1-4x)/(1-13x).