A010684 -id:A010684 - cp's OEIS frontend

A062172 Table T(n,k) by antidiagonals of n^(k-1) mod k [n,k > 0].

0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 3, 1, 1, 0, 1, 2, 1, 0, 1, 0, 0, 1, 1, 3, 1, 1, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 0, 0, 1, 4, 3, 1, 5, 1, 3, 1, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 3, 7, 5, 1, 1, 1, 1, 1, 1, 0, 1, 8, 1, 4, 7, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1, 3, 1, 5, 0, 7, 1, 3, 0, 3, 0, 1, 0

Offset: 1

Henry Bottomley, Jun 12 2001

			T(5,3)=5^(3-1) mod 3=25 mod 3=1. Rows start (0,1,1,1,1,...), (0,0,1,0,1,...), (0,1,0,3,1...), (0,0,1,0,1,...), (0,1,1,1,0,...), ...

Cf. A002997, A060154. Rows include A057427, A062173, A062174, A062175, A062176. Columns include A000004, A000035, A011655, A010684 with interleaved 0's, A011558, A010875. Diagonals include all the rows again and A000004 and A009001 unsigned.

A282124 Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 430", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 3, 15, 11, 63, 43, 255, 171, 1023, 683, 4095, 2731, 16383, 10923, 65535, 43691, 262143, 174763, 1048575, 699051, 4194303, 2796203, 16777215, 11184811, 67108863, 44739243, 268435455, 178956971, 1073741823, 715827883, 4294967295, 2863311531, 17179869183

Offset: 0

Robert Price, Feb 06 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A282121, A282122, A282123.

Cf. A001045, A010684, A024036, A081631, A098646.

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 430; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i ,1, stages - 1}]

Conjectures from Colin Barker, Feb 07 2017: (Start)
a(n) = (-1 + 2*(-1)^n - (-1)^n*2^(1+n) + 2^(2+n)) / 3.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
G.f.: (1 + 3*x - 2*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
(End)
Conjectures from Paul Curtz, Jun 10 2019: (Start)
a(n) = A001045(n+1)*(period 2: repeat[1, 3]).
a(n+4) = a(n) + 10*A081631(n).
a(2*n+1) = 2^(2*n+2) -1.
a(n+2) = a(n) + A098646(n+1).
(End)

A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

Original entry on oeis.org

1, 4, 2, 3, 5, 8, 6, 7, 9, 12, 10, 11, 13, 16, 14, 15, 17, 20, 18, 19, 21, 24, 22, 23, 25, 28, 26, 27, 29, 32, 30, 31, 33, 36, 34, 35, 37, 40, 38, 39, 41, 44, 42, 43, 45, 48, 46, 47, 49, 52, 50, 51, 53, 56, 54, 55, 57, 60, 58, 59, 61, 64, 62, 63, 65, 68, 66, 67

Offset: 1

Guenther Schrack, Mar 24 2017

Partition the natural number sequence into quadruples starting with (1, 2, 3, 4); swap the third and fourth element, then swap the second and third element; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Inverse: A056699.
Subsequences:
  elements with odd index: A042963(n), n > 0
  elements with even index: A014601(A103889(n)), n > 0
  odd elements: A005408(n-1), n > 0
  indices of odd elements: A042948(n), n > 0
  even elements: 2*A103889(n), n > 0
  indices of even elements: A042964(n), n > 0
Sequence of fixed points: A016813(n-1), n > 0
Every fourth element starting at:
  n=1: a(4n-3) = 4n-3 = A016813(n-1), n > 0
  n=2: a(4n-2) = 4n = A008586(n), n > 0
  n=3: a(4n-1) = 4n-2 = A016825(n-1), n > 0
  n=4: a(4n) = 4n-1 = A004767(n-1), n > 0
Difference between pairs of elements:
  a(2n+1)-a(2n-1) = A010684(n-1), n > 0
Compositions:
  a(n) = A133256(A116966(n-1)), n > 0
  a(A042948(n)) = A005408(n-1), n > 0
  A067060(a(n)) = A092486(n), n > 0

a = [1 4 2 3];
max = (specify);
for n = 5:max
   a(n) = a(n-4) + 4;
end;

Table[n + ((-1)^n - (-1)^(n (n - 1)/2) (1 + 2 (-1)^n))/2, {n, 68}] (* Michael De Vlieger, Mar 28 2017 *)
LinearRecurrence[{1,0,0,1,-1},{1,4,2,3,5},70] (* or *) {#[[1]],#[[4]], #[[2]],#[[3]]}&/@Partition[Range[70],4]//Flatten(* Harvey P. Dale, Sep 27 2017 *)

for(n=1, 68, print1(n + ((-1)^n - (-1)^(n*(n - 1)/2)*(1 + 2*(-1)^n))/2,", ")) \\ Indranil Ghosh, Mar 29 2017

a(1)=1, a(2)=4, a(3)=2, a(4)=3, a(n) = a(n-4) + 4, n > 4.
O.g.f.: (x^4 + x^3 - 2*x^2 + 3x - 1)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^n - (-1)^(n*(n-1)/2)*(1 + 2*(-1)^n))/2.
a(n) = n + (-1)^n*(1 - (-1)^(n*(n-1)/2) - (i^n - (-i)^n))/2.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5), n > 5.
First differences, periodic: (3, -2, 1, 2), repeat.
a(n) = (2*n - 3*cos(n*Pi/2) + cos(n*Pi) + sin(n*Pi/2))/2. - Wesley Ivan Hurt, Apr 01 2017

A346741 Irregular triangle read by rows which is constructed in row n replacing the first A000070(n-1) terms of A336811 with their divisors.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 5, 1, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 2, 1, 1, 5, 1, 3, 1, 2, 1, 1, 1, 2, 3, 6, 1, 2, 4, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 5, 1, 2, 4, 1, 3, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Jul 31 2021

The terms in row n are also all parts of all partitions of n.
The terms of row n in nonincreasing order give the n-th row of A302246.
The terms of row n in nondecreasing order give the n-th row of A302247.
For further information about the correspondence divisor/part see A336811 and A338156.

			Triangle begins:
[1];
[1],[1, 2];
[1],[1, 2],[1, 3],[1];
[1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1];
[1],[1, 2],[1, 3],[1],[1, 2, 4],[1, 2],[1],[1, 5],[1, 3],[1, 2],[1],[1];
...
Below the table shows the correspondence divisor/part.
|---|-----------------|-----|-------|---------|-----------|-------------|
| n |                 |  1  |   2   |    3    |     4     |      5      |
|---|-----------------|-----|-------|---------|-----------|-------------|
| P |                 |     |       |         |           |             |
| A |                 |     |       |         |           |             |
| R |                 |     |       |         |           |             |
| T |                 |     |       |         |           |  5          |
| I |                 |     |       |         |           |  3 2        |
| T |                 |     |       |         |  4        |  4 1        |
| I |                 |     |       |         |  2 2      |  2 2 1      |
| O |                 |     |       |  3      |  3 1      |  3 1 1      |
| N |                 |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |                 |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
----|-----------------|-----|-------|---------|-----------|-------------|
.
|---|-----------------|-----|-------|---------|-----------|-------------|
|   |         A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |                 |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I |         A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |                 |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K |         A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |                 |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   |         A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|-----------------|-----|-------|---------|-----------|-------------|
.
.   |-------|
.   |Section|
|---|-------|---------|-----|-------|---------|-----------|-------------|
|   |   1   | A000012 |  1  |  1    |  1      |  1        |  1          |
|   |-------|---------|-----|-------|---------|-----------|-------------|
|   |   2   | A000034 |     |  1 2  |  1 2    |  1 2      |  1 2        |
|   |-------|---------|-----|-------|---------|-----------|-------------|
| D |   3   | A010684 |     |       |  1   3  |  1   3    |  1   3      |
| I |       | A000012 |     |       |  1      |  1        |  1          |
| V |-------|---------|-----|-------|---------|-----------|-------------|
| I |   4   | A069705 |     |       |         |  1 2   4  |  1 2   4    |
| S |       | A000034 |     |       |         |  1 2      |  1 2        |
| O |       | A000012 |     |       |         |  1        |  1          |
| R |-------|---------|-----|-------|---------|-----------|-------------|
| S |   5   | A010686 |     |       |         |           |  1       5  |
|   |       | A010684 |     |       |         |           |  1   3      |
|   |       | A000034 |     |       |         |           |  1 2        |
|   |       | A000012 |     |       |         |           |  1          |
|   |       | A000012 |     |       |         |           |  1          |
|---|-------|---------|-----|-------|---------|-----------|-------------|
.
In the above table both the zone of partitions and the "Link" zone are the same zones as in the table of the example section of A338156, but here in the lower zone the divisors are ordered in accordance with the sections of the set of partitions of n.
The number of rows in the j-th section of the lower zone is equal to A000041(j-1).
The divisors of the j-th section are also the parts of the j-th section of the set of partitions of n.

Another version of A338156.
Row n has length A006128(n).
The sum of row n is A066186(n).
The product of row n is A007870(n).
Row n lists the first n rows of A336812.
The number of parts k in row n is A066633(n,k).
The sum of all parts k in row n is A138785(n,k).
The number of parts >= k in row n is A181187(n,k).
The sum of all parts >= k in row n is A206561(n,k).
The number of parts <= k in row n is A210947(n,k).
The sum of all parts <= k in row n is A210948(n,k).
Cf. A000012, A000034, A000041, A000070, A002260, A010684, A010686, A027750, A066633, A069705, A135010, A138785, A181187, A221529, A221649, A237593, A302246, A302247, A336811, A340011, A340031, A340032, A340035, A340056, A340057.

A060589 a(n) = 2(2^n-1)3^(n-1).

Original entry on oeis.org

0, 2, 18, 126, 810, 5022, 30618, 185166, 1115370, 6705342, 40271418, 241746606, 1450833930, 8706066462, 52239587418, 313447090446, 1880711240490, 11284353536382, 67706379498618, 406239051832686, 2437436635519050, 14624626786683102, 87747781640805018

Offset: 0

Henry Bottomley, Apr 05 2001

a(n)/3^n is the expected time to finish a random Tower of Hanoi problem with n disks using optimal moves.

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9,-18).

Cf. A007798, A060586, A060590.

[2*(2^n - 1)*3^(n - 1): n in [0..30]]; // Vincenzo Librandi, Jul 03 2018

Table[2 (2^n - 1) 3^(n - 1), {n, 0, 50}] (* or *) LinearRecurrence[{9, -18}, {0, 2}, 40] (* Vincenzo Librandi, Jul 03 2018 *)

a(n)={2*(2^n - 1)*3^(n - 1)} \\ Harry J. Smith, Jul 07 2009

a(n) = Sum_{j<2^n} j*A001316(j) = 6*a(n-1) + A008776(n-1) = 4*A000400(n-1) - A008776(n-1) = A000244(n)*A060590(n)/A010684(n).

G.f.: 2*x/((3*x-1)*(6*x-1)). [Colin Barker, Dec 26 2012]

Corrected by T. D. Noe, Nov 07 2006

A204689 a(n) = n^n (mod 4).

Original entry on oeis.org

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

Offset: 0

José María Grau Ribas, Jan 18 2012

Apart from a(0), the same as A109718. [Joerg Arndt, Sep 17 2013]

Periodic for n>0 with period 4 = A174824(4): repeat [1, 0, 3, 0].

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A000007, A010684, A109718, A174824.

[1] cat &cat [[1, 0, 3, 0]^^30]; // Wesley Ivan Hurt, Jun 15 2016

A204689:=n->n^n mod 4: seq(A204689(n), n=0..150); # Wesley Ivan Hurt, Jun 15 2016

Mathematica
```
Table[PowerMod[n, n, 4], {n,0,140}]
```

From Bruno Berselli, Jan 18 2012: (Start)
G.f.: (1+x+3x^3-x^4)/(1-x^4).
a(n) = (1-(-1)^n)*(2+i^(n+1))/2 with i=sqrt(-1), a(0)=1.
a(n) = A109718(n) for n>0. (End)
a(2k) = A000007(k), a(2k+1) = A010684(k). - Wesley Ivan Hurt, Jun 15 2016

A207607 Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 1, 9, 9, 2, 1, 14, 25, 13, 2, 1, 20, 55, 49, 17, 2, 1, 27, 105, 140, 81, 21, 2, 1, 35, 182, 336, 285, 121, 25, 2, 1, 44, 294, 714, 825, 506, 169, 29, 2, 1, 54, 450, 1386, 2079, 1716, 819, 225, 33, 2, 1, 65, 660, 2508, 4719, 5005, 3185, 1240, 289, 37, 2

Offset: 1

Clark Kimberling, Feb 19 2012

Subtriangle of the triangle T(n,k) given by (1, 0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

			First five rows:
  1;
  1,  2;
  1,  5,  2;
  1,  9,  9,  2;
  1, 14, 25, 13,  2;
Triangle (1, 0, 1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1, 0, 0, 0, 0, ...) begins:
  1;
  1,  0;
  1,  2,  0;
  1,  5,  2,  0;
  1,  9,  9,  2,  0;
  1, 14, 25, 13,  2,  0;
  1, 20, 55, 49, 17,  2,  0;
  ...
1 = 2*1 - 1, 20 = 2*14 + 1 - 9, 55 = 2*25 + 14 - 9, 49 = 2*13 + 25 - 2, 17 = 2*2 + 1 - 0, 2 = 2*0 + 2 - 0. - _Philippe Deléham_, Mar 03 2012

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A207606.

A207607:= (n,k) -> `if`(k=1, 1, binomial(n+k-3, 2*k-2) + 2*binomial(n+k-3, 2*k-3) ); seq(seq(A207607(n, k), k = 1..n), n = 1..10); # G. C. Greubel, Mar 15 2020

(* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207606 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207607 *)
(* Second program *)
Table[If[k==1, 1, Binomial[n+k-3, 2*k-2] + 2*Binomial[n+k-3, 2*k-3]], {n, 10}, {k, n}]//Flatten (* G. C. Greubel, Mar 15 2020 *)

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else x*u(n - 1, x) + (x + 1)*v(n - 1, x)
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

def T(n, k):
    if k == 1: return 1
    else: return binomial(n+k-3, 2*k-2) + 2*binomial(n+k-3, 2*k-3)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 15 2020

u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = x*u(n-1,x) + (x+1)v(n-1,x), where u(1,x)=1, v(1,x)=1.
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Mar 03 2012
G.f.: (1-x+y*x)/(1-(y+2)*x+x^2). - Philippe Deléham, Mar 03 2012
For n >= 1, Sum{k=0..n} T(n,k)*x^k = A000012(n), A001906(n), A001834(n-1), A055271(n-1), A038761(n-1), A056914(n-1) for x = 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Mar 03 2012
T(n,k) = C(n+k-1,2*k) + 2*C(n+k-1,2*k-1). where C is binomial. - Yuchun Ji, May 23 2019
T(n,k) = T(n-1,k) + A207606(n,k-1). - Yuchun Ji, May 28 2019
Sum_{k=1..n} T(n, k)*x^k = { 4*(-1)^(n-1)*A016921(n-1) (x=-4), 3*(-1)^(n-1) * A130815(n-1) (x=-3), 2*(-1)^(n-1)*A010684(n-1) (x=-2), A057079(n+1) (x=-1), 0 (x=0), A001906(n) = Fibonacci(2*n) (x=1), 2*A001834(n-1) (x=2), 3*A055271(n-1) (x=3), 4*A038761(n-1) (x=4) }. - G. C. Greubel, Mar 15 2020

A053438 Expansion of (1+x+2x^3)/((1-x)(1-x^2)).

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, 31, 34, 35, 38, 39, 42, 43, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 67, 70, 71, 74, 75, 78, 79, 82, 83, 86, 87, 90, 91, 94, 95, 98, 99, 102, 103, 106, 107, 110, 111, 114, 115, 118, 119, 122

Offset: 0

N. J. A. Sloane, Jan 12 2000

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A010684 (first differences), A263511 (partial sums).

I:=[2,3,6]; [1] cat [n le 3 select I[n] else Self(n-1) +Self(n-2) -Self(n-3): n in [1..30]]; // G. C. Greubel, May 26 2018

A053438 := proc(n)
    if n > 0 then
        2*n -(1+(-1)^n)/2 ;
    else
        1 ;
    end if
end proc:
seq(A053438(n),n=0..30) ; # R. J. Mathar, Oct 27 2020

CoefficientList[Series[(1+x+2*x^3)/((1-x)*(1-x^2)), {x, 0, 50}], x] (* or *) Join[{1}, LinearRecurrence[{1,1,-1}, {2,3,6}, 50]] (* G. C. Greubel, May 26 2018 *)

a(n)=abs(n\2*4+n%2*3-1) \\ Charles R Greathouse IV, Dec 08 2011

a(n) = 2*n -(1+(-1)^n)/2 if n>=1. - Frank Ellermann, Feb 12 2002
a(n) = A042964(n), n>0. - R. J. Mathar, Oct 13 2008
a(n) = A014601(n) - 1 for n>0. - Hugo Pfoertner, Oct 26 2020

A154890 Jacobsthal numbers A001045 alternatingly incremented by 3 and 5.

Original entry on oeis.org

3, 6, 4, 8, 8, 16, 24, 48, 88, 176, 344, 688, 1368, 2736, 5464, 10928, 21848, 43696, 87384, 174768, 349528, 699056, 1398104, 2796208, 5592408, 11184816, 22369624, 44739248, 89478488, 178956976, 357913944, 715827888, 1431655768, 2863311536, 5726623064

Offset: 0

Paul Curtz, Jan 17 2009

a(2n+1) = 2*a(2n).
a(n) = A153643(n)+A010684(n).
a(n+2) = 4*A128209(n).
a(n) = 2*a(n-1)+a(n-2)-2*a(n-3). G.f.: (3-11x^2)/((1-x)(1+x)(1-2x)). [R. J. Mathar, Jan 23 2009]

Edited and extended by R. J. Mathar, Jan 23 2009

A203231 (n-1)-st elementary symmetric function of the first n terms of the periodic sequence (3,1,3,1,3,1,3,1,...).

Original entry on oeis.org

1, 4, 15, 24, 81, 108, 351, 432, 1377, 1620, 5103, 5832, 18225, 20412, 63423, 69984, 216513, 236196, 728271, 787320, 2421009, 2598156, 7971615, 8503056, 26040609, 27634932, 84499119, 89282088, 272629233, 286978140, 875283327, 918330048

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A010684, A203230, A120908 (bisection).

r = {3, 1, 3, 1, 3, 1};
s = Flatten[{r, r, r, r, r, r, r, r, r}];
t[n_] := Part[s, Range[n]]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 32}]     (* A203231 *)

Conjecture: a(n) = 6*a(n-2)-9*a(n-4) with G.f. x*(1+4*x+9*x^2) / (-1+3*x^2)^2 . - R. J. Mathar, Oct 15 2013

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