A069705 -id:A069705 - cp's OEIS frontend

A153112 a(0) = 0 and a(1)=a(2)=1; a(n) = a(a(n-1)) + a(n-a(n-1)) unless floor( sum_{i=0..n-1} a(i)/2) mod 16*A069705(n) = 1 in which case a(n) = A010882(n).

0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 8, 8, 9, 10, 11, 2, 12, 12, 12, 13, 13, 2, 14, 14, 14, 14, 15, 16, 16, 17, 18, 18, 19, 19, 10, 19, 20, 20, 20, 21, 21, 21, 10, 24, 24, 13, 24, 25, 16, 26, 26, 26, 27, 27, 28, 28, 1, 2, 2, 3, 3, 3, 4, 5, 5, 5, 2, 6, 7, 7, 8, 8, 8, 8, 5

Offset: 0

Roger L. Bagula, Dec 18 2008

nonn

Per Bak, "How nature works, the science of self-organized criticality", Springer, New York (1996), pp. 49-64.

Cf. A004001, A092550, A136640.

A069705 := proc(n) op(1+(n mod 3), [1,2,4]) ; end proc:
A010882 := proc(n) op(1+(n mod 3), [1,2,3]) ; end proc:
A153112 := proc(n) option remember; local psu ; if n=0 then 0; elif n<=2 then 1; else psu := add( procname(i),i=0..n-1) ; if floor(psu/2) mod (16*A069705(n)) = 1 then A010882(n) ; else procname(procname(n-1)) +procname(n-procname(n-1)) ; end if; end if; end proc:
seq(A153112(n),n=0..100) ; # R. J. Mathar, Jun 24 2011

Clear[f, n]; f[0] = 0; f[1] = 1; f[2] = 1;
f[n_] := f[n] = If[Mod[ Floor[Sum[f[i], {i,0, n - 1}]/2], 2^(4 + Mod[n, 3])] == 1, 1 + Mod[n, 3],
f[f[n - 1]] + f[n - f[n - 1]]]; a = Table[f[n], {n, 0, 200}]

Definition cleaned up. - R. J. Mathar, Jun 24 2011

A144437 Period 3: repeat [3, 3, 1].

Original entry on oeis.org

3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3, 1, 3, 3

Offset: 1

Paul Curtz, Oct 05 2008

The sequence is generated from numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...
Conjecture: a(n) is the separatix. See A045944.
Also the decimal expansion of the constant 3310/999. - R. J. Mathar, May 21 2009
Continued fraction expansion of A171417.
Greatest common divisor of (n+1)^2-1 and (n+1)^2+2. - Bruno Berselli, Mar 08 2017

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

Cf. A000217, A045944, A109007, A164306, A167192, A169609, A256095.

Numerators in the energy differences of the hydrogen spectrum: A005563(1), A061037(4), A061039(6), A061041(8), A061043(10), A061045(12), A061047(14), A061049(16), ...

&cat [[3, 3, 1]^^30]; // Wesley Ivan Hurt, Jul 02 2016

seq(op([3, 3, 1]), n=1..50); # Wesley Ivan Hurt, Jul 02 2016

A144437[n_]:=Denominator[n/3]; Array[A144437,100] (* Enrique Pérez Herrero, Oct 05 2011 *)
CoefficientList[Series[(3 + 3 x + x^2)/(1 - x^3), {x, 0, 120}], x] (* Michael De Vlieger, Jul 02 2016 *)
Table[Mod[2*n^2 + 1, 3,1], {n,1,50}] (* G. C. Greubel, Aug 24 2017 *)

a(n)=if(n%3,3,1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = (7-4*cos(2*Pi*n/3))/3. - Jaume Oliver Lafont, Nov 23 2008
G.f.: x*(3 + 3*x + x^2)/((1 - x)*(1 + x + x^2)). - R. J. Mathar, May 21 2009
a(n) = 3/gcd(n,3). - Reinhard Zumkeller, Oct 30 2009
a(n) = denominator(n^k/3), where k>0 is an integer. - Enrique Pérez Herrero, Oct 05 2011
a(n) = gcd(T(n+1), T(2)) = A256095(n+1, 2), with the triangular numbers T = A000217, for n >= 1. - Wolfdieter Lang, Mar 17 2015
a(n) = a(n-3) for n>3; a(n) = A169609(n) for n>0. - Wesley Ivan Hurt, Jul 02 2016
E.g.f.: (1/3)*(7*exp(x) - 4*exp(-x/2)*cos(sqrt(3)*x/2) - 3). - G. C. Greubel, Aug 24 2017
From Nicolas Bělohoubek, Nov 11 2021: (Start)
a(n) = 9/(a(n-2)*a(n-1)).
a(n) = 7 - a(n-2) - a(n-1). See also A052901 or A069705. (End)

Edited by R. J. Mathar, May 21 2009

A201908 Irregular triangle of 2^k mod (2n-1).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 11, 1, 2, 4, 8, 16, 9

Offset: 1

T. D. Noe, Dec 07 2011

The length of the rows is given by A002326. For n > 1, the first term of row n is 1 and the last term is n. Many sequences are in this one: starting at A036117 (mod 11) and A070335 (mod 23).

Row n, for n >= 2, divided elementwise by (2*n-1) gives the cycles of iterations of the doubling function D(x) = 2*x or 2*x-1 if 0 <= x < 1/2 or , 1/2 <= x < 1, respectively, with seed 1/(2*n-1). See the Devaney reference, pp. 25-26. D^[k](x) = frac(2^k/(2*n-1)), for k = 0, 1, ..., A002326(n-1) - 1. E.g., n = 3: 1/5, 2/5, 4/5, 3/5. - Gary W. Adamson and Wolfdieter Lang, Jul 29 2020.

			The irregular triangle T(n, k) begins:
n\k  0 1 2 3  4  5  6  7 8  9 10 11 12 13 14 15 16 17 ...
---------------------------------------------------------
1:   0
2:   1 2
3:   1 2 4 3
4:   1 2 4
5:   1 2 4 8  7  5
6:   1 2 4 8  5 10  9  7 3  6
7:   1 2 4 8  3  6 12 11 9  5 10  7
8:   1 2 4 8
9:   1 2 4 8 16 15 13  9
10:  1 2 4 8 16 13  7 14 9 18 17 15 11  3  6 12  5 10
... reformatted by _Wolfdieter Lang_, Jul 29 2020.

Robert L. Devaney, A First Course in Chaotic Dynamical Systems, Addison-Wesley., 1992. pp. 24-25

T. D. Noe, Rows n = 1..100, flattened

Cf. A002326, A201909 (3^k), A201910 (5^k), A201911 (7^k).

Cf. A000034 (3), A070402 (5), A069705 (7), A036117 (11), A036118 (13), A062116 (17), A036120 (19), A070347 (21), A070335 (23), A070336 (25), A070337 (27), A036122 (29), A070338 (33), A070339 (35), A036124 (37), A070340 (39), A070348 (41), A070349 (43), A070350 (45), A070351 (47), A036128 (53), A036129 (59), A036130 (61), A036131 (67), A036135 (83), A036138 (101), A036140 (107), A201920 (125), A036144 (131), A036146 (139), A036147 (149), A036150 (163), A036152 (173), A036153 (179), A036154 (181), A036157 (197), A036159 (211), A036161 (227).

R:=List([0..72],n->OrderMod(2,2*n+1));;
Flat(Concatenation([0],List([2..11],n->List([0..R[n]-1],k->PowerMod(2,k,2*n-1))))); # Muniru A Asiru, Feb 02 2019

nn = 30; p = 2; t = p^Range[0, nn]; Flatten[Table[If[IntegerQ[Log[p, n]], {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, 1, nn, 2}]]

T(n, k) = 2^k mod (2*n-1), n >= 1, k = 0, 1, ..., A002326(n-1) - 1.

T(n, k) = (2*n-1)*frac(2^k/(2*n-1)), n >= 1, k = 0, 1, ..., A002326(n-1) - 1, with the fractional part frac(x) = x - floor(x). - Wolfdieter Lang, Jul 29 2020

A052901 Periodic with period 3: a(3n)=3, a(3n+1)=a(3n+2)=2.

Original entry on oeis.org

3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Continued fraction expansion of (15 + sqrt(365))/10 = A176979. - Klaus Brockhaus, Apr 30 2010
First differences of A047390. - Tom Edgar, Jul 17 2014
Also decimal expansion of 322/999. - Nicolas Bělohoubek, Nov 11 2021

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 878
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A176979 (decimal expansion of (15+sqrt(365))/10).

Cf. A208131 (partial products).

a052901 n = a052901_list !! n
a052901_list = cycle [3,2,2]  -- Reinhard Zumkeller, Apr 08 2012

spec := [S,{S=Union(Sequence(Z),Sequence(Z),Sequence(Prod(Z,Z,Z)))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

PadRight[{},110,{3,2,2}] (* Harvey P. Dale, Mar 19 2013 *)
LinearRecurrence[{0, 0, 1},{3, 2, 2},105] (* Ray Chandler, Aug 25 2015 *)

Vec((2*x^2+2*x+3)/(1-x^3)+O(x^99)) \\ Charles R Greathouse IV, Apr 08 2012

G.f.: (2*x^2 + 2*x + 3)/(1-x^3).
a(n) = Sum((1/3)*(2*alpha^2 + 3*alpha + 2)*alpha^(-1-n), where alpha = RootOf(-1+x^3)).
a(n) = ceiling(7*(n+1)/3) - ceiling(7*n/3). - Tom Edgar, Jul 17 2014
From Nicolas Bělohoubek, Nov 11 2021: (Start)
a(n) = 12/(a(n-2)*a(n-1)).
a(n) = 7 - a(n-2) - a(n-1). See also A069705 or A144437. (End)

More terms from James Sellers, Jun 06 2000

A179132 Denominators of A178381(4n+1)/A178381(4n).

Original entry on oeis.org

1, 3, 14, 36, 47, 246, 644, 843, 4414, 11556, 15127, 79206, 207364, 271443, 1421294, 3720996, 4870847, 25504086, 66770564, 87403803, 457652254, 1198149156, 1568397607, 8212236486, 21499914244, 28143753123, 147362604494

Offset: 0

Johannes W. Meijer, Jul 01 2010

For the numerators see A179131.

Index entries for linear recurrences with constant coefficients, signature (0,0,18,0,0,-1).

Cf. A128052 and A179133.

with(GraphTheory): nmax:=116; P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): for n from 0 to nmax do B(n):=A^n; A178381(n):=add(B(n)[1,k],k=1..P); od: for n from 0 to nmax-1 do a(n):= denom(A178381(4*n+1)/A178381(4*n)) od: seq(a(n),n=0..nmax/4-1);

LinearRecurrence[{0,0,18,0,0,-1},{1,3,14,36,47,246,644},30] (* Harvey P. Dale, Jun 11 2022 *)

a(n) = A069705(n-1)*A128052(n) for n>=1.
Limit(A179131(n)/A179132(n), n =infinity) = 1+cos(Pi/5) = A296182.
a(n) = 18*a(n-3)-a(n-6) for n>6. G.f.: -(3*x^6+6*x^5+7*x^4-18*x^3-14*x^2-3*x-1) / ((x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)). - Colin Barker, Jun 27 2013

A201912 Irregular triangle of 2^k mod prime(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12, 1, 2, 4, 8, 16, 3, 6, 12, 24

Offset: 1

T. D. Noe, Dec 17 2011

The row lengths are in A014664. For n > 1, the first term of each row is 1 and the last term is 2*prime(n)-1, which is A006254. Many sequences are in this one.

			The first 11 rows are:
2:  0;
3:  1, 2;
5:  1, 2, 4, 3;
7:  1, 2, 4;
11: 1, 2, 4, 8,  5, 10,  9,  7,  3,  6;
13: 1, 2, 4, 8,  3,  6, 12, 11,  9,  5, 10,  7;
17: 1, 2, 4, 8, 16, 15, 13,  9;
19: 1, 2, 4, 8, 16, 13,  7, 14,  9, 18, 17, 15, 11,  3,  6, 12,  5, 10;
23: 1, 2, 4, 8, 16,  9, 18, 13,  3,  6, 12;
29: 1, 2, 4, 8, 16,  3,  6, 12, 24, 19,  9, 18,  7, 14, 28, 27, 25, 21, 13, 26, 23, 17, 5, 10, 20, 11, 22, 15;
31: 1, 2, 4, 8, 16;

T. D. Noe, Rows n = 1..60, flattened

Cf. A062117, A201908, A201909, A201910, A201911.

Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), A269266 (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67), A036135 (p=83), A036138 (p=101), A036140 (p=107), A036144 (p=131), A036146 (p=139), A036147 (p=149), A036150 (p=163), A036152 (p=173), A036153 (p=179), A036154 (p=181), A036157 (p=197), A036159 (p=211), A036161 (p=227).

P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(2,P[n]));;
Flat(Concatenation([0],List([2..10],n->List([0..R[n]-1],k->PowerMod(2,k,P[n]))))); # Muniru A Asiru, Feb 01 2019

nn = 10; p = 2; t = p^Range[0,Prime[nn]]; Flatten[Table[If[Mod[n, p] == 0, {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, Prime[Range[nn]]}]]

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.

A133145 Period 4: repeat [1, 2, 4, 8].

Original entry on oeis.org

1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8, 1, 2, 4, 8

Offset: 0

Paul Curtz, Dec 16 2007

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

Cf. A069705. [Jaume Oliver Lafont, Mar 27 2009]

Cf. A000079, A160700.

&cat [[1, 2, 4, 8]^^30]; // Wesley Ivan Hurt, Jul 09 2016

seq(op([1, 2, 4, 8]), n=0..50); # Wesley Ivan Hurt, Jul 09 2016

PadRight[{}, 100, {1, 2, 4, 8}] (* Wesley Ivan Hurt, Jul 09 2016 *)
Table[First@ IntegerDigits[2^n, 16], {n, 0, 120}] (* Michael De Vlieger, Jul 09 2016 *)

a(n)=2^(n%4) \\ Jaume Oliver Lafont, Mar 27 2009

[power_mod(2,n,15) for n in range(0,80)] # Zerinvary Lajos, Nov 03 2009

a(n) == 2*a(n-1) mod 15.
a(n) = 2^(n mod 4). - Jaume Oliver Lafont, Mar 27 2009
a(n) = A160700(A000079(n)). [Reinhard Zumkeller, Jun 10 2009]
a(n) = 2^n (mod 15). G.f.: (1+2*x)*(4*x^2+1)/ ((1-x)*(1+x)*(x^2+1)). [R. J. Mathar, Apr 13 2010]
From Wesley Ivan Hurt, Jul 09 2016: (Start)
a(n) = a(n-4) for n>3.
a(n) = (15-6*cos(n*Pi/2)-5*cos(n*Pi)-12*sin(n*Pi/2)-5*I*sin(n*Pi))/4. (End)

A145643 Cubefree part of n!!.

Original entry on oeis.org

1, 2, 3, 1, 15, 6, 105, 6, 35, 60, 385, 90, 5005, 1260, 75075, 315, 1276275, 210, 24249225, 525, 509233725, 11550, 11712375675, 34650, 2342475135, 900900, 2342475135, 3153150, 67931778915, 28028, 2105885146365, 14014, 2573859623335

Offset: 1

Artur Jasinski, Oct 15 2008

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A004709, A050985, A006882, A069705 (cubefree part of 2^n), A145642.

CubefreePart[n_Integer?Positive] := Times @@ Power @@@ ({#[[1]], Mod[ #[[2]], 3]} & /@ FactorInteger[n]); Table[CubefreePart[n!! ], {n, 1, 40}]

a(n) = my(f = factor(prod(i = 0, (n-1)\2, n - 2*i))); prod(i = 1, #f~, f[i, 1]^(f[i, 2] % 3)); \\ Amiram Eldar, Sep 07 2024

from sympy import factorint, prod
def A145643(n):
    return 1 if n <= 1 else prod(p**(e % 3) for p, e in factorint(prod(range(n,0,-2))).items())
# Chai Wah Wu, Feb 04 2015

a(n) = A050985(A006882(n)). - Michel Marcus, May 11 2020

A178141 Period 6: repeat [4, -1, 2, -4, 1, 2].

Original entry on oeis.org

4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2, 4, -1, 2, -4, 1, 2

Offset: 0

Paul Curtz, May 21 2010

Differences of the period 6: repeat [1, 5, 4, 6, 2, 3] (A070365).

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

Cf. A069705, A070365, A132954, A153727.

&cat [[4, -1, 2, -4, 1, 2]^^20]; // Wesley Ivan Hurt, Jun 23 2016

A178141:=n->[4, -1, 2, -4, 1, 2][(n mod 6)+1]: seq(A178141(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

PadRight[{}, 100, {4, -1, 2, -4, 1, 2}] (* Wesley Ivan Hurt, Jun 23 2016 *)

a(n)=[1,5,4,6,2,3][n%5+1] \\ Charles R Greathouse IV, Jun 02 2011

Mix A153727(n+1) with -A153727(n).
From Wesley Ivan Hurt, Jun 23 2016: (Start)
G.f.: (4-x+2*x^2-4*x^3+x^4+2*x^5)/(1-x^6).
a(n) = a(n-6) for n>5.
a(n) = (2 + 5*cos(n*Pi) + 7*cos(n*Pi/3) - 2*cos(2*n*Pi/3) - sqrt(3)*sin(n*Pi/3) - 2*sqrt(3)*sin(2*n*Pi/3))/3. (End)

New name from Wesley Ivan Hurt, Jun 23 2016

cp's OEIS Frontend

A153112 a(0) = 0 and a(1)=a(2)=1; a(n) = a(a(n-1)) + a(n-a(n-1)) unless floor( sum_{i=0..n-1} a(i)/2) mod 16*A069705(n) = 1 in which case a(n) = A010882(n).

Views

Author

Keywords

References

Crossrefs

Programs

Maple

Mathematica

Extensions

A144437 Period 3: repeat [3, 3, 1].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A201908 Irregular triangle of 2^k mod (2n-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Mathematica

Formula

A052901 Periodic with period 3: a(3n)=3, a(3n+1)=a(3n+2)=2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A179132 Denominators of A178381(4*n+1)/A178381(4*n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A201912 Irregular triangle of 2^k mod prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Mathematica

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

Views

Author

Keywords

Comments

Examples

A179132 Denominators of A178381(4n+1)/A178381(4n).