A080425 -id:A080425 - cp's OEIS frontend

A061891 a(0) = 1; for n>0, a(n) = a(n-1) if n is already in the sequence, a(n) = a(n-1) + 3 otherwise.

1, 1, 4, 7, 7, 10, 13, 13, 16, 19, 19, 22, 25, 25, 28, 31, 31, 34, 37, 37, 40, 43, 43, 46, 49, 49, 52, 55, 55, 58, 61, 61, 64, 67, 67, 70, 73, 73, 76, 79, 79, 82, 85, 85, 88, 91, 91, 94, 97, 97, 100, 103, 103, 106, 109, 109, 112, 115, 115, 118, 121, 121, 124

Offset: 0

N. J. A. Sloane and Benoit Cloitre, Apr 01 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, arXiv:math/0305308 [math.NT], 2003.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A080578.

LinearRecurrence[{1, 0, 1, -1}, {1, 1, 4, 7}, 63] (* Jean-François Alcover, Jan 07 2019 *)

a(n) = 2*n-1 if n == 1 (mod 3), 2*n if n == 2 (mod 3), 2*n + 1 if n == 0 (mod 3).
Differences are periodic with period 3.
From Colin Barker, Jun 20 2013: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4).
G.f.: (2*x^3 + 3*x^2 + 1) / ((x - 1)^2*(x^2 + x +1)). (End)
a(n) = 2*n + 1 - A080425(n) = 2*n - 1 + A010872(n+1). [Wesley Ivan Hurt, Jul 07 2013]

A173178 Numbers k such that 2k+3 is a prime of the form 3A024893(m) + 2.

Original entry on oeis.org

1, 4, 7, 10, 13, 19, 22, 25, 28, 34, 40, 43, 49, 52, 55, 64, 67, 73, 82, 85, 88, 94, 97, 112, 115, 118, 124, 127, 130, 133, 139, 145, 154, 157, 172, 175, 178, 190, 193, 199, 208, 214, 220, 223, 229, 232, 238, 244, 250, 253, 259, 277, 280, 283, 292, 295, 298, 307, 319

Offset: 1

Eric Desbiaux, Feb 11 2010

With the Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
   k   a b OR a b
  --   - -    - -
   0   0 0    0 0
   1   - -    - -
   2   1 0    1 0
   3   0 1    0 1
   4   2 0    2 0
   5   1 1    1 1
   6   0 2    3 0
   7   2 1    2 1
   8   1 2    4 0
   9   0 3    3 1
  10   2 2    5 0
  11   1 3    4 1
  12   0 4    6 0
  13   2 3    5 1
  14   1 4    7 0
  15   0 5    6 1
  ...
  2* 1 + 3 OR 3* 1 + 2 =  5;
  2* 4 + 3 OR 3* 3 + 2 = 11;
  2* 7 + 3 OR 3* 5 + 2 = 17;
  2*10 + 3 OR 3* 7 + 2 = 23;
  2*13 + 3 OR 3* 9 + 2 = 29;
  2*19 + 3 OR 3*13 + 2 = 41;
  2*22 + 3 OR 3*15 + 2 = 47;
  2*25 + 3 OR 3*17 + 2 = 53;
  2*28 + 3 OR 3*19 + 2 = 59.
A024893 Numbers k such that 3k+2 is prime.
A007528 Primes of the form 6k-1.
A024898 Positive integers k such that 6k-1 is prime.
1, 4, 7, 10, 13, 19, ... = (3*(4*A024898 - A024893) - 7)/2 = (A112774 - 3*A024893 - 5)/2 = A003627 - (3*A024893 - 5)/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?, Frequently asked questions about primes.

Cf. A067076, A024893, A007528, A024898, A059325.

Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* Amiram Eldar, Jul 30 2024 *)

a(n) = 3*A059325(n) + 1. - Amiram Eldar, Jul 30 2024

Data corrected and extended by Amiram Eldar, Jul 30 2024

A131380 a(3n) = 2n, a(3n+1) = 2n+2, a(3n+2) = 2n+1.

Original entry on oeis.org

0, 2, 1, 2, 4, 3, 4, 6, 5, 6, 8, 7, 8, 10, 9, 10, 12, 11, 12, 14, 13, 14, 16, 15, 16, 18, 17, 18, 20, 19, 20, 22, 21, 22, 24, 23, 24, 26, 25, 26, 28, 27, 28, 30, 29, 30, 32, 31, 32, 34, 33, 34, 36, 35, 36, 38, 37, 38, 40, 39, 40, 42, 41, 42, 44, 43, 44, 46, 45, 46, 48, 47, 48, 50

Offset: 0

Paul Curtz, Oct 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A002264, A080425.

[(-n mod 3) + 2*Floor(n/3) : n in [0..100]]; // Wesley Ivan Hurt, Aug 20 2014

I:=[0,2,1,2]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..100]]; // Vincenzo Librandi, Sep 27 2017

A131380:=n->(-n mod 3) + 2*floor(n/3): seq(A131380(n), n=0..100); # Wesley Ivan Hurt, Aug 20 2014

Table[Mod[-n, 3] + 2 Floor[n/3], {n, 0, 100}] (* Wesley Ivan Hurt, Aug 20 2014 *)
CoefficientList[Series[x*(2 - x + x^2)/((x - 1)^2 (1 + x + x^2)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Aug 20 2014 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 2, 1, 2}, 200] (* Vincenzo Librandi, Sep 27 2017 *)

G.f.: x*(2-x+x^2)/((x-1)^2*(1+x+x^2)); a(n) = a(n-1)+a(n-3)-a(n-4); a(n) = (-n mod 3) + 2*floor(n/3) = A080425(n) + 2*A002264(n). - Wesley Ivan Hurt, Aug 20 2014
E.g.f.: ((2*z+1)/3)*exp(z)+((5/9)*sqrt(3)*sin(sqrt(3)*z/2)-(1/3)*cos(sqrt(3)*z/2))*exp(-z/2). - Robert Israel, Aug 21 2014
a(n) = (6*n+3-6*cos(2*(n+4)*Pi/3)-4*sqrt(3)*sin(2*(n+4)*Pi/3))/9. - Wesley Ivan Hurt, Sep 26 2017

A132798 Period 6: repeat [0, 2, 1, 0, -2, -1].

Original entry on oeis.org

0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1, 0, 2, 1, 0, -2, -1

Offset: 0

Paul Curtz, Nov 21 2007

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

Cf. A002264, A080425, A117373, A129339, A130151.

[(-n mod 3)*(-1)^Floor(n/3) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

A132798:=n->(-n mod 3)*(-1)^floor(n/3); seq(A132798(n), n=0..50); # Wesley Ivan Hurt, Jun 20 2014

Table[Mod[-n, 3]*(-1)^Floor[n/3], {n, 0, 50}] (* Wesley Ivan Hurt, Jun 20 2014 *)

G.f.: x*(2+x)/((x+1)*(x^2-x+1)) = (1/3)*(4*x+1)/(x^2-x+1)-(1/3)/(x+1). - R. J. Mathar, Nov 28 2007
a(n) + a(n+1) = A117373(n+4). - R. J. Mathar, Jul 22 2009
a(n) = (-n mod 3) * (-1)^floor(n/3) = A080425(n) * (-1)^A002264(n) = A080425(n) * A130151(n). - Wesley Ivan Hurt, Jun 20 2014
From Wesley Ivan Hurt, Jun 21 2016: (Start)
a(n) + a(n-3) = 0 for n>2.
a(n) = sin(n*Pi/3) * (3*sqrt(3) + 2*sin(2*n*Pi/3))/3. (End)

A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

Original entry on oeis.org

2, 5, 8, 14, 17, 20, 29, 32, 35, 38, 47, 50, 53, 62, 68, 74, 77, 80, 89, 95, 98, 104, 110, 113, 119, 134, 137, 140, 152, 155, 164, 167, 173, 182, 185, 188, 197, 203, 209, 215, 218, 227, 230, 242, 248, 260, 269, 272, 284, 287, 299

Offset: 1

Eric Desbiaux, Feb 11 2010

With Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR
second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
   k   a b OR a b
  --   - -    - -
   0   0 0    0 0
   1   - -    - -
   2   1 0    1 0
   3   0 1    0 1
   4   2 0    2 0
   5   1 1    1 1
   6   0 2    3 0
   7   2 1    2 1
   8   1 2    4 0
   9   0 3    3 1
  10   2 2    5 0
  11   1 3    4 1
  12   0 4    6 0
  13   2 3    5 1
  14   1 4    7 0
  15   0 5    6 1
  ...
  2* 2 + 3 OR 3* 1 + 4 =  7;
  2* 5 + 3 OR 3* 3 + 4 = 13;
  2* 8 + 3 OR 3* 5 + 4 = 19;
  2*14 + 3 OR 3* 9 + 4 = 31;
  2*17 + 3 OR 3*11 + 4 = 37;
  2*20 + 3 OR 3*13 + 4 = 43;
  2*29 + 3 OR 3*19 + 4 = 61;
  2*32 + 3 OR 3*21 + 4 = 67;
  2*35 + 3 OR 3*23 + 4 = 73.
A034936 Numbers k such that 3k+4 is prime.
A002476 Primes of the form 6k+1.
A024899 Nonnegative integers k such that 6k+1 is prime.
2, 5, 8, 14, 17, 20, ... = (3*(4*A024899 - A034936) - 5)/2.

Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?

Cf. A067076, A034936, A002476, A024899.

Select[Range[300],PrimeQ[2#+3]&&Divisible[2#-1,3]&] (* Harvey P. Dale, Aug 25 2016 *)

More terms from Harvey P. Dale, Aug 25 2016

A180964 a(0)=1; for n>0, a(n) = 1 + 3*A117571(n-1).

Original entry on oeis.org

1, 4, 4, 10, 13, 13, 19, 22, 22, 28, 31, 31, 37, 40, 40, 46, 49, 49, 55, 58, 58, 64, 67, 67, 73, 76, 76, 82, 85, 85, 91, 94, 94, 100, 103, 103, 109, 112, 112, 118, 121, 121, 127, 130, 130, 136, 139, 139, 145, 148, 148, 154, 157, 157, 163, 166, 166, 172

Offset: 0

Bruno Berselli, Sep 28 2010 - Oct 01 2010

B. Berselli, Table of n, a(n) for n = 0..10000.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A061347.

I:=[1, 4, 4, 10]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Mar 26 2013

m:=60; S:=series( (1+3*x+5*x^3)/((1-x)^2*(1+x+x^2)), x, m+1):
seq(coeff(S, x, j), j=1..m); # G. C. Greubel, Apr 06 2021

CoefficientList[Series[(1 +3x +5x^3)/((1-x)^2(1+x+x^2)), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{1,0,1,-1},{1,4,4,10},60] (* Harvey P. Dale, Aug 05 2020 *)

[3*n +chebyshev_U(n, -1/2) +2*chebyshev_U(n-1, -1/2) for n in (0..60)] # G. C. Greubel, Apr 06 2021

G.f.: (1 +3*x +5*x^3)/((1-x)^2*(1+x+x^2)).
a(n) = a(n-1) +a(n-3) -a(n-4) for n>3.
a(n) = (n-1)*(n mod 3) +(n+1)*(n+1 mod 3) +n*(n+2 mod 3).
a(n) = 3*n +sqrt(3)*cos((4*n-3)*Pi/6) -sin((4*n-3)*Pi/6).
a(n) - a(n-1) = 3*A080425(n+1) for n>0.
From G. C. Greubel, Apr 06 2021: (Start)
a(n) = 3*n - 2*cos(2*Pi*(n+1)/3) = 3*n + A061347(n+1).
a(n) = 3*n + ChebyshevU(n, -1/2) + 2*ChebyshevU(n-1, -1/2). (End)

A191272 Expansion of x(4+5x)/( (1-4x)(1 + x + x^2) ).

Original entry on oeis.org

0, 4, 17, 63, 256, 1025, 4095, 16384, 65537, 262143, 1048576, 4194305, 16777215, 67108864, 268435457, 1073741823, 4294967296, 17179869185, 68719476735, 274877906944, 1099511627777, 4398046511103, 17592186044416, 70368744177665

Offset: 0

Paul Curtz, May 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..175
Index entries for linear recurrences with constant coefficients, signature (3,3,4).

m:=24; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(4+5*x)/((1-4*x)*(1+x+x^2))));  // Bruno Berselli, Jul 04 2011

LinearRecurrence[{3,3,4},{0,4,17},30] (* or *) CoefficientList[ Series[ x (4+5x)/((1-4x)(1+x+x^2)),{x,0,30}],x] (* Harvey P. Dale, Jun 19 2011 *)

makelist(coeff(taylor(x*(4+5*x)/((1-4*x)*(1+x+x^2)), x, 0, n), x, n), n, 0, 23);  /* Bruno Berselli, Jun 06 2011 */

a(n)=4^n-[1,0,-1][n%3+1] \\ Charles R Greathouse IV, Jun 06 2011

G.f.:  x*(4+5*x)/(1 - 3*x - 3*x^2 - 4*x^3).
a(n) = 4^n-A057078(n) = 4^n - (n-th element of periodic length 3 repeat 1, 0, -1)
a(n) = A024495(2*n) + A024495(1+2*n).
a(n+1) = 4*a(n) + (n-th element of periodic length 3 repeat 4, 1, -5).
a(n) = A052539(n) - A080425(n+1). - Bruno Berselli, Jun 06 2011
a(0)=0, a(1)=4, a(2)=17, a(n) = 3*a(n-1) + 3*a(n-2) + 4*a(n-3). - Harvey P. Dale, Jun 19 2011
a(n) = 4*a(n-1) + a(n-3) - 4*(n-4) (n>3). - Bruno Berselli, Jul 04 2011

A271390 a(n) = (2n + 1)^(2floor((n-1)/2) + 1).

Original entry on oeis.org

1, 3, 5, 343, 729, 161051, 371293, 170859375, 410338673, 322687697779, 794280046581, 952809757913927, 2384185791015625, 4052555153018976267, 10260628712958602189, 23465261991844685929951, 59938945498865420543457, 177482997121587371826171875, 456487940826035155404146917

Offset: 0

Ilya Gutkovskiy, Apr 06 2016

All members are odd, therefore:
........................
|   k   |  a(n) mod k  |
|.......|..............|
|  n+1  |  A001477(n)  |
| 2*n+2 |  A005408(n)  |
|   2   |  A000012(n)  |
|   3   |  A080425(n+2)|
|   4   |  A010684(n)  |
|   6   |  A130793(n)  |
........................
Final digit of (2*n + 1)^(2*floor((n-1)/2) + 1) gives periodic sequence -> period 20: repeat [1,3,5,3,9,1,3,5,3,9,1,7,5,7,9,1,7,5,7,9], defined by the recurrence relation b(n) = b(n-2) - b(n-4) + b(n+5) + b(n+6) - b(n-7) - b(n-8) + b(n-9) - b(n-11) + b(n-13).

			a(0) =  1;
a(1) =  3^1 = 3;
a(2) =  5^1 = 5;
a(3) =  7^3 = 343;
a(4) =  9^3 = 729;
a(5) = 11^5 = 161051;
a(6) = 13^5 = 371293;
a(7) = 15^7 = 170859375;
a(8) = 17^7 = 410338673;
...
a(10000) = 1.644...*10^43006;
...
a(100000) = 8.235...*10^530097, etc.
This sequence can be represented as a binary tree:
                                    1
                 ................../ \..................
                3^1                                   5^1
     7^3......../ \......9^3                11^5....../ \.......13^5
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
15^7    17^7        19^9    21^9        23^11   25^11       27^13   29^13

Ilya Gutkovskiy, Table of n, a(n) for n = 0..75

Cf. A005408, A092503, A109613.

A271390:=n->(2*n + 1)^(n - 1/2 - (-1)^n/2): seq(A271390(n), n=0..30); # Wesley Ivan Hurt, Apr 10 2016

Table[(2 n + 1)^(2 Floor[(n - 1)/2] + 1), {n, 0, 18}]
Table[(2 n + 1)^(n - 1 + (1 + (-1)^(n - 1))/2), {n, 0, 18}]

a(n) = (2*n + 1)^(2*((n-1)\2) + 1); \\ Altug Alkan, Apr 06 2016

for n in range(0,10**3):print((int)((2*n+1)**(2*floor((n-1)/2)+1)))
# Soumil Mandal, Apr 10 2016

a(n) = (2*n + 1)^(n - 1 + (1 + (-1)^(n-1))/2).
a(n) = A005408(n)^A109613(n-1).
a(n) = (2*n + 1)^(n - 1/2 - (-1)^n/2). - Wesley Ivan Hurt, Apr 10 2016

A174012 a(n) = 3 * A064680(n).

Original entry on oeis.org

0, 6, 3, 18, 6, 30, 9, 42, 12, 54, 15, 66, 18, 78, 21, 90, 24, 102, 27, 114, 30, 126, 33, 138, 36, 150, 39, 162, 42, 174, 45, 186, 48, 198, 51, 210, 54, 222, 57, 234, 60, 246, 63, 258, 66, 270, 69, 282, 72, 294, 75, 306, 78, 318, 81, 330, 84, 342, 87, 354, 90, 366, 93, 378, 96

Offset: 0

Paul Curtz, Mar 05 2010

Cf. A008585, A017593, A064680, A080425, A165988.

a(n) = A064680(3*n), similar to A165988.
a(n) mod 9 = 3*A080425(n) (period length 3).
a(2n+1) = A017593(n).
a(2n) = A008585(n).

a(0) = 0 prepended by Georg Fischer, Jul 01 2020

cp's OEIS Frontend

A061891 a(0) = 1; for n>0, a(n) = a(n-1) if n is already in the sequence, a(n) = a(n-1) + 3 otherwise.

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A173178 Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.

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A131380 a(3n) = 2n, a(3n+1) = 2n+2, a(3n+2) = 2n+1.

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A132798 Period 6: repeat [0, 2, 1, 0, -2, -1].

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A173177 Numbers k such that 2k+3 is a prime of the form 3*A034936(m) + 4.

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A180964 a(0)=1; for n>0, a(n) = 1 + 3*A117571(n-1).

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A191272 Expansion of x*(4+5*x)/( (1-4*x)*(1 + x + x^2) ).

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A271390 a(n) = (2*n + 1)^(2*floor((n-1)/2) + 1).

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A173178 Numbers k such that 2k+3 is a prime of the form 3A024893(m) + 2.

A191272 Expansion of x(4+5x)/( (1-4x)(1 + x + x^2) ).

A271390 a(n) = (2n + 1)^(2floor((n-1)/2) + 1).