A047464 -id:A047464 - cp's OEIS frontend

A047395 Numbers that are congruent to {0, 2, 6} mod 8.

0, 2, 6, 8, 10, 14, 16, 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 46, 48, 50, 54, 56, 58, 62, 64, 66, 70, 72, 74, 78, 80, 82, 86, 88, 90, 94, 96, 98, 102, 104, 106, 110, 112, 114, 118, 120, 122, 126, 128, 130, 134, 136, 138, 142, 144, 146, 150, 152, 154, 158

Offset: 1

N. J. A. Sloane

The members of this sequence together with the members of A017113 give the even numbers. - Wesley Ivan Hurt, Apr 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A017113, A042965, A047407, A047410, A047464.

[n : n in [0..150] | n mod 8 in [0, 2, 6]]; // Wesley Ivan Hurt, Jun 13 2016

A047395:=n->2*floor((4*n-3)/3); seq(A047395(n), n=1..100); # Wesley Ivan Hurt, Apr 01 2014

Table[2 Floor[(4 n - 3)/3], {n, 100}] (* Wesley Ivan Hurt, Apr 01 2014 *)
Flatten[Table[8n + {0, 2, 6}, {n, 0, 19}]] (* Alonso del Arte, Apr 11 2014 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 2, 6, 8}, 100] (* Vincenzo Librandi, Jun 14 2016 *)

From R. J. Mathar, Dec 05 2011: (Start)
G.f.: 2*x^2*(1+x)^2 / ((1+x+x^2)*(x-1)^2).
a(n) = 2 * A042965(n). (End)
From Wesley Ivan Hurt, Jun 13 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 2*(12*n-12+3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-2, a(3k-1) = 8k-6, a(3k-2) = 8k-8. (End)
a(n) = 2*(n - 1 + floor(n/3)). - Wolfdieter Lang, Sep 11 2021
Sum_{n>=2} (-1)^n/a(n) = sqrt(2)*log(sqrt(2)+2)/4 - (sqrt(2)-1)*log(2)/8. - Amiram Eldar, Dec 19 2021

A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3floor(4n/3).

Original entry on oeis.org

0, 3, 6, 12, 15, 18, 24, 27, 30, 36, 39, 42, 48, 51, 54, 60, 63, 66, 72, 75, 78, 84, 87, 90, 96, 99, 102, 108, 111, 114, 120, 123, 126, 132, 135, 138, 144, 147, 150, 156, 159, 162, 168, 171, 174, 180, 183, 186, 192, 195, 198, 204, 207, 210, 216, 219, 222, 228

Offset: 0

Jianing Song, Sep 19 2018

Key-numbers of the pitches of a diminished chord on a standard chromatic keyboard, with root = 0.

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

Cf. A004773, A047464.
A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): A083120
Dorian mode (D): A083033
Aeolian mode (A): A060107 (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): this sequence
Seventh chords:
Major seventh chord (F,C): A319280
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): A319279
Half-diminished seventh chord (B): A319452

Filtered([0..230],n->n mod 12 = 0 or n mod 12 = 3 or n mod 12 = 6); # Muniru A Asiru, Oct 24 2018

[n : n in [0..150] | n mod 12 in [0, 3, 6]]

seq(3*floor(4*n/3),n=0..60); # Muniru A Asiru, Oct 24 2018

Select[Range[0, 200], MemberQ[{0, 3, 6}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 1, -1}, {0, 3, 6, 12}, 100]
Table[4n-1+Sin[Pi/3(2n+1)]/Sin[Pi/3],{n,0,99}] (* Federico Provvedi, Oct 23 2018 *)

PARI
```
a(n)=3*(4*n\3)
```

for n in range(0,60): print(3*int(4*n/3), end=", ") # Stefano Spezia, Dec 07 2018

a(n) = a(n-3) + 12 for n > 2.
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 3.
G.f.: 3*(1 + x + 2*x^2)/((1 - x)*(1 - x^3)).
a(n) = 3*A004773(n) = 3*(floor(n/3) + n).
a(n) = 4*n - 1 + sin((Pi/3)*(2*n + 1))/sin(Pi/3). - Federico Provvedi, Oct 23 2018
E.g.f.: (3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/(3*exp(x/2)) - exp(x)*(1 - 4*x). - Franck Maminirina Ramaharo, Nov 27 2018
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/24 + (2-sqrt(2))*log(2)/24 + sqrt(2)*log(2+sqrt(2))/12. - Amiram Eldar, Dec 30 2021

A047410 Numbers that are congruent to {2, 4, 6} mod 8.

Original entry on oeis.org

2, 4, 6, 10, 12, 14, 18, 20, 22, 26, 28, 30, 34, 36, 38, 42, 44, 46, 50, 52, 54, 58, 60, 62, 66, 68, 70, 74, 76, 78, 82, 84, 86, 90, 92, 94, 98, 100, 102, 106, 108, 110, 114, 116, 118, 122, 124, 126, 130, 132, 134, 138, 140, 142, 146, 148, 150, 154, 156, 158

Offset: 1

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 17 ).

William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A042968, A047395, A047407, A047464.

[n : n in [0..150] | n mod 8 in [2, 4, 6]]; // Wesley Ivan Hurt, Jun 09 2016

A047410:=n->2*(12*n-6-3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9: seq(A047410(n), n=1..100); # Wesley Ivan Hurt, Jun 09 2016

With[{upto=140},Complement[2*Range[upto/2],8*Range[upto/8]]] (* or *) LinearRecurrence[{1,0,1,-1}, {2,4,6,10}, 60] (* Harvey P. Dale, Oct 06 2014 *)

a(n) = 2*floor((n-1)/3) + 2*n. - Gary Detlefs, Mar 18 2010
From R. J. Mathar, Dec 05 2011: (Start)
G.f.: 2*x*(1+x)*(1+x^2) / ( (1+x+x^2)*(x-1)^2 ).
a(n) = 2*A042968(n). (End)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4, with a(1)=2, a(2)=4, a(3)=6, a(4)=10. - Harvey P. Dale, Oct 06 2014
From Wesley Ivan Hurt, Jun 09 2016: (Start)
a(n) = 2*(12*n-6-3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-2, a(3k-1) = 8k-4, a(3k-2) = 8k-6. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 19 2021
E.g.f.: 2*(9 + 6*exp(x)*(2*x - 1) - exp(-x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/9. - Stefano Spezia, Oct 17 2022

A047407 Numbers that are congruent to {0, 4, 6} mod 8.

Original entry on oeis.org

0, 4, 6, 8, 12, 14, 16, 20, 22, 24, 28, 30, 32, 36, 38, 40, 44, 46, 48, 52, 54, 56, 60, 62, 64, 68, 70, 72, 76, 78, 80, 84, 86, 88, 92, 94, 96, 100, 102, 104, 108, 110, 112, 116, 118, 120, 124, 126, 128, 132, 134, 136, 140, 142, 144, 148, 150, 152, 156

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A004772, A047395, A047410, A047464.

[n : n in [0..160] | n mod 8 in [0, 4, 6]]; // Vincenzo Librandi, May 02 2016

A047407:=n->2*(12*n-9-2*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047407(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016

Select[Range[0,200], MemberQ[{0,4,6}, Mod[#,8]]&] (* or *) LinearRecurrence[{1,0,1,-1}, {0,4,6,8}, 70] (* Harvey P. Dale, Apr 20 2016 *)

a(n)=n\3*8+[-2,0,4][n%3+1] \\ Charles R Greathouse IV, May 02 2016

From R. J. Mathar, Dec 05 2011: (Start)
a(n) = 2*A004772(n).
G.f.: 2*x^2*(2+x+x^2) / ((1+x+x^2)*(x-1)^2). (End)
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 2*(12*n-9-2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-2, a(3k-1) = 8k-4, a(3k-2) = 8k-8. (End)
a(n) = 2*(n - 1 + floor((n + 1)/3)). - Wolfdieter Lang, Sep 11 2021
Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8 - (sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 19 2021

A236535 a(n)*Pi is the total length of irregular spiral (center points: 2, 3, 1; pattern 1) after n rotations.

Original entry on oeis.org

2, 5, 8, 10, 13, 16, 18, 21, 24, 26, 29, 32, 34, 37, 40, 42, 45, 48, 50, 53, 56, 58, 61, 64, 66, 69, 72, 74, 77, 80, 82, 85, 88, 90, 93, 96, 98, 101, 104, 106, 109, 112, 114, 117, 120, 122, 125, 128, 130, 133, 136, 138, 141, 144, 146, 149, 152, 154, 157, 160, 162, 165, 168, 170, 173, 176, 178, 181, 184, 186, 189

Offset: 1

Kival Ngaokrajang, Jan 28 2014

nonn

Let points 2, 3, & 1 be placed on a straight line at intervals of 1 unit. At point 1 make a half unit circle then at point 2 make another half circle; by selecting radius point on the left hand side of point 1 (pattern 1); at point 3 make another half circle and maintain continuity of circumferences. Continue using this procedure at point 1, 2, 3, ... and so on.
Conjecture: All forms of 3 center points are non-expanded loops.
There are other sets of center points that give the same sequence, e.g.: [2,3,1,4]; [3,2,4,1]; [3,2,4,1,5]; [2,3,1,4,5,7,6]; [2,3,1,7,4,6,5]; [3,4,2,5,1,6,7]; [4,3,5,6,2,7,1]; [4,5,3,2,1,6,7]; [5,4,6,3,2,7,1].
Also, there are some similar patterns that give difference sequences, e.g.:
   A047622: [1,2,7,3,4,6,5]; [1,2,7,6,3,5,4]...
   A047399: [1,2,7,3,6,4,5]; [1,2,7,6,5,3,4]...
   A047395: [2,3,1,4 7,5,6]; [2,3,1,7,6,4,5]...
   A047464: [4,5,3,6,2,7,1]; [1,8,2,7,3,6,4,5];
            [9,1,8,2,7,3,6,4,5].
See illustration in links.
Appears to be basically a duplicate of A047618. - R. J. Mathar, Feb 03 2014

Kival Ngaokrajang, Illustration of irregular spirals (Center points 3..9)

Cf. A014105 (2 center points); A234902, A234903, A234904 (3 center points); A235088, A235089 (4 center points); A236326, A236327 (5 center points).

Conjecture from Colin Barker, Jul 12 2014: (Start)
a(n) = a(n-1)+a(n-3)-a(n-4).
G.f.: x*(3*x^2+3*x+2) / ((x-1)^2*(x^2+x+1)). (End)

A189786 a(n) = n + [nr/t] + [ns/t]; r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).

Original entry on oeis.org

2, 4, 8, 10, 12, 16, 18, 20, 24, 26, 28, 32, 34, 36, 40, 42, 44, 48, 50, 52, 56, 58, 60, 64, 66, 68, 72, 74, 76, 80, 82, 84, 88, 90, 92, 96, 98, 100, 104, 106, 108, 112, 114, 116, 120, 122, 124, 128, 130, 132, 136, 138, 140, 144, 146, 148, 152, 154, 156, 160, 162, 164, 168, 170, 172, 176, 178, 180, 184, 186, 188, 192, 194, 196, 200, 202, 204

Offset: 1

Clark Kimberling, Apr 27 2011

See A189785.
Conjecture: Sequence consists of all the positive even numbers except numbers of the form 8*x+6, x >= 0. - Harvey P. Dale, Dec 07 2018
Contains numbers like a(143)=382, a(146)=390, a(149)=398, a(152)=406,... which are not in A047464. - R. J. Mathar, Aug 25 2025
For n<143, a(n) = n+A047217(n+1), but then this formula becomes invalid. - R. J. Mathar, Aug 25 2025

Cf. A189785, A189787.

(See A189785.)
With[{t=ArcSin[12/13]},Table[n+Floor[(n*Pi/2)/t]+Floor[(n*ArcSin[5/13])/t],{n,80}]] (* Harvey P. Dale, Dec 07 2018 *)

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A047395 Numbers that are congruent to {0, 2, 6} mod 8.

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A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3*floor(4*n/3).

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A047410 Numbers that are congruent to {2, 4, 6} mod 8.

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A047407 Numbers that are congruent to {0, 4, 6} mod 8.

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A236535 a(n)*Pi is the total length of irregular spiral (center points: 2, 3, 1; pattern 1) after n rotations.

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A189786 a(n) = n + [nr/t] + [ns/t]; r=Pi/2, s=arcsin(5/13), t=arcsin(12/13).

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A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3floor(4n/3).