A047410 -id:A047410 - 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

A047464 Numbers that are congruent to {0, 2, 4} mod 8.

Original entry on oeis.org

0, 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

Offset: 1

N. J. A. Sloane

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

Cf. A004773, A047395, A047407, A047410.

[n : n in [0..150] | n mod 8 in [0, 2, 4]]; // Wesley Ivan Hurt, Jun 10 2016

A047464:=n->2*(12*n-15-3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9: seq(A047464(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016

Flatten[#+{0,2,4}&/@(8Range[0,20])] (* or *) LinearRecurrence[{1,0,1,-1}, {0,2,4,8}, 80] (* Harvey P. Dale, May 04 2013 *)

a(n) = 2*floor((n-1)/3)+2*n-2. - Gary Detlefs, Mar 18 2010
a(n) = 2*A004773(n-1). G.f.: 2*x^2*(1+x+2*x^2)/((1+x+x^2)*(x-1)^2). - R. J. Mathar, Mar 29 2010
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-15-3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-4, a(3k-1) = 8k-6, a(3k-2) = 8k-8. (End)
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/16 + (2-sqrt(2))*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 19 2021
a(n) = A047217(n)+n-1. - R. J. Mathar, Aug 25 2025

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

A357837 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a fishbone pattern using symmetric L-shaped tiles with side length 2.

Original entry on oeis.org

0, 4, 10, 20, 32, 46, 64, 84, 106, 132, 160, 190, 224, 260, 298, 340, 384, 430, 480, 532, 586, 644, 704, 766, 832, 900, 970, 1044, 1120, 1198, 1280, 1364, 1450, 1540, 1632, 1726, 1824, 1924, 2026, 2132, 2240, 2350, 2464, 2580, 2698, 2820, 2944, 3070, 3200, 3332

Offset: 0

Stefano Spezia, Oct 17 2022

			Illustrations for n = 1..8:
        _           _ _          _ _ _
       |_|         |  _|        |  _|_|
                   |_|_|        |_|  _|
                                |_|_|_|
    a(1) = 4     a(2) = 10     a(3) = 20
     _ _ _ _     _ _ _ _ _    _ _ _ _ _ _
    |  _|_| |   |  _|_|  _|  |  _|_|  _|_|
    |_|  _|_|   |_|  _|_| |  |_|  _|_|  _|
    |_|_|  _|   |_|_|  _|_|  |_|_|  _|_| |
    |_ _|_|_|   |  _|_|  _|  |  _|_|  _|_|
                |_|_ _|_|_|  |_|  _|_|  _|
                             |_|_|_ _|_|_|
    a(4) = 32    a(5) = 46     a(6) = 64
      _ _ _ _ _ _ _      _ _ _ _ _ _ _ _
     |  _|_|  _|_| |    |  _|_|  _|_|  _|
     |_|  _|_|  _|_|    |_|  _|_|  _|_| |
     |_|_|  _|_|  _|    |_|_|  _|_|  _|_|
     |  _|_|  _|_| |    |  _|_|  _|_|  _|
     |_|  _|_|  _|_|    |_|  _|_|  _|_| |
     |_|_|  _|_|  _|    |_|_|  _|_|  _|_|
     |_ _|_|_ _|_|_|    |  _|_|  _|_|  _|
                        |_|_ _|_|_ _|_|_|
        a(7) = 84           a(8) = 106

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A002264, A002522, A005843, A047410 (first differences), A071619, A211547.

Cf. A345118.

Table[2(Ceiling[2(n+1)^2/3]-1),{n,0,49}]

a(n) = 2*(ceiling(2*(n+1)^2/3) - 1).
a(n) = 2*(A071619(n+1) - 1).
a(n) = 2*(1 + n^2 - 2*(n - 2)*floor((n - 1)/3) + 3*floor((n - 1)/3)^2) for n > 0.
a(n) = Sum_{k=1..n} A047410(k+1) for n > 0.
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 4.
O.g.f.: 2*x*(2 + x + 2*x^2 - x^3)/((1 - x)^3*(1 + x + x^2)).
E.g.f.: 2*exp(-x/2)*(exp(3*x/2)*(6*x*(3 + x) - 1) + cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9.

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

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A047464 Numbers that are congruent to {0, 2, 4} mod 8.

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

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A357837 a(n) is the sum of the lengths of all the segments used to draw a square of side n representing a fishbone pattern using symmetric L-shaped tiles with side length 2.

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