A047514 Numbers that are congruent to {3, 4, 6, 7} mod 8.

3, 4, 6, 7, 11, 12, 14, 15, 19, 20, 22, 23, 27, 28, 30, 31, 35, 36, 38, 39, 43, 44, 46, 47, 51, 52, 54, 55, 59, 60, 62, 63, 67, 68, 70, 71, 75, 76, 78, 79, 83, 84, 86, 87, 91, 92, 94, 95, 99, 100, 102, 103, 107, 108, 110, 111, 115, 116, 118, 119, 123, 124

Offset

1,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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

Formula

From Wesley Ivan Hurt, May 27 2016: (Start)

G.f.: x*(3+x+2*x^2+x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).

a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.

a(n) = (1+i)*(4*n-4*n*i+(i-1)*i^(2*n)+i^(1-n)-i^n)/4 where i=sqrt(-1).

a(2k) = A047535(k), a(2k-1) = A047398(k). (End)

E.g.f.: (2 + sin(x) - cos(x) + (4*x + 1)*sinh(x) + (4*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 27 2016

From Wesley Ivan Hurt, Aug 10 2016: (Start)

a(n) = a(n-4) + 8 for n > 4.

a(4*k) = 8*k-1, a(4*k-1) = 8*k-2, a(4*k-2) = 8*k-4, a(4*k-3) = 8*k-5. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 - (sqrt(2)-1)*log(2)/8 + sqrt(2)*log(2-sqrt(2))/4. - Amiram Eldar, Dec 26 2021

Maple

A047514:=n->(1+I)*(4*n-4*n*I+(I-1)*I^(2*n)+I^(1-n)-I^n)/4: seq(A047514(n), n=1..100); # Wesley Ivan Hurt, May 27 2016

Mathematica

Table[(1+I)*(4n-4n*I+(I-1)*I^(2n)+I^(1-n)-I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 27 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {3, 4, 6, 7, 11}, 100] (* Vincenzo Librandi, Aug 11 2016 *)

Prog

(Magma) [n : n in [0..150] | n mod 8 in [3, 4, 6, 7]]; // Wesley Ivan Hurt, May 27 2016

Crossrefs

Cf. A047398, A047535.

Sequence in context: A294488 A364341 A335059 * A345716 A011975 A202112

Adjacent sequences: A047511 A047512 A047513 * A047515 A047516 A047517

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved