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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A047398, A047535.

Programs

Magma

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

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 *)

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