A047533 Numbers that are congruent to {1, 2, 3, 7} mod 8.

1, 2, 3, 7, 9, 10, 11, 15, 17, 18, 19, 23, 25, 26, 27, 31, 33, 34, 35, 39, 41, 42, 43, 47, 49, 50, 51, 55, 57, 58, 59, 63, 65, 66, 67, 71, 73, 74, 75, 79, 81, 82, 83, 87, 89, 90, 91, 95, 97, 98, 99, 103, 105, 106, 107, 111, 113, 114, 115, 119, 121, 122, 123

Offset

1,2

Links

G. C. Greubel, 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 29 2016: (Start)

G.f.: x*(1+x+x^2+4*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) = (8*n-7+i^(2*n)+(1+2*i)*i^(-n)+(1-2*i)*i^n)/4 where i=sqrt(-1).

a(2k) = A047524(k), a(2k-1) = A047471(k). (End)

E.g.f.: (2 + 2*sin(x) + cos(x) + 4*(x - 1)*sinh(x) + (4*x - 3)*cosh(x))/2. - Ilya Gutkovskiy, May 29 2016

Sum_{n>=1} (-1)^(n+1)/a(n) = 3*sqrt(2)*Pi/16 + log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 23 2021

Maple

A047533:=n->(8*n-7+I^(2*n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4: seq(A047533(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Mathematica

Table[(8n-7+I^(2n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 29 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 7, 9}, 50] (* G. C. Greubel, May 29 2016 *)

Prog

(Magma) [n : n in [0..150] | n mod 8 in [1, 2, 3, 7]]; // Wesley Ivan Hurt, May 29 2016

Crossrefs

Cf. A047471, A047524.

Sequence in context: A349641 A140221 A046668 * A060525 A152863 A047359

Adjacent sequences: A047530 A047531 A047532 * A047534 A047535 A047536

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved