A047499 Numbers that are congruent to {3, 4, 5, 7} mod 8.

3, 4, 5, 7, 11, 12, 13, 15, 19, 20, 21, 23, 27, 28, 29, 31, 35, 36, 37, 39, 43, 44, 45, 47, 51, 52, 53, 55, 59, 60, 61, 63, 67, 68, 69, 71, 75, 76, 77, 79, 83, 84, 85, 87, 91, 92, 93, 95, 99, 100, 101, 103, 107, 108, 109, 111, 115, 116, 117, 119, 123, 124

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A047535, A047621.

Programs

Magma

[n : n in [0..150] | n mod 8 in [3, 4, 5, 7]]; // Wesley Ivan Hurt, May 27 2016

Maple

A047499:=n->(8*n-1-I^(2*n)-(1-2*I)*I^(-n)-(1+2*I)*I^n)/4: seq(A047499(n), n=1..100); # Wesley Ivan Hurt, May 27 2016

Mathematica

Table[(8n-1-I^(2n)-(1-2*I)*I^(-n)-(1+2*I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 27 2016 *)
LinearRecurrence[{1,0,0,1,-1},{3,4,5,7,11},70] (* Harvey P. Dale, May 16 2025 *)

Formula

G.f.: x*(3+x+x^2+2*x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Nov 06 2015

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

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

a(n) = (8*n-1-i^(2*n)-(1-2*i)*i^(-n)-(1+2*i)*i^n)/4 where i=sqrt(-1).

a(2k) = A047535(k), a(2k-1) = A047621(k). (End)

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

Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+1)*Pi/16 + (4-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(2-sqrt(2))/8. - Amiram Eldar, Dec 26 2021