A047544 Numbers that are congruent to {1, 3, 4, 7} mod 8.

1, 3, 4, 7, 9, 11, 12, 15, 17, 19, 20, 23, 25, 27, 28, 31, 33, 35, 36, 39, 41, 43, 44, 47, 49, 51, 52, 55, 57, 59, 60, 63, 65, 67, 68, 71, 73, 75, 76, 79, 81, 83, 84, 87, 89, 91, 92, 95, 97, 99, 100, 103, 105, 107, 108, 111, 113, 115, 116, 119, 121, 123, 124

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A004767, A047461.

Programs

Magma

[n : n in [0..150] | n mod 8 in [1, 3, 4, 7]]; // Wesley Ivan Hurt, May 29 2016

Maple

A047544:=n->(8*n-5+I^(2*n)+I^(1-n)-I^(1+n))/4: seq(A047544(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Mathematica

Table[(8n-5+I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 29 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 4, 7, 9}, 50] (* G. C. Greubel, May 29 2016 *)

Formula

From Wesley Ivan Hurt, May 29 2016: (Start)

G.f.: x*(1+2*x+x^2+3*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-5+i^(2*n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).

a(2k) = A004767(k-1) for n>0, a(2k-1) = A047461(k). (End)

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

Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)+3)*Pi/16 - log(2)/4 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 24 2021