A047425 Numbers that are congruent to {3, 4, 5, 6} mod 8.

3, 4, 5, 6, 11, 12, 13, 14, 19, 20, 21, 22, 27, 28, 29, 30, 35, 36, 37, 38, 43, 44, 45, 46, 51, 52, 53, 54, 59, 60, 61, 62, 67, 68, 69, 70, 75, 76, 77, 78, 83, 84, 85, 86, 91, 92, 93, 94, 99, 100, 101, 102, 107, 108, 109, 110, 115, 116, 117, 118, 123, 124

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Complement of numbers congruent to {0, 1, 2, 7} mod 8. - Jaroslav Krizek, Dec 19 2009

In general, sequences congruent to {a, a + i, a + 2i, ..., a + pi} mod k and a + p*i < k have a general form of (k - i*p)*floor(n/p) + i*n + a, from offset 0. - Gary Detlefs, Oct 20 2013

Links

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

Crossrefs

Cf. A047406, A047621.

Programs

Magma

[n : n in [0..150] | n mod 8 in [3, 4, 5, 6]]; // Wesley Ivan Hurt, May 31 2016

Maple

A047425:=n->8*floor((n-1)/4)+((n-1) mod 4)+3: seq(A047425(n), n=1..100); # Wesley Ivan Hurt, May 31 2016

Mathematica

Flatten[# + {3, 4, 5, 6} &/@(8*Range[0, 15])] (* Harvey P. Dale, Jun 26 2011 *)

Formula

G.f.: x*(3+x+x^2+x^3+2*x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

a(n) = 8*floor((n-1)/4) + ((n-1) mod 4) + 3.

a(n) = OR(n-1, 1) + OR(n-1, 2). - Gary Detlefs, Oct 20 2013

From Wesley Ivan Hurt, May 31 2016: (Start)

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

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

a(2k) = A047406(k), a(2k-1) = A047621(k). (End)

E.g.f.: 2 + sin(x) - cos(x) + 2*x*sinh(x) + (2*x - 1)*cosh(x). - Ilya Gutkovskiy, May 31 2016

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