A047624 Numbers that are congruent to {0, 1, 3, 5} mod 8.

0, 1, 3, 5, 8, 9, 11, 13, 16, 17, 19, 21, 24, 25, 27, 29, 32, 33, 35, 37, 40, 41, 43, 45, 48, 49, 51, 53, 56, 57, 59, 61, 64, 65, 67, 69, 72, 73, 75, 77, 80, 81, 83, 85, 88, 89, 91, 93, 96, 97, 99, 101, 104, 105, 107, 109, 112, 113, 115, 117, 120, 121, 123

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. A004526, A016813, A042948, A047470, A140081, A140201, A173562.

Programs

Magma

[n : n in [0..150] | n mod 8 in [0, 1, 3, 5]]; // Wesley Ivan Hurt, Jun 01 2016

Maple

A047624:=n->(8*n-11-I^(2*n)+I^(1-n)-I^(1+n))/4: seq(A047624(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Mathematica

Table[(8n-11-I^(2n)+I^(1-n)-I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, Jun 01 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,3,5,8},100] (* G. C. Greubel, Jun 01 2016 *)

Formula

From Reinhard Zumkeller, Feb 21 2010: (Start)

a(n+1) = A173562(n) - A173562(n-1);

a(n+1) - a(n) = A140081(n-1) + 1;

a(n) = A140201(n-1) + A042948(A004526(n-1)). (End)

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

From Wesley Ivan Hurt, Jun 01 2016: (Start)

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

a(n) = (8*n-11-i^(2*n)+i^(1-n)-i^(1+n))/4 where i=sqrt(-1).

a(2k) = A016813(k-1) for k>0, a(2k-1) = A047470(k). (End)

E.g.f.: (6 + sin(x) + (4*x - 5)*sinh(x) + (4*x - 6)*cosh(x))/2. - Ilya Gutkovskiy, Jun 01 2016

Sum_{n>=2} (-1)^n/a(n) = (3-sqrt(2))*Pi/16 + (8-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - Amiram Eldar, Dec 20 2021