A047397 Numbers that are congruent to {0, 1, 2, 6} mod 8.

0, 1, 2, 6, 8, 9, 10, 14, 16, 17, 18, 22, 24, 25, 26, 30, 32, 33, 34, 38, 40, 41, 42, 46, 48, 49, 50, 54, 56, 57, 58, 62, 64, 65, 66, 70, 72, 73, 74, 78, 80, 81, 82, 86, 88, 89, 90, 94, 96, 97, 98, 102, 104, 105, 106, 110, 112, 113, 114, 118, 120, 121, 122

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. A047452, A047467.

Programs

Magma

[n : n in [0..150] | n mod 8 in [0, 1, 2, 6]]; // Wesley Ivan Hurt, May 24 2016

Maple

A047397:=n->(8*n-11+I^(2*n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4: seq(A047397(n), n=1..100); # Wesley Ivan Hurt, May 24 2016

Mathematica

Table[(8n-11+I^(2n)+(1+2*I)*I^(-n)+(1-2*I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 24 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,1,2,6,8},70] (* Harvey P. Dale, Dec 31 2017 *)

Formula

G.f.: x^2*(1+x+4*x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011

From Wesley Ivan Hurt, May 24 2016: (Start)

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

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

a(2k) = A047452(k), a(2k-1) = A047467(k). (End)

E.g.f.: (4 + 2*sin(x) + cos(x) + (4*x - 6)*sinh(x) + (4*x - 5)*cosh(x))/2. - Ilya Gutkovskiy, May 25 2016

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

Extensions

More terms from Wesley Ivan Hurt, May 24 2016