A047551 Numbers that are congruent to {0, 1, 6, 7} mod 8.

0, 1, 6, 7, 8, 9, 14, 15, 16, 17, 22, 23, 24, 25, 30, 31, 32, 33, 38, 39, 40, 41, 46, 47, 48, 49, 54, 55, 56, 57, 62, 63, 64, 65, 70, 71, 72, 73, 78, 79, 80, 81, 86, 87, 88, 89, 94, 95, 96, 97, 102, 103, 104, 105, 110, 111, 112, 113, 118, 119, 120, 121, 126

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. A010873, A030308, A047451, A047522.

Programs

Magma

[n : n in [0..150] | n mod 8 in [0, 1, 6, 7]]; // Wesley Ivan Hurt, May 29 2016

Maple

A047551:=n->(4*n-3-I^(2*n)+(1-I)*I^(-n)+(1+I)*I^n)/2: seq(A047551(n), n=1..100); # Wesley Ivan Hurt, May 29 2016

Mathematica

Table[(4n-3-I^(2n)+(1-I)*I^(-n)+(1+I)*I^n)/2, {n, 80}] (* Wesley Ivan Hurt, May 29 2016 *)

Formula

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=6 and b(k)=2^(k+1) for k>1. - Philippe Deléham, Oct 19 2011

a(n) = 2n - A010873(n+1). - Wesley Ivan Hurt, Jul 07 2013

G.f.: x^2*(1+5*x+x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Jul 14 2013

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

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

a(n) = (4*n-3-i^(2*n)+(1-i)*i^(-n)+(1+i)*i^n)/2 where i=sqrt(-1).

a(2k) = A047522(k), a(2k-1) = A047451(k). (End)

E.g.f.: 1 - sin(x) + cos(x) + (2*x - 1)*sinh(x) + 2*(x - 1)*cosh(x). - Ilya Gutkovskiy, May 29 2016

Sum_{n>=2} (-1)^n/a(n) = Pi/16 + (5-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021