A047530 Numbers that are congruent to {0, 1, 3, 7} mod 8.

0, 1, 3, 7, 8, 9, 11, 15, 16, 17, 19, 23, 24, 25, 27, 31, 32, 33, 35, 39, 40, 41, 43, 47, 48, 49, 51, 55, 56, 57, 59, 63, 64, 65, 67, 71, 72, 73, 75, 79, 80, 81, 83, 87, 88, 89, 91, 95, 96, 97, 99, 103, 104, 105, 107, 111, 112, 113, 115, 119, 120, 121, 123

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Numbers n such that the n-th homotopy group of the topological group O(oo) does not vanish [see Baez]. Cf. A195679.

The a(n+1) determine the maximal number of linearly independent smooth nowhere zero vector fields on a (2n+1)-sphere, see A053381. - Johannes W. Meijer, Jun 07 2011

Links

• John C. Baez, The Octonions, Bull. Amer. Math. Soc., Vol. 39, No. 2 (2002), pp. 145-205; Errata, ibid., Vol. 42, No. 2 (2005), p. 213; alternative link.
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A008621. - Johannes W. Meijer, Jun 07 2011

Cf. A047470, A047522, A053381, A195679.

Programs

Maple

A047530 := proc(n): ceil(n/4) + 2*ceil((n-1)/4) + 4*ceil((n-2)/4) + ceil((n-3)/4) end: seq(A047530(n), n=0..47); # Johannes W. Meijer, Jun 07 2011
A047530:=n->(8*n-9+I^(2*n)+(2+I)*I^(-n)+(2-I)*I^n)/4: seq(A047530(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Mathematica

Table[(8n-9+I^(2n)+(2+I)*I^(-n)+(2-I)*I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)

PARI

a(n)=n>>2<<3+[-1,0,1,3][n%4+1] \\ Charles R Greathouse IV, Jun 09 2011

Formula

From Johannes W. Meijer, Jun 07 2011: (Start)

a(n) = ceiling(n/4) + 2*ceiling((n-1)/4) + 4*ceiling((n-2)/4) + ceiling((n-3)/4).

a(n+1) = A053381(2^p). (End)

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

From Wesley Ivan Hurt, May 21 2016: (Start)

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

a(n) = (8n-9+i^(2n)+(2+i)*i^(-n)+(2-i)*i^n)/4, where i=sqrt(-1).

a(2n) = A047522(n), a(2n-1) = A047470(n). (End)

E.g.f.: (2 + sin(x) + 2*cos(x) + (4*x - 5)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 21 2016

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

Extensions

More terms from Wesley Ivan Hurt, May 21 2016