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

1,3

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

Table of n, a(n) for n=1..63.

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).

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

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 *)

Prog

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

Crossrefs

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

Cf. A047470, A047522, A053381, A195679.

Sequence in context: A116034 A364166 A122987 * A346300 A265350 A096315

Adjacent sequences: A047527 A047528 A047529 * A047531 A047532 A047533

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Wesley Ivan Hurt, May 21 2016

Status

approved