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A047530 Numbers that are congruent to {0, 1, 3, 7} mod 8. 3
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 (list; graph; refs; listen; history; text; internal format)
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., 39 (2002), 145-205.

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

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: A091679 A116034 A122987 * A265350 A096315 A286395

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

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Last modified January 29 11:16 EST 2020. Contains 331337 sequences. (Running on oeis4.)