A047538 Numbers that are congruent to {0, 1, 4, 7} mod 8.

0, 1, 4, 7, 8, 9, 12, 15, 16, 17, 20, 23, 24, 25, 28, 31, 32, 33, 36, 39, 40, 41, 44, 47, 48, 49, 52, 55, 56, 57, 60, 63, 64, 65, 68, 71, 72, 73, 76, 79, 80, 81, 84, 87, 88, 89, 92, 95, 96, 97, 100, 103, 104, 105, 108, 111, 112, 113, 116, 119, 120, 121, 124

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Related to a Chebyshev transform of A046055. See A074231. - Paul Barry, Oct 27 2004

Starting (1, 4, 7, ...) = partial sums of (1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, ...). - Gary W. Adamson, Jun 19 2008

The product of any two terms belongs to the sequence and therefore also a(n)^2, a(n)^3, a(n)^4 etc. - Bruno Berselli, Nov 28 2012

Nonnegative m such that floor(k*(m/4)^2) = k*floor((m/4)^2), where k can assume the values from 4 to 15. See also the second comment in A047513. - Bruno Berselli, Dec 03 2015

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Crossrefs

Cf. A047404, A047431, A047546, A047557, A047578, A047620, A056594.

Cf. A046055, A074231.

Programs

Magma

[2*n-2-(1+(-1)^n)*(-1)^((2*n-3) div 4-(-1)^n div 4) / 2 : n in [1..80]]; // Wesley Ivan Hurt, Sep 22 2015

Magma

[n: n in [0..150] | n mod 8 in {0,1,4,7}]; // Vincenzo Librandi, Sep 23 2015

Maple

A047538:=n->2*n-2-sin(Pi*(n-1)/2): seq(A047538(n), n=1..80); # Wesley Ivan Hurt, Sep 22 2015

Mathematica

Table[2n-2-Sin[Pi*(n-1)/2], {n, 80}] (* Wesley Ivan Hurt, Sep 22 2015 *)
Select[Range[0, 150], MemberQ[{0, 1, 4, 7}, Mod[#, 8]] &] (* Vincenzo Librandi, Sep 23 2015 *)
LinearRecurrence[{2,-2,2,-1},{0,1,4,7},100] (* Harvey P. Dale, Aug 12 2016 *)

PARI

a(n) = (-4+(-I)^n+I^n+4*n)/2 \\ Colin Barker, Oct 18 2015

PARI

concat(0, Vec(x^2*(1+x)^2/((1+x^2)*(1-2*x+x^2)) + O(x^100))) \\ Colin Barker, Oct 18 2015

Sage

[lucas_number1(n,0,1)+2*n-4 for n in (2..57)] # Zerinvary Lajos, Jul 06 2008

Formula

From Paul Barry, Oct 27 2004: (Start)

G.f.: x^2*(1+x)^2 / ((1+x^2)*(1-2*x+x^2)).

E.g.f.: 2*x*exp(x)-sin(x).

a(n) = 2*n-2-sin(Pi*(n-1)/2).

a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>4. (End)

a(n) = 2*n-2-(1+(-1)^n)*(-1)^((2*n-3)/4-(-1)^n/4)/2. - Wesley Ivan Hurt, Sep 22 2015

a(n) = (-4+(-i)^n+i^n+4*n)/2, where i = sqrt(-1). - Colin Barker, Oct 18 2015

Sum_{n>=2} (-1)^n/a(n) = (6-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021

Extensions

More terms from Wesley Ivan Hurt, Sep 22 2015

G.f. adapted to offset by Colin Barker, Oct 18 2015