|
%I #50 Sep 08 2022 08:44:57
%S 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,
%T 49,52,55,56,57,60,63,64,65,68,71,72,73,76,79,80,81,84,87,88,89,92,95,
%U 96,97,100,103,104,105,108,111,112,113,116,119,120,121,124
%N Numbers that are congruent to {0, 1, 4, 7} mod 8.
%C Related to a Chebyshev transform of A046055. See A074231. - _Paul Barry_, Oct 27 2004
%C 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
%C 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
%C 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
%H Colin Barker, <a href="/A047538/b047538.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F From _Paul Barry_, Oct 27 2004: (Start)
%F G.f.: x^2*(1+x)^2 / ((1+x^2)*(1-2*x+x^2)).
%F E.g.f.: 2*x*exp(x)-sin(x).
%F a(n) = 2*n-2-sin(Pi*(n-1)/2).
%F a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4) for n>4. (End)
%F a(n) = 2*n-2-(1+(-1)^n)*(-1)^((2*n-3)/4-(-1)^n/4)/2. - _Wesley Ivan Hurt_, Sep 22 2015
%F a(n) = (-4+(-i)^n+i^n+4*n)/2, where i = sqrt(-1). - _Colin Barker_, Oct 18 2015
%F 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
%p A047538:=n->2*n-2-sin(Pi*(n-1)/2): seq(A047538(n), n=1..80); # _Wesley Ivan Hurt_, Sep 22 2015
%t Table[2n-2-Sin[Pi*(n-1)/2], {n, 80}] (* _Wesley Ivan Hurt_, Sep 22 2015 *)
%t Select[Range[0, 150], MemberQ[{0, 1, 4, 7}, Mod[#, 8]] &] (* _Vincenzo Librandi_, Sep 23 2015 *)
%t LinearRecurrence[{2,-2,2,-1},{0,1,4,7},100] (* _Harvey P. Dale_, Aug 12 2016 *)
%o (Sage) [lucas_number1(n,0,1)+2*n-4 for n in (2..57)] # _Zerinvary Lajos_, Jul 06 2008
%o (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
%o (Magma) [n: n in [0..150] | n mod 8 in {0,1,4,7}]; // _Vincenzo Librandi_, Sep 23 2015
%o (PARI) a(n) = (-4+(-I)^n+I^n+4*n)/2 \\ _Colin Barker_, Oct 18 2015
%o (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
%Y Cf. A047404, A047431, A047546, A047557, A047578, A047620, A056594.
%Y Cf. A046055, A074231.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_
%E More terms from _Wesley Ivan Hurt_, Sep 22 2015
%E G.f. adapted to offset by _Colin Barker_, Oct 18 2015
|
