A047615 - OEIS

%I #66 Mar 30 2024 11:38:43

%S 0,5,8,13,16,21,24,29,32,37,40,45,48,53,56,61,64,69,72,77,80,85,88,93,

%T 96,101,104,109,112,117,120,125,128,133,136,141,144,149,152,157,160,

%U 165,168,173,176,181,184,189,192,197,200,205,208,213,216,221,224,229,232

%N Numbers that are congruent to {0, 5} mod 8.

%H Colin Barker, <a href="/A047615/b047615.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F a(n) = 8*n-a(n-1)-11 (with a(1)=0). - _Vincenzo Librandi_, Aug 06 2010

%F a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=5 and b(k)=2^(k+2) for k>0. - _Philippe Deléham_, Oct 17 2011

%F From _Wesley Ivan Hurt_, Mar 26 2015: (Start)

%F a(n) = a(n-1)+a(n-2)-a(n-3).

%F a(n) = (8n - 7 + (-1)^n)/2. (End)

%F G.f.: x^2*(5+3*x) / ((1-x)^2*(1+x)). - _Colin Barker_, Aug 25 2016

%F From _Franck Maminirina Ramaharo_, Jul 23 2018: (Start)

%F a(n) = A047470(n) - (-1)^(n - 1) + 1.

%F E.g.f.: (6 + exp(-x) + (8*x - 7)*exp(x))/2. (End)

%F Sum_{n>=2} (-1)^n/a(n) = log(2)/2 - (sqrt(2)-1)*Pi/16 - sqrt(2)*log(sqrt(2)+1)/8. - _Amiram Eldar_, Dec 18 2021

%p a:=n->add(4-(-1)^j, j=1..n): seq(a(n), n=0..59); # _Zerinvary Lajos_, Dec 13 2008

%t Table[(8 n - 7 + (-1)^n)/2, {n, 1, 40}] (* _Wesley Ivan Hurt_, Mar 26 2015 *)

%t Rest@ CoefficientList[Series[x^2*(5 + 3 x)/((1 - x)^2*(1 + x)), {x, 0, 59}], x] (* _Michael De Vlieger_, Aug 25 2016 *)

%t Rest@(Range[0, 60]! CoefficientList[ Series[(6 + Exp[-x] + (8 x - 7)*Exp[x])/2, {x, 0, 60}], x]) (* or *)

%t LinearRecurrence[{1, 1, -1}, {0, 5, 8}, 60] (* _Robert G. Wilson v_, Jul 23 2018 *)

%o (PARI) forstep(n=0,200,[5,3],print1(n", ")) \\ _Charles R Greathouse IV_, Oct 17 2011

%o (PARI) concat(0, Vec(x^2*(5+3*x)/((1-x)^2*(1+x)) + O(x^100))) \\ _Colin Barker_, Aug 25 2016

%o (Magma) [(8*n - 7 + (-1)^n)/2 : n in [1..50]]; // _Wesley Ivan Hurt_, Mar 26 2015

%o (GAP) Filtered([0..250], n->n mod 8=0 or n mod 8=5); # _Muniru A Asiru_, Jul 23 2018

%o (Python)

%o def A047615(n): return (n<<2)-3-(n&1) # _Chai Wah Wu_, Mar 30 2024

%Y Union of A008590 and A004770.

%Y Cf. A047398, A047452, A047461, A047470, A047524, A047535, A047617.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Vincenzo Librandi_, Aug 06 2010