A047385 - OEIS

%I #43 Nov 22 2024 06:26:55

%S 2,5,9,12,16,19,23,26,30,33,37,40,44,47,51,54,58,61,65,68,72,75,79,82,

%T 86,89,93,96,100,103,107,110,114,117,121,124,128,131,135,138,142,145,

%U 149,152,156,159,163,166,170

%N Numbers that are congruent to {2, 5} mod 7.

%C Also, numbers n such that (n^2+3)/7 is a nonnegative integer. - _Bruno Berselli_, Jan 16 2016

%H David Lovler, <a href="/A047385/b047385.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 G.f.: x*(2 + 3*x + 2*x^2) / ( (1+x)*(x-1)^2 ). - _R. J. Mathar_, Dec 05 2011

%F a(n) = (14*n - (-1)^n - 7)/4. - _Bruno Berselli_, Dec 05 2011

%F a(n) = 4*n - 2 - floor(n/2). - _Wesley Ivan Hurt_, Oct 16 2013

%F E.g.f.: 2 + ((14*x - 7)*exp(x) - exp(-x))/4. - _David Lovler_, Sep 01 2022

%F From _Amiram Eldar_, Sep 26 2022: (Start)

%F a(n) = A113804(n)/2.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/14)*Pi/7. (End)

%F From _Amiram Eldar_, Nov 22 2024: (Start)

%F Product_{n>=1} (1 - (-1)^n/a(n)) = 2*sin(3*Pi/14) (A255249).

%F Product_{n>=1} (1 + (-1)^n/a(n)) = 1/(2*cos(Pi/7)) (A255240). (End)

%p A047385:=n->4*n-2-floor(n/2); seq(A047385(k),k=1..100); # _Wesley Ivan Hurt_, Oct 16 2013

%t Table[4 n - 2 - Floor[n/2], {n,100}] (* _Wesley Ivan Hurt_, Oct 16 2013 *)

%t #+{2,5}&/@(7*Range[0,30])//Flatten (* _Harvey P. Dale_, Jul 15 2017 *)

%o (PARI) a(n)=(14*n-6)>>2 \\ _Charles R Greathouse IV_, Dec 05 2011

%Y Cf. A113804, A255240, A255249.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_