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A032769 Numbers that are congruent to {0, 1, 2, 4} mod 5.

%I A032769 #42 Jan 02 2025 09:30:12
%S A032769 0,1,2,4,5,6,7,9,10,11,12,14,15,16,17,19,20,21,22,24,25,26,27,29,30,
%T A032769 31,32,34,35,36,37,39,40,41,42,44,45,46,47,49,50,51,52,54,55,56,57,59,
%U A032769 60,61,62,64,65,66,67,69,70,71,72,74,75,76,77,79,80,81,82,84,85
%N A032769 Numbers that are congruent to {0, 1, 2, 4} mod 5.
%C A032769 Also, numbers m such that m*(m+1)*(m+2)*(m+3)*(m+4)/(m+(m+1)+(m+2)+(m+3)+(m+4)) is an integer.
%H A032769 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A032769 a(n) = (1/8)*(10*n-11+(-1)^n+2*(-1)^floor(n/2)). - _Ralf Stephan_, Jun 09 2005
%F A032769 a(n) = floor((5*n-4)/4). - _Gary Detlefs_, Mar 06 2010
%F A032769 G.f.: x^2*(1+x+2*x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - _R. J. Mathar_, Oct 08 2011
%F A032769 From _Wesley Ivan Hurt_, May 30 2016: (Start)
%F A032769 a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F A032769 a(n) = (10*n-11+i^(2*n)+(1+i)*I^(-n)+(1-i)*i^n)/8 where i=sqrt(-1).
%F A032769 a(2k) = A047209(k), a(2k-1) = A047215(k). (End)
%F A032769 E.g.f.: (4 + sin(x) + cos(x) + (5*x - 6)*sinh(x) + 5*(x - 1)*cosh(x))/4. - _Ilya Gutkovskiy_, May 31 2016
%F A032769 Sum_{n>=2} (-1)^n/a(n) = log(5)/4 + 3*sqrt(5)*log(phi)/10 - sqrt(1-2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - _Amiram Eldar_, Dec 10 2021
%p A032769 seq(floor((5*n-4)/4), n=1..69); # _Gary Detlefs_, Mar 06 2010
%t A032769 Table[Floor[(5n - 4)/4], {n, 80}] (* _Wesley Ivan Hurt_, May 30 2016 *)
%o A032769 (Magma) [Floor((5*n - 4)/4) : n in [1..80]]; // _Wesley Ivan Hurt_, May 30 2016
%o A032769 (PARI) a(n)=5*n\4-1 \\ _Charles R Greathouse IV_, Jan 02 2025
%Y A032769 Cf. A001622, A032768, A032770, A047209, A047215.
%K A032769 nonn,easy
%O A032769 1,3
%A A032769 _Patrick De Geest_, May 15 1998
%E A032769 Better description from _Michael Somos_, Jun 08 2000