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A047215 Numbers that are congruent to {0, 2} mod 5. 34

%I #76 Aug 23 2022 10:03:10

%S 0,2,5,7,10,12,15,17,20,22,25,27,30,32,35,37,40,42,45,47,50,52,55,57,

%T 60,62,65,67,70,72,75,77,80,82,85,87,90,92,95,97,100,102,105,107,110,

%U 112,115,117,120,122,125,127,130,132,135,137,140,142,145,147,150,152,155,157

%N Numbers that are congruent to {0, 2} mod 5.

%C Number of partitions of 5n into exactly 2 parts. - _Colin Barker_, Mar 23 2015

%C Numbers k such that k^2/5 + k*(k + 1)/5 = k*(2*k + 1)/5 is a nonnegative integer. - _Bruno Berselli_, Feb 14 2017

%H David Lovler, <a href="/A047215/b047215.txt">Table of n, a(n) for n = 0..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) = floor(5*n/2).

%F From _R. J. Mathar_, Sep 23 2008: (Start)

%F G.f.: x*(2 + 3*x)/((1 + x)*(1 - x)^2).

%F a(n) = 5*n/2 + ((-1)^n-1)/4.

%F a(n+1) - a(n) = A010693(n+1). (End)

%F a(n) = 5*n - a(n-1) - 8 with a(1)=0. - _Vincenzo Librandi_, Aug 05 2010

%F a(n+1) = Sum_{k>=0} A030308(n,k)*A084215(k+1). - _Philippe Deléham_, Oct 17 2011

%F a(n) = 2*n + floor(n/2). - _Arkadiusz Wesolowski_, Sep 19 2012

%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(5)/4 - sqrt(5)*log(phi)/10 + sqrt(1-2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - _Amiram Eldar_, Dec 07 2021

%F E.g.f.: (5*x*exp(x) - sinh(x))/2. - _David Lovler_, Aug 22 2022

%t Table[Floor[5 n/2], {n, 0, 100}] (* or *) LinearRecurrence[{1, 1, -1}, {0, 2, 5},101] (* _Vladimir Joseph Stephan Orlovsky_, Jan 28 2012 *)

%o (PARI) a(n)=5*n\2 \\ _Charles R Greathouse IV_, Oct 17 2011

%Y Different from A038126.

%Y Cf. A001622, A249547 (partial sums), A010693 (1st differences).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

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