A146763 - OEIS

%I #41 Sep 17 2023 10:00:11

%S 0,4,10,14,20,24,30,34,40,44,50,54,60,64,70,74,80,84,90,94,100,104,

%T 110,114,120,124,130,134,140,144,150,154,160,164,170,174,180,184,190,

%U 194,200,204,210,214,220,224,230,234,240,244,250,254,260,264,270,274

%N Rank of terms ending in 0 in A061039.

%C From Paschen spectrum of hydrogen.

%C Numbers that are congruent to 0 or 4 mod 10. - _Philippe Deléham_, Oct 18 2011

%H G. C. Greubel, <a href="/A146763/b146763.txt">Table of n, a(n) for n = 0..5000</a>

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

%F a(n) = 10*n - 6 - a(n-1) (with a(0)=0). - _Vincenzo Librandi_, Nov 26 2010

%F a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=4 and b(k) = 5*2^k = A020714(k) for k>0. - _Philippe Deléham_, Oct 18 2011

%F From _Colin Barker_, May 14 2012: (Start)

%F a(n) = (-1 + (-1)^n + 10*n)/2.

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

%F G.f.: x*(4+6*x)/((1-x)^2*(1+x)). (End)

%F E.g.f.: 1/2 (exp(-x) - (1 - 10*x)*exp(x)). - _G. C. Greubel_, Mar 10 2022

%F Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(1-2/sqrt(5))*Pi/20 - log(phi)/(4*sqrt(5)) + log(5)/8, where phi is the golden ratio (A001622). - _Amiram Eldar_, Sep 17 2023

%t Select[Range[0, 100], MemberQ[{0,4}, Mod[#, 10]] &] (* _K G Teal_, Dec 02 2014 *)

%o (Magma) [5*n - (n mod 2): n in [0..60]]; // _G. C. Greubel_, Mar 10 2022

%o (Sage) [5*n - (n%2) for n in (0..60)] # _G. C. Greubel_, Mar 10 2022

%Y Cf. A001622, A020714, A030308, A061039.

%K nonn,base,easy

%O 0,2

%A _Paul Curtz_, Nov 02 2008