A215414 - OEIS

%I #27 Feb 27 2017 02:56:14

%S 0,31536000,63072000,94694400,126230400,157766400,189302400,220924800,

%T 252460800,283996800,315532800,347155200,378691200,410227200,

%U 441763200,473385600,504921600,536457600,567993600,599616000,631152000,662688000,694224000,725846400,757382400

%N Unix epoch timestamp for start of year, beginning with 1970.

%C This is based on a naive multiplication of A033172 with a fixed number of seconds per year, 24*3600 = 86400. It ignores that leap years are not regularly occurring after 4 years (but after 400 years, note the formula that relates a(n+4) to a(n) and also the simple Mma implementation), ignores leap seconds, and any other influences that align the slowing down of the Earth rotation in an astronomical fixed coordinate system measured relative to atomic clocks. In summary, the use of "year" in the definition is not commensurate with years in standard astronomical or earth observational terms. - _R. J. Mathar_, Aug 21 2012

%H G. C. Greubel, <a href="/A215414/b215414.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unix_time">Unix time</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Leap_seconds">Leap second</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Revised_Julian_calendar">Revised Julian Calendar</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/IERS">International Earth Rotation and Reference Systems Service</a>

%F From _Alexander R. Povolotsky_, Aug 20 2012: (Start)

%F a(n) = 10800*(2922*n + (-1)^n + (1+i)*(-i)^n + (1-i)*i^n - 2923).

%F a(n+4) = a(n) + 126230400.

%F G.f.: 86400*(365*x +365*x^2 +366*x^3 +365*x^4)/((1-x)^2*(1+x+x^2+x^3)). (End)

%t lst = {}; t = 86400; Do[e = t*(365*(n - 1) + Ceiling[n/4]); If[! Mod[n, 4] == 0, e = e - t]; AppendTo[lst, e], {n, 25}]; lst (* _Arkadiusz Wesolowski_, Aug 20 2012 *)

%t CoefficientList[Series[86400*(365*x + 365*x^2 + 366*x^3 + 365*x^4)/((x - 1)^2*(1 +x +x^2 +x^3)), {x,0,50}], x] (* _G. C. Greubel_, Feb 26 2017 *)

%o (PARI) x='x+O('x^50); Vec(86400*(365*x +365*x^2 +366*x^3 +365*x^4)/((1-x)^2*(1+x+x^2+x^3))) \\ _G. C. Greubel_, Feb 26 2017

%Y Cf. A193910.

%K easy,nonn,less

%O 1,2

%A _Kyle Stern_, Aug 09 2012

%E a(11)-a(25) from _Arkadiusz Wesolowski_, Aug 20 2012