This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A178242 #48 Aug 16 2022 03:07:55
%S A178242 0,3,7,6,9,25,33,21,26,63,75,44,51,117,133,75,84,187,207,114,125,273,
%T A178242 297,161,174,375,403,216,231,493,525,279,296,627,663,350,369,777,817,
%U A178242 429,450,943,987,516,539,1125,1173,611,636,1323,1375,714,741,1537,1593
%N A178242 Numerator of n*(5+n)/((n+1)*(n+4)).
%C A178242 Sequence of differences denominator(n) - numerator(n) = 1,2,2,1... = A014695(n).
%C A178242 Denominator: A160050(n+2).
%H A178242 Vincenzo Librandi, <a href="/A178242/b178242.txt">Table of n, a(n) for n = 0..10000</a>
%H A178242 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quasi-polynomial">Quasi-polynomial</a>.
%H A178242 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-6,10,-12,12,-10,6,-3,1).
%F A178242 a(n) = numerator(A176027(n)/A001793(n+1)).
%F A178242 a(n) = A060819(n)*A060819(n+5).
%F A178242 a(n) = +3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*a(n-6) +6*a(n-7) -3*a(n-8) +a(n-9).
%F A178242 a(n) = 3*a(n-4) -3*a(n-8) +a(n-12).
%F A178242 G.f.: x*(-3+2*x-3*x^2-3*x^3+x^7) / ( (x-1)^3*(x^2+1)^3 ).
%F A178242 a(n) = n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8 = n*(n+5)*(3-i^(n*(n+1)))/8, where i=sqrt(-1); also a(n) = a(n-4)*A028557(n)/A028557(n-4) for n>4. - _Bruno Berselli_, Dec 30 2010
%F A178242 From _Peter Bala_, Aug 07 2022: (Start)
%F A178242 a(n) = numerator of n*(n+5)/4.
%F A178242 a(n) is quasi-polynomial in n: a(4*n) = n*(4*n+5) = A343560(n+1); a(4*n+1) = (2*n+3)*(4*n+1); a(4*n+2) = (2*n+1)*(4*n+7); a(4*n+3) = (n+2)*(4*n+3) = A180863(n+2). (End)
%F A178242 Sum_{n>=1} 1/a(n) = 112/75 - Pi/10. - _Amiram Eldar_, Aug 16 2022
%e A178242 The reduced fractions are 0, 3/5, 7/9, 6/7, 9/10, 25/27, 33/35, 21/22, 26/27, 63/65, 75/77, 44/45, ..
%p A178242 A178242 := proc(n) n*(5+n)/(n+1)/(n+4) ; numer(%) ;end proc:
%p A178242 seq(A178242(n),n=0..80) ; # _R. J. Mathar_, Dec 20 2010
%t A178242 f[n_] := n/GCD[n, 4]; Array[f[#] f[# + 5] &, 50, 0]
%t A178242 Table[Numerator[n*(5+n)/((n+1)*(n+4))], {n,0,50}] (* _G. C. Greubel_, Sep 21 2018 *)
%o A178242 (Magma) [Floor(n*(n+5)*((-1)^((2*n-(-1)^n-3)/4)+3)/8) : n in [0..50]]; // _Vincenzo Librandi_, Oct 08 2011
%o A178242 (PARI) vector(50, n, n--; numerator(n*(5+n)/((n+1)*(n+4)))) \\ _G. C. Greubel_, Sep 21 2018
%Y A178242 Cf. A001793, A014695, A028557, A060819, A160050, A176027.
%K A178242 nonn,easy,frac
%O A178242 0,2
%A A178242 _Paul Curtz_, Dec 20 2010