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 A247829 #53 Oct 08 2023 04:45:50
%S A247829 0,-1,-3,2,3,5,6,15,21,12,35,45,20,63,77,30,99,117,42,143,165,56,195,
%T A247829 221,72,255,285,90,323,357,110,399,437,132,483,525,156,575,621,182,
%U A247829 675,725,210,783,837,240,899,957,272,1023,1085,306,1155,1221,342,1295
%N A247829 a(3*k) = k*(k+1), a(3*k+1) = (2*k-1)*(2*k+1), a(3*k+2) = (2*k-1)*(2*k+3).
%C A247829 A permutation of A061037(n) = -1, -3, 0, 5, 3, 21, 2, 45, 15, 77, 6, ... and of A214297(n) = -1, 0, -3, 2, 3, 6, 5, 12, ... .
%C A247829 Among consequences: b(n) = 4*a(n) + (sequence of period 3:repeat 1, 4, 16) = 1, 0, 4, 9, 16, 36, 25, 64, 100, ... , is a permutation of the squares of the nonnegative integers A000290(n).
%C A247829 And a(n)/b(n) = 0/1, -1/0, -3/4, 2/9, 3/16, 5/36, ... is a permutation of the Balmer series A061037(n)/A061038(n) = -1/0, -3/4, 0/1, 5/36, 3/16, ... .
%C A247829 a(5n) is divisible by 5.
%H A247829 Colin Barker, <a href="/A247829/b247829.txt">Table of n, a(n) for n = 0..1000</a>
%H A247829 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,3,0,0,-3,0,0,1).
%F A247829 a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9).
%F A247829 a(3*k) + a(3*k+1) + a(3*k+2) = 9*k^2 + 5*k - 4.
%F A247829 G.f.: x*(3*x^7 - 3*x^6 - 14*x^4 - 6*x^3 - 2*x^2 + 3*x + 1)/((x-1)^3*(x^2 +x+1)^3). - _Robert Israel_, Dec 01 2014
%F A247829 a(n) = -(n^2 + n + floor(n/3)*(27*floor(n/3)^3 - 18*(n+1)*floor(n/3)^2 + (3*n^2 + 21*n - 14)*floor(n/3) - (5*n^2 - n + 5)))/2. - _Luce ETIENNE_, Mar 13 2017
%F A247829 From _Amiram Eldar_, Oct 08 2023: (Start)
%F A247829 Sum_{n>=1} 1/a(n) = 1/2.
%F A247829 Sum_{n>=1} (-1)^n/a(n) = Pi/4 + 1 - 2*log(2). (End)
%p A247829 seq(op([k*(k+1),(2*k-1)*(2*k+1),(2*k-1)*(2*k+3)]), k=0..100); # _Robert Israel_, Dec 01 2014
%t A247829 Table[Sequence @@ {n*(n+1), (2*n-1)*(2*n+1), (2*n-1)*(2*n+3)}, {n, 0, 18}] (* _Jean-François Alcover_, Dec 16 2014 *)
%o A247829 (PARI) concat(0, Vec(x*(3*x^7-3*x^6-14*x^4-6*x^3-2*x^2+3*x+1)/((x-1)^3*(x^2+x+1)^3) + O(x^100))) \\ _Colin Barker_, Dec 02 2014
%o A247829 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(3*x^7-3*x^6-14*x^4-6*x^3-2*x^2+3*x+1)/((x-1)^3*(x^2 +x+1)^3))); // _G. C. Greubel_, Sep 20 2018
%Y A247829 Cf. A142717 (2 terms out of 3), A000290, A000466, A002378, A061037/A061038, A078371, A214297.
%K A247829 sign,easy
%O A247829 0,3
%A A247829 _Paul Curtz_, Dec 01 2014