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 A184218 #31 Sep 08 2022 08:45:55
%S A184218 0,0,0,0,9,14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230,
%T A184218 252,275,299,324,350,377,405,434,464,495,527,560,594,629,665,702,740,
%U A184218 779,819,860,902,945,989,1034,1080,1127,1175,1224,1274,1325,1377
%N A184218 a(n) = largest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.
%C A184218 From the definition, a(n) = A000217(n) - (n + 1) if A000217(n) - (n + 1) > (n + 1), or 0 otherwise, where A000217 are the triangular numbers.
%H A184218 G. C. Greubel and Vincenzo Librandi, <a href="/A184218/b184218.txt">Table of n, a(n) for n = 1..10000</a> [Originally computed by Remi Eismann]
%H A184218 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A184218 a(n) = (n+1)*(n-2)/2 = A000096(n-2) for n >= 5 and a(n) = 0 for n <= 4. - _M. F. Hasler_, Jan 10 2011
%F A184218 From _Chai Wah Wu_, Jun 21 2016: (Start)
%F A184218 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 7.
%F A184218 G.f.: x^5*(5*x^2 - 13*x + 9)/(1 - x)^3. (End)
%e A184218 For n = 3 we have A000217(3) = 6, A000217(4) = 10; there is no k such that 10 - 6 = 4 = (6 mod k), hence a(3) = 0.
%e A184218 For n = 5 we have A000217(5) = 15, A000217(6) = 21; 9 is the largest k such that 21 - 15 = 6 = (15 mod k), hence a(5) = 9; a(5) = A000217(5) - (5 + 1) = 15 - 6 = 9.
%e A184218 For n = 24 we have A000217(24) = 300, A000217(25) = 325; 275 is the largest k such that 325 - 300 = 25 = (300 mod k), hence a(24) = 275; a(24) = A000217(24) - (24 + 1) = 275.
%t A184218 Join[{0, 0, 0, 0}, LinearRecurrence[{3, -3, 1}, {9, 14, 20}, 100]] (* _G. C. Greubel_, Jun 22 2016 *)
%t A184218 lim = 10^4; Table[SelectFirst[Reverse@ Range@ lim, Function[k, PolygonalNumber[n + 1] == # + Mod[#, k] &@ PolygonalNumber@ n]], {n, 53}] /. {k_ /; MissingQ@ k -> 0, k_ /; k == lim -> 0} (* _Michael De Vlieger_, Jun 30 2016, Version 10.4 *)
%o A184218 (Magma) [0,0,0,0] cat [(n+1)*(n-2)/2: n in [5..60]]; // _Vincenzo Librandi_, Jun 22 2016
%Y A184218 Cf. essentially the same as A000096, A000217, A000027, A130703, A184219, A118534, A117078, A117563, A001223.
%K A184218 nonn,easy
%O A184218 1,5
%A A184218 _Rémi Eismann_, Jan 10 2011