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 A129109 #28 Nov 17 2024 10:12:42
%S A129109 7,22,49,88,139,202,277,364,463,574,697,832,979,1138,1309,1492,1687,
%T A129109 1894,2113,2344,2587,2842,3109,3388,3679,3982,4297,4624,4963,5314,
%U A129109 5677,6052,6439,6838,7249,7672,8107,8554,9013,9484,9967,10462,10969,11488,12019,12562
%N A129109 Sums of three consecutive hexagonal numbers.
%C A129109 Arises in hexagonal number analog to A129803 Triangular numbers which are the sum of three consecutive triangular numbers. What are the hexagonal numbers which are the sum of three consecutive hexagonal numbers? Prime for a(0) = 7, a(4) = 139, a(6) = 277, a(8) = 463, a(18) = 2113, a(22) = 3109, a(26) = 4297, a(38) = 9013, a(40) = 9967.
%H A129109 Vincenzo Librandi, <a href="/A129109/b129109.txt">Table of n, a(n) for n = 0..1000</a>
%H A129109 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A129109 a(n) = H(n) + H(n+1) + H(n+2) where H(n) = A000384(n) = n*(2*n-1).
%F A129109 a(n) = 6*n^2 + 9*n + 7.
%F A129109 From _Colin Barker_, Feb 20 2012: (Start)
%F A129109 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A129109 G.f.: (7 + x + 4*x^2)/(1-x)^3. (End)
%F A129109 E.g.f.: (7 + 15*x + 6*x^2)*exp(x). - _Elmo R. Oliveira_, Nov 16 2024
%e A129109 a(0) = H(0) + H(1) + H(2) = 0 + 1 + 6 = 7 = 6*0^2 + 9*0 + 7.
%e A129109 a(1) = H(1) + H(2) + H(3) = 1 + 6 + 15 = 22 = 6*1^2 + 9*1 + 7.
%e A129109 a(2) = H(2) + H(3) + H(4) = 6 + 15 + 28 = 49 = 6*2^2 + 9*2 + 7.
%t A129109 LinearRecurrence[{3,-3,1},{7,22,49},50] (* _Vincenzo Librandi_, Feb 20 2012 *)
%t A129109 Total/@Partition[PolygonalNumber[6,Range[0,50]],3,1] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Mar 14 2020 *)
%o A129109 (Magma) I:=[7,22,49]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 20 2012
%o A129109 (PARI) a(n)=6*n^2+9*n+7 \\ _Charles R Greathouse IV_, Feb 20 2012
%Y A129109 Cf. A000384, A007667, A034961, A129803, A129863.
%K A129109 easy,nonn
%O A129109 0,1
%A A129109 _Jonathan Vos Post_, May 24 2007