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 A378569 #52 Feb 08 2025 04:56:14
%S A378569 7,13,25,43,67,97,133,175,223,277,337,403,475,553,637,727,823,925,
%T A378569 1033,1147,1267,1393,1525,1663,1807,1957,2113,2275,2443,2617,2797,
%U A378569 2983,3175,3373,3577,3787,4003,4225,4453,4687,4927,5173,5425,5683,5947,6217,6493,6775,7063,7357,7657,7963,8275,8593,8917,9247,9583,9925
%N A378569 a(n) = 3*n*(n+1) + 7.
%C A378569 The terms a(1) = 13 through a(7) = 175, coincide with A102724(2..8), cumulative sums of pairs of primes. From a(8) = 223, it differs from A102724(9) = 227.
%H A378569 Vincenzo Librandi, <a href="/A378569/b378569.txt">Table of n, a(n) for n = 0..10000</a>
%H A378569 Paul Bourgarel, <a href="https://dn790006.ca.archive.org/0/items/s2journaldemathm03pari/s2journaldemathm03pari.pdf">Journal de mathématiques élémentaires</a>, (1884), p. 111.
%H A378569 Antreas Hatzipolakis, <a href="https://groups.google.com/g/seqfan/c/jf28ZmsvpeQ/m/yzcKlGt-BgAJ">A sequence</a>, mail to the SeqFan list / google group, Jan. 31, 2025.
%H A378569 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A378569 a(n) = A102724(n+1) for 1 <= n <= 7, where A102724 = partial sums of A001043(n) = prime(n)+prime(n+1).
%F A378569 From _Vincenzo Librandi_, Feb 06 2025: (Start)
%F A378569 a(n) = 2* a(n-1) - a(n-2) + 6.
%F A378569 G.f.: (7-8x+7x^2)/ (1-3x+3x^2-x^3). (End)
%F A378569 E.g.f.: exp(x)*(7 + 6*x + 3*x^2). - _Stefano Spezia_, Feb 06 2025
%t A378569 LinearRecurrence[{3,-3,1},{7, 13, 25 }, 58] (* _James C. McMahon_, Feb 08 2025 *)
%o A378569 (PARI) A378569(n)=3*n*(n+1)+7;
%o A378569 apply(A378569, [0..55])
%Y A378569 Cf. A001043, A102724.
%K A378569 nonn,easy
%O A378569 0,1
%A A378569 _M. F. Hasler_, Feb 04 2025