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 A248917 #58 Sep 08 2022 08:46:10
%S A248917 1,3,17,73,257,801,2305,6273,16385,41473,102401,247809,589825,1384449,
%T A248917 3211265,7372801,16777217,37879809,84934657,189267969,419430401,
%U A248917 924844033,2030043137,4437573633,9663676417,20971520001,45365592065,97844723713,210453397505,451508436993
%N A248917 a(n) = 2^n * n^2 + 1.
%C A248917 Binomial transform of A118239 (Engel expansion of cosh(1)).
%C A248917 Table of successive differences of a(n):
%C A248917 1, 3, 17, 73, 257, 801, 2305,...
%C A248917 2, 14, 56, 184, 544, 1504,...
%C A248917 12, 42, 128, 360, 960,...
%C A248917 30, 86, 232, 600,...
%C A248917 56, 146, 368,...
%C A248917 90, 222,...
%C A248917 132,...
%C A248917 etc.
%C A248917 Via b(n) = 0, 0, 0 followed by A055580(n), i.e., 0, 0, 0, 1, 7, 31, 111, ... (the main sequence for the recurrence), a link can be found between a(n) and A002064(n): c(n) = b(n+1) - 2*b(n) = 0, 0, 1, 5, 17, 49, 129, ... (the main sequence for the signature (5, -8, 4)).
%H A248917 G. C. Greubel, <a href="/A248917/b248917.txt">Table of n, a(n) for n = 0..1000</a>
%H A248917 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-18,20,-8).
%F A248917 a(n) = 4*a(n-1) - 4*a(n-2) + 2^(n+1) + 1.
%F A248917 a(n) = A007758(n) + 1.
%F A248917 a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 8*a(n-4). - _Jean-François Alcover_, Oct 22 2014
%F A248917 G.f.: -(12*x^3-14*x^2+4*x-1) / ((x-1)*(2*x-1)^3). - _Colin Barker_, Oct 22 2014
%F A248917 E.g.f.: exp(x) + 2*x*(1 + 2*x)*exp(2*x). - _G. C. Greubel_, Oct 28 2016
%e A248917 a(3) = 9 * 8 + 1 = 73.
%e A248917 a(4) = 16 * 16 + 1 = 257.
%e A248917 a(5) = 25 * 32 + 1 = 801.
%t A248917 Table[n^2 * 2^n + 1, {n, 0, 31}] (* _Alonso del Arte_, Oct 22 2014 *)
%t A248917 LinearRecurrence[{7,-18,20,-8}, {1,3,17,73}, 25] (* _G. C. Greubel_, Oct 28 2016 *)
%o A248917 (PARI) Vec(-(12*x^3-14*x^2+4*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ _Colin Barker_, Oct 22 2014
%o A248917 (PARI) a(n)=n^2<<n + 1 \\ _Charles R Greathouse IV_, Oct 22 2014
%o A248917 (Magma) [2^n*n^2+1: n in [0..30]]; // _Vincenzo Librandi_, Oct 29 2016
%Y A248917 Cf. A000225, A002064 (Cullen numbers), A006784, A007758, A055580, A118239, A168298.
%K A248917 nonn,easy
%O A248917 0,2
%A A248917 _Paul Curtz_, Oct 22 2014