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 A110523 #48 Jul 27 2025 15:37:14
%S A110523 1,0,-3,3,6,-15,-3,48,-39,-105,222,93,-759,480,1797,-3237,-2154,11865,
%T A110523 -5403,-30192,46401,44175,-183378,50853,499281,-651840,-846003,
%U A110523 2801523,-263514,-8141055,8931597,15491568,-42286359,-4188345,131047422,-118482387,-274659879,630107040,193872597
%N A110523 Expansion of (1 + x)/(1 + x + 3*x^2).
%C A110523 Row sums of number triangle A110522.
%C A110523 The sequence a(n) is conjugate with A214733 since the following alternative relations: either ((-1 + i*sqrt(11))/2)^n = a(n) + A214733(n)*(-1 + i*sqrt(11))/2 or ((-1 - i*sqrt(11))/2)^n = a(n) + A214733(n)*(-1 - i*sqrt(11))/2. We have a(n+1) = -3*A214733(n), A214733(n+1) = a(n) - A214733(n). We note that sequences A110512 and A001607 are conjugated in a similar way. The above relations are connected with the Gauss sums, for example if e := exp(i*2Pi/11) then e + e^3 + e^4 + e^5 + e^9 = (-1 + i*sqrt(11))/2, and e^2 + e^6 + e^7 + e^8 + e^10 = (-1 - i*sqrt(11))/2, which is equivalent to the system of sums: Sum_{k=1..5} cos(2Pi*k/11) = -1/2 and Sum_{k=1..5} sin(2Pi*k/11) = sqrt(11)/2, and which is equivalent to the system of products: P_{k=1..5} cos(2Pi*k/11) = -1/32 and P_{k=1..5} sin(2Pi*k/11) = sqrt(11)/32 - for details see Witula's book. Lastly we note that ((-1 + i*sqrt(11))/2)^n + ((-1 - i*sqrt(11))/2)^n = 2*a(n) - A214733(n). - _Roman Witula_, Jul 27 2012
%D A110523 Roman Witula, On Some Applications of Formulae for Unimodular Complex Numbers, Jacek Skalmierski's Press, Gliwice 2011.
%H A110523 G. C. Greubel, <a href="/A110523/b110523.txt">Table of n, a(n) for n = 0..1000</a>
%H A110523 Taras Goy and Mark Shattuck, <a href="https://doi.org/10.2478/amsil-2023-0027">Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries</a>, Ann. Math. Silesianae, Vol. 38, No. 2 (2024), pp. 284-313. See p. 298.
%H A110523 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-3).
%F A110523 a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-3)^(j-k)*C(k, j-k).
%F A110523 From _Roman Witula_, Jul 27 2012: (Start)
%F A110523 a(n+2) + a(n+1) + 3*a(n) = 0.
%F A110523 a(n+1) = (-1)^n*(3*i*sqrt(11)/11)*(((1 + i*sqrt(11))/2)^(n-1) - ((1 - i*sqrt(11))/2)^(n-1)). (End)
%F A110523 From _G. C. Greubel_, Dec 28 2023: (Start)
%F A110523 a(n) = (-1)^n*3^((n-1)/2)*( sqrt(3)*ChebyshevU(n, 1/(2*sqrt(3))) - ChebyshevU(n-1, 1/(2*sqrt(3))) ).
%F A110523 a(n) = A106852(n) - A106852(n-1).
%F A110523 a(n) = (-1)^n*( A214733(n+1) + A214733(n) ). (End)
%F A110523 E.g.f.: exp(-x/2)*(sqrt(11)*cos(sqrt(11)*x/2) + sin(sqrt(11)*x/2))/sqrt(11). - _Stefano Spezia_, Jul 27 2025
%t A110523 CoefficientList[Series[(1+x)/(1+x+3*x^2), {x,0,50}], x] (* _G. C. Greubel_, Aug 30 2017 *)
%t A110523 LinearRecurrence[{-1,-3},{1,0},40] (* _Harvey P. Dale_, Jul 02 2022 *)
%o A110523 (PARI) my(x='x+O('x^50)); Vec((1+x)/(1+x+3*x^2)) \\ _G. C. Greubel_, Aug 30 2017
%o A110523 (Magma) [n le 2 select 2-n else -(Self(n-1) +3*Self(n-2)): n in [1..50]]; // _G. C. Greubel_, Dec 28 2023
%o A110523 (SageMath)
%o A110523 @CachedFunction # a = A110523
%o A110523 def a(n): return 1-n if n<2 else -a(n-1) -3*a(n-2)
%o A110523 [a(n) for n in range(41)] # _G. C. Greubel_, Dec 28 2023
%Y A110523 Cf. A001607, A106852, A110512, A110522, A214733.
%K A110523 easy,sign
%O A110523 0,3
%A A110523 _Paul Barry_, Jul 24 2005