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A219233 Alternating row sums of Riordan triangle A110162.

%I A219233 #44 Jun 18 2025 23:33:05
%S A219233 1,-3,7,-18,47,-123,322,-843,2207,-5778,15127,-39603,103682,-271443,
%T A219233 710647,-1860498,4870847,-12752043,33385282,-87403803,228826127,
%U A219233 -599074578,1568397607,-4106118243,10749957122,-28143753123,73681302247,-192900153618,505019158607
%N A219233 Alternating row sums of Riordan triangle A110162.
%C A219233 If a(0) is put to 2 instead of 1 this becomes a(n) = (-1)^n*A005248(n), n >= 0. These are then the alternating row sums of triangle A127677.
%C A219233 Also abs(a(n)) is the number of rounded area of pentagon or pentagram in series arrangement. - _Kival Ngaokrajang_, Mar 27 2013
%H A219233 Colin Barker, <a href="/A219233/b219233.txt">Table of n, a(n) for n = 0..1000</a>
%H A219233 Richard M. Low and Ardak Kapbasov, <a href="https://www.emis.de/journals/JIS/VOL20/Low/low2.html">Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 8.
%H A219233 Kival Ngaokrajang, <a href="/A219233/a219233.jpg">Pentagram for n = 1..6</a>
%H A219233 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pentagram.html">Pentagram</a>
%H A219233 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-3,-1).
%F A219233 a(0) = 1 and a(n) = (-1)^n*(F(2*(n+1)) - F(2*(n-1))) = (-1)^n*L(2*n), n>=1, with F=A000045 (Fibonacci) and L=A000032 (Lucas).
%F A219233 O.g.f.: (1-x^2)/(1+3*x+x^2).
%F A219233 G.f.: (W(0) -6)/(5*x) -1 , where W(k) = 5*x*k + x + 6 - 6*x*(5*k-9)/W(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Aug 19 2013
%F A219233 From _Colin Barker_, Oct 14 2015: (Start)
%F A219233 a(n) = -3*a(n-1) - a(n-2) for n>2.
%F A219233 a(n) = (1/2*(-3-sqrt(5)))^n + (1/2*(-3+sqrt(5)))^n for n>0. (End)
%F A219233 E.g.f.: 2*exp(-3*x/2)*cosh(sqrt(5)*x/2) - 1. - _Stefano Spezia_, Dec 26 2021
%F A219233 From _G. C. Greubel_, Jun 13 2025: (Start)
%F A219233 a(-n) = a(n).
%F A219233 a(n) = (-1)^n*A001254(n) - 2 - [n=0] = A075150(n) - 2 - [n=0]. (End)
%t A219233 A219233[n_]:= (-1)^n*LucasL[2*n] - Boole[n==0]; (* _G. C. Greubel_, Jun 13 2025 *)
%o A219233 (PARI) Vec((1-x^2)/(1+3*x+x^2) + O(x^40)) \\ _Colin Barker_, Oct 14 2015
%o A219233 (Magma)
%o A219233 A219233:= func< n | n eq 0 select 1 else (-1)^n*Lucas(2*n) >; // _G. C. Greubel_, Jun 13 2025
%o A219233 (SageMath)
%o A219233 def A219233(n): return (-1)**n*lucas_number2(2*n,1,-1) - int(n==0) # _G. C. Greubel_, Jun 13 2025
%Y A219233 Cf. A099837 (row sums of A110162).
%Y A219233 Cf. A000032, A000045, A001254, A005248, A075150, A127677.
%K A219233 sign,easy
%O A219233 0,2
%A A219233 _Wolfdieter Lang_, Nov 16 2012