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A144472 Negative values along the main diagonal of the array defined by A020806 and its differences.

%I A144472 #11 Nov 06 2017 06:22:57
%S A144472 -1,2,9,13,31,57,119,233,471,937,1879,3753,7511,15017,30039,60073,
%T A144472 120151,240297,480599,961193,1922391,3844777,7689559,15379113,
%U A144472 30758231,61516457,123032919,246065833,492131671,984263337,1968526679,3937053353,7874106711
%N A144472 Negative values along the main diagonal of the array defined by A020806 and its differences.
%H A144472 Colin Barker, <a href="/A144472/b144472.txt">Table of n, a(n) for n = 1..1000</a>
%H A144472 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).
%F A144472 a(n+1) - 2*a(n) = (-1)^n*A010716(n), n>1, period 2.
%F A144472 G.f.: x*(1-3*x-9*x^2) / ((1+x)*(2*x-1)). - _R. J. Mathar_, Oct 24 2008
%F A144472 a(n) = 11*2^(n-2)/3 - 5*(-1)^n/3, n>1. - _R. J. Mathar_, Oct 24 2008
%F A144472 From _Colin Barker_, Nov 06 2017: (Start)
%F A144472 a(n) = (11*2^n - 20) / 12 for n>1 and even.
%F A144472 a(n) = (11*2^n + 20) / 12 for n>1 and odd.
%F A144472 a(n) = a(n-1) + 2*a(n-2) for n>3.
%F A144472 (End)
%e A144472 A020806 and its repeated differences in the next rows start as follows:
%e A144472 ..1,..4,..2,..8,..5,..7,..1,..4,..2,..8, <- A020806
%e A144472 ..3,.-2,..6,.-3,..2,.-6,..3,.-2,..6,.-3, <- A131969
%e A144472 .-5,..8,.-9,..5,.-8,..9,.-5,..8,.-9,..5,
%e A144472 .13,-17,.14,-13,.17,-14,.13,-17,.14,-13,
%e A144472 -30,.31,-27,.30,-31,.27,-30,.31,-27,.30,
%e A144472 .61,-58,.57,-61,.58,-57,.61,-58,.57,-61,
%e A144472 The diagonal is 1,-2,-9,-13,-31,... which yields a(n) after signs are flipped.
%t A144472 Join[{-1}, LinearRecurrence[{1, 2}, {2, 9}, 40]] (* _Jean-François Alcover_, Nov 06 2017 *)
%o A144472 (PARI) Vec(-x*(1 - 3*x - 9*x^2) / ((1 + x)*(1 - 2*x)) + O(x^50)) \\ _Colin Barker_, Nov 06 2017
%K A144472 sign,easy
%O A144472 1,2
%A A144472 _Paul Curtz_, Oct 10 2008, Oct 14 2008
%E A144472 Edited and extended by _R. J. Mathar_, Oct 24 2008