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%I #18 Sep 08 2022 08:45:34
%S 0,1,1,3,5,9,15,27,49,91,169,317,599,1143,2197,4251,8269,16161,31711,
%T 62435,123273,243963,483745,960725,1910503,3803295,7577933,15109499,
%U 30143973,60166553,120136687,239955563,479396897,957961755,1914577241
%N a(n) = A000045(n) + A113405(n).
%C The inverse binomial transform yields the sequence (-1)^(n+1)*a(n). This property is inherited from the A000045 and A113405 sequences, which have the same property individually. The same sign flipping behavior under inverse binomial transform is found in A001045 and for the sequence with two zeros followed by A000975.
%C This is often, but not here, related to the recurrences a(n)=2a(n-1)+a(n-2)-2a(n-3) associated with denominators 1-2x-x^2+2x^3=(x-1)(2x-1)(x+1) in the o.g.f., which transform into the similar -(x-1)(2x+1)/(1+x)^4 under the inverse binomial transform, see A137241.
%H G. C. Greubel, <a href="/A140428/b140428.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,3,-1,-2).
%F O.g.f.: -x*(1-2*x-3*x^4+x^2)/((1-x-x^2)*(2*x-1)*(1+x)*(x^2-x+1)). - _R. J. Mathar_, Jul 10 2008
%F a(n)= -A128834(n)/3 + 2^n/9 + A000045(n) - (-1)^n/9. - _R. J. Mathar_, Jul 10 2008
%e a(n) and the repeated differences in the followup rows are:
%e 0, 1, 1, 3, 5, 9, 15, ...
%e 1, 0, 2, 2, 4, 6, 12, ...
%e -1, 2, 0, 2, 2, 6, 10, ...
%e 3, -2, 2, 0, 4, 4, 10, ...
%e -5, 4, -2, 4, 0, 6, 6, ...
%e 9, -6, 6, -4, 6, 0, 12, ...
%e -15, 12, -10, 10, -6, -12, 0, ...
%e The main diagonal consists of zeros.
%t CoefficientList[Series[-x (1 - 2 x - 3 x^4 + x^2)/((1 - x - x^2) (2 x - 1) (1 + x) (x^2 - x + 1)), {x, 0, 50}], x] (* _G. C. Greubel_, Oct 11 2017 *)
%t LinearRecurrence[{3,-1,-3,3,-1,-2}, {0,1,1,3,5,9}, 30] (* _G. C. Greubel_, Jan 15 2018 *)
%o (PARI) a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -2,-1,3,-3,-1,3]^n*[0;1;1;3;5;9])[1,1] \\ _Charles R Greathouse IV_, Oct 03 2016
%o (Magma) I:=[0,1,1,3,5,9]; [n le 6 select I[n] else 3*Self(n-1)-Self(n-2) -3*Self(n-3)+3*Self(n-4)-Self(n-5)-2*Self(n-6): n in [1..30]]; // _G. C. Greubel_, Jan 15 2018
%K nonn,easy
%O 0,4
%A _Paul Curtz_, Jun 19 2008
%E Edited and extended by _R. J. Mathar_, Jul 10 2008
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