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 A140428 #18 Sep 08 2022 08:45:34
%S A140428 0,1,1,3,5,9,15,27,49,91,169,317,599,1143,2197,4251,8269,16161,31711,
%T A140428 62435,123273,243963,483745,960725,1910503,3803295,7577933,15109499,
%U A140428 30143973,60166553,120136687,239955563,479396897,957961755,1914577241
%N A140428 a(n) = A000045(n) + A113405(n).
%C A140428 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 A140428 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 A140428 G. C. Greubel, <a href="/A140428/b140428.txt">Table of n, a(n) for n = 0..1000</a>
%H A140428 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,3,-1,-2).
%F A140428 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 A140428 a(n)= -A128834(n)/3 + 2^n/9 + A000045(n) - (-1)^n/9. - _R. J. Mathar_, Jul 10 2008
%e A140428 a(n) and the repeated differences in the followup rows are:
%e A140428 0, 1, 1, 3, 5, 9, 15, ...
%e A140428 1, 0, 2, 2, 4, 6, 12, ...
%e A140428 -1, 2, 0, 2, 2, 6, 10, ...
%e A140428 3, -2, 2, 0, 4, 4, 10, ...
%e A140428 -5, 4, -2, 4, 0, 6, 6, ...
%e A140428 9, -6, 6, -4, 6, 0, 12, ...
%e A140428 -15, 12, -10, 10, -6, -12, 0, ...
%e A140428 The main diagonal consists of zeros.
%t A140428 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 A140428 LinearRecurrence[{3,-1,-3,3,-1,-2}, {0,1,1,3,5,9}, 30] (* _G. C. Greubel_, Jan 15 2018 *)
%o A140428 (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 A140428 (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 A140428 nonn,easy
%O A140428 0,4
%A A140428 _Paul Curtz_, Jun 19 2008
%E A140428 Edited and extended by _R. J. Mathar_, Jul 10 2008