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%I #20 Oct 01 2021 15:02:19
%S 0,0,2,3,2,0,3,14,26,27,22,44,123,234,310,363,586,1224,2259,3382,4642,
%T 7227,13070,23092,36555,54450,85022,143883,245282,396720,616803,
%U 973214,1600106,2664027,4334662,6887804,10970523,17828154,29272390,47634603,76493626
%N a(n) = 2*a(n-1)-2*a(n-2)+a(n-3)+2*a(n-4) with a(0)=a(1)=0, a(2)=2, a(3)=3.
%C This is an autosequence of the second kind, the companion to A192395.
%C The array D(n, k) of successive differences begins:
%C 0, 0, 2, 3, 2, 0, 3, 14, 26, 27, ...
%C 0, 2, 1, -1, -2, 3, 11, 12, 1, -5, ...
%C 2, -1, -2, -1, 5, 8, 1, -11, -6, 27, ...
%C -3, -1, 1, 6, 3, -7, -12, 5, 33, 30, ...
%C 2, 2, 5, -3, -10, -5, 17, 28, -3, -55, ...
%C 0, 3, -8, -7, 5, 22, 11, -31, -52, 13, ...
%C ...
%C The main diagonal (0,2,-2,6,-10,22,...) is essentially the same as A151575.
%C It can be seen that abs(D(n, 1)) = D(1, n).
%C The diagonal starting from the third 0 is -(-1)^n*11*A001045(n), inverse binomial transform of 11*A001045(n).
%H Colin Barker, <a href="/A307688/b307688.txt">Table of n, a(n) for n = 0..1000</a>
%H OEIS Wiki, <a href="https://oeis.org/wiki/Autosequence">Autosequence</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,1,2).
%F G.f.: x^2*(2 - x) / ((1 - x - x^2)*(1 - x + 2*x^2)). - _Colin Barker_, Apr 22 2019
%t a[0] = a[1] = 0; a[2] = 2; a[3] = 3; a[n_] := a[n] = 2*a[n-1] - 2*a[n-2] + a[n-3] + 2*a[n-4]; Table[a[n], {n, 0, 40}]
%t LinearRecurrence[{2,-2,1,2},{0,0,2,3},50] (* _Harvey P. Dale_, Oct 01 2021 *)
%o (PARI) concat([0,0], Vec(x^2*(2 - x) / ((1 - x - x^2)*(1 - x + 2*x^2)) + O(x^40))) \\ _Colin Barker_, Apr 22 2019
%Y Cf. A151575, A192395.
%Y Cf. A001045 (first and fifth upper diagonals), A014551 (second upper diagonal), A115102 (third), A155980 (fourth).
%K nonn,easy
%O 0,3
%A _Jean-François Alcover_ and _Paul Curtz_, Apr 22 2019
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