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A307688
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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.
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1
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0, 0, 2, 3, 2, 0, 3, 14, 26, 27, 22, 44, 123, 234, 310, 363, 586, 1224, 2259, 3382, 4642, 7227, 13070, 23092, 36555, 54450, 85022, 143883, 245282, 396720, 616803, 973214, 1600106, 2664027, 4334662, 6887804, 10970523, 17828154, 29272390, 47634603, 76493626
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OFFSET
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0,3
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COMMENTS
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This is an autosequence of the second kind, the companion to A192395.
The array D(n, k) of successive differences begins:
0, 0, 2, 3, 2, 0, 3, 14, 26, 27, ...
0, 2, 1, -1, -2, 3, 11, 12, 1, -5, ...
2, -1, -2, -1, 5, 8, 1, -11, -6, 27, ...
-3, -1, 1, 6, 3, -7, -12, 5, 33, 30, ...
2, 2, 5, -3, -10, -5, 17, 28, -3, -55, ...
0, 3, -8, -7, 5, 22, 11, -31, -52, 13, ...
...
The main diagonal (0,2,-2,6,-10,22,...) is essentially the same as A151575.
It can be seen that abs(D(n, 1)) = D(1, n).
The diagonal starting from the third 0 is -(-1)^n*11*A001045(n), inverse binomial transform of 11*A001045(n).
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LINKS
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FORMULA
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G.f.: x^2*(2 - x) / ((1 - x - x^2)*(1 - x + 2*x^2)). - Colin Barker, Apr 22 2019
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MATHEMATICA
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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}]
LinearRecurrence[{2, -2, 1, 2}, {0, 0, 2, 3}, 50] (* Harvey P. Dale, Oct 01 2021 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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