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A332863
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Total binary weight squared of all A005251(n) binary sequences of length n not containing any isolated 1's.
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2
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0, 0, 4, 17, 46, 116, 288, 683, 1548, 3403, 7320, 15461, 32146, 65954, 133800, 268804, 535434, 1058533, 2078732, 4057858, 7878814, 15223495, 29285368, 56109673, 107108104, 203766859, 386443052, 730768044, 1378180568, 2592664120, 4866008208, 9112796113
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f.: x^2*(4-7*x+4*x^2+3*x^3-x^4)/(1-2*x+x^2-x^3)^3.
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EXAMPLE
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The only two 2-bitstrings without isolated 1's are 00 and 11. The bitsums squared of these are 0 and 4. Adding these give a(2)=4.
The only four 3-bitstrings without isolated 1's are 000, 011, 110 and 111. The bitsums squared of these are 0, 4, 4 and 9. Adding these give a(3)=17.
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MATHEMATICA
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LinearRecurrence[{6, -15, 23, -27, 24, -16, 9, -3, 1}, {0, 0, 4, 17, 46, 116, 288, 683, 1548}, 40] (* G. C. Greubel, Apr 13 2022 *)
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PROG
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(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0, 0] cat Coefficients(R!( x^2*(4-7*x+4*x^2+3*x^3-x^4)/(1-2*x+x^2-x^3)^3 )); // G. C. Greubel, Apr 13 2022
(SageMath)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x^2*(4-7*x+4*x^2+3*x^3-x^4)/(1-2*x+x^2-x^3)^3 ).list()
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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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