A332863 Total binary weight squared of all A005251(n) binary sequences of length n not containing any isolated 1's.

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

Offset: 0

Author

Steven Finch, Feb 27 2020

Keywords

• nonn
• easy

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,23,-27,24,-16,9,-3,1).

Crossrefs

Cf. A005251, A097230, A259966.

Programs

Magma

R:=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

Mathematica

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 *)

SageMath

def A332863_list(prec):
    P. = 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()
A332863_list(40) # G. C. Greubel, Apr 13 2022

Formula

G.f.: x^2*(4-7*x+4*x^2+3*x^3-x^4)/(1-2*x+x^2-x^3)^3.

a(n) = Sum_{k=1..n} k^2 * A097230(n,k). - Alois P. Heinz, Mar 03 2020