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

0,3

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).

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A005251, A097230, A259966.

Sequence in context: A147656 A095667 A212577 * A119949 A213499 A213502

Adjacent sequences: A332860 A332861 A332862 * A332864 A332865 A332866

Keyword

nonn,easy

Author

Steven Finch, Feb 27 2020

Status

approved