A259966 Total binary weight (cf. A000120) of all A005251(n) binary sequences of length n not containing any isolated 1's.

0, 0, 2, 7, 16, 34, 72, 149, 300, 593, 1158, 2239, 4292, 8168, 15450, 29072, 54456, 101597, 188878, 350038, 646880, 1192415, 2192956, 4024583, 7371884, 13479421, 24607048, 44853552, 81645236, 148424000, 269497614, 488784787, 885571340, 1602879242, 2898512344

Offset

0,3

References

R. K. Guy, Letter to N. J. A. Sloane, Feb 05 1986.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Félix Balado and Guénolé C. M. Silvestre, Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings, arXiv:2602.10005 [math.CO], 2026. See p. 114.

Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.

R. K. Guy, Letter to N. J. A. Sloane, Feb 1986

Index entries for linear recurrences with constant coefficients, signature (4,-6,6,-5,2,-1).

Formula

a(n) = a(n-1)+a(n-2)+2*b(n)+a(n-4)+3*b(n-2), where b() is A005251().

G.f.: -x^2*(x-2) / (x^3-x^2+2*x-1)^2. - Colin Barker, Jul 21 2015

a(n) = Sum_{k=1..n} k * 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 of these are 0 and 2.  Adding these give a(2)=2.
The only four 3-bitstrings without isolated 1's are 000, 011, 110 and 111.  The bitsums of these are 0, 2, 2 and 3.  Adding these give a(3)=7.

Prog

(Haskell)
a259966 n = a259966_list !! n
a259966_list = 0 : 0 : 2 : 7 : zipWith (+)
   (zipWith3 (((+) .) . (+))
             a259966_list (drop 2 a259966_list) (drop 3 a259966_list))
   (drop 2 $ zipWith (+)
             (map (* 2) $ drop 2 a005251_list) (map (* 3) a005251_list))
-- Reinhard Zumkeller, Jul 13 2015
(PARI) concat([0, 0], Vec(-x^2*(x-2)/(x^3-x^2+2*x-1)^2 + O(x^50))) \\ Colin Barker, Jul 21 2015

Crossrefs

Cf. A005251, A097230.

Sequence in context: A023612 A192952 A132738 * A283500 A394192 A097442

Adjacent sequences: A259963 A259964 A259965 * A259967 A259968 A259969

Keyword

nonn,easy

Author

N. J. A. Sloane, Jul 11 2015

Extensions

Edited by Reinhard Zumkeller, Jul 13 2015

Status

approved