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

Author

N. J. A. Sloane, Jul 11 2015

Keywords

• nonn
• easy

Examples

			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.
		

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
• 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).

Crossrefs

Cf. A005251, A097230.

Programs

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

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

Extensions

Edited by Reinhard Zumkeller, Jul 13 2015