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A259966
Total binary weight (cf. A000120) of all A005251(n) binary sequences of length n not containing any isolated 1's.
3
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
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
Sequence in context: A023612 A192952 A132738 * A283500 A097442 A345025
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jul 11 2015
EXTENSIONS
Edited by Reinhard Zumkeller, Jul 13 2015
STATUS
approved