A224227 a(n) = (1/50)*((15*n^2-20*n+4)*Fibonacci(n) - (5*n^2-6*n)*A000032(n)).

0, 0, 0, 1, 2, 7, 16, 38, 82, 173, 352, 701, 1368, 2628, 4980, 9329, 17302, 31811, 58040, 105178, 189446, 339373, 604964, 1073593, 1897488, 3341160, 5863080, 10256065, 17888138, 31115071, 53985856, 93447278, 161397754, 278184461, 478550344, 821734901, 1408610088, 2410719084, 4119433884, 7029086705, 11977419742, 20382654971, 34643298728, 58811818210

Offset: 0

Author

N. J. A. Sloane, Apr 09 2013

Keywords

• nonn
• easy

Comments

The right-hand side of a binomial-coefficient identity.

From Steven Finch, Mar 22 2020: (Start)

a(n+2) is the total binary weight squared of all A000045(n+2) binary sequences of length n not containing any adjacent 1's.

The only three 2-bitstrings without adjacent 1's are 00, 01 and 10. The bitsums squared of these are 0, 1 and 1. Adding these give a(4)=2.

The only five 3-bitstrings without adjacent 1's are 000, 001, 010, 100 and 101. The bitsums squared of these are 0, 1, 1, 1 and 4. Adding these give a(5)=7.

The only eight 4-bitstrings without adjacent 1's are 0000, 0001, 0010, 0100, 1000, 0101, 1010 and 1001. The bitsums squared of these are 0, 1, 1, 1, 1, 4, 4, and 4. Adding these give a(6)=16. (End)

Links

• Steven Finch, Cantor-solus and Cantor-multus Distributions, arXiv:2003.09458 [math.CO], 2020.
• N. Gauthier (Proposer), Problem H-703, Fib. Quart., 50 (2012), 379-381.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Crossrefs

Cf. A000032, A000045, A001628.

Programs

Mathematica

LinearRecurrence[{3,0,-5,0,3,1},{0,0,0,1,2,7},50] (* Harvey P. Dale, Jan 22 2016 *)
Table[((15 n^2 - 20 n + 4) Fibonacci[n] - (5 n - 6) n LucasL[n])/50, {n, 0, 30}] (* Vladimir Reshetnikov, Oct 10 2016 *)

PARI

concat([0,0,0],Vec((x^2-x+1)/(x^2+x-1)^3+O(x^96))) \\ Charles R Greathouse IV, Mar 19 2014

Formula

a(n) = Sum_{k=0..n-1} k^2*binomial(n-k-1,k).

G.f.: -x^3*(x^2-x+1)/(x^2+x-1)^3. - Mark van Hoeij, Apr 10 2013

a(n+3) = A001628(n) - A001628(n-1) + A001628(n-2). - R. J. Mathar, May 23 2014

E.g.f.: 2*exp(x/2)*(sqrt(5)*(2 + 5*x^2)*sinh(sqrt(5)*x/2) - 5*x*cosh(sqrt(5)*x/2))/125. - Stefano Spezia, Mar 20 2023