A258085 Strictly increasing list of F and F - 1, where F = A000045, the Fibonacci numbers.

0, 1, 2, 3, 4, 5, 7, 8, 12, 13, 20, 21, 33, 34, 54, 55, 88, 89, 143, 144, 232, 233, 376, 377, 609, 610, 986, 987, 1596, 1597, 2583, 2584, 4180, 4181, 6764, 6765, 10945, 10946, 17710, 17711, 28656, 28657, 46367, 46368, 75024, 75025, 121392, 121393, 196417

Offset: 1

Author

Clark Kimberling, Jun 27 2015

Keywords

• nonn
• easy

Comments

Beginning with a(4) = 3, these are the numbers m such that if r = golden ratio and the fractional parts {r}, {2 r}, ..., {mr} are arranged in increasing order, then the set of differences {kr} - {(k - 1)r}, for k = 2..m, consists of exactly two numbers.

Examples

			F = (1,1,2,3,5,8,13,...); F-1 = (0,0,1,2,4,7,12,...), so that the ordered list of F and F-1 is (0,1,2,3,4,5,7,8,...).
Regarding the fractional parts in Comment, for r = golden ratio and m = 7, the fractional parts are ordered as follows: -8+r, -3+2r, -11+7r, -6+4 r,-1+r, -9+6r, -4+3r. The set of differences is {5-3r, -8+5r}, so that 7 is a term in A258085.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,0,0,-1).

Crossrefs

Cf. A000035, A000045, A004526.

Programs

Magma

[0,1] cat &cat[[Fibonacci(n)-1, Fibonacci(n)]: n in [4..40]]; // Vincenzo Librandi, Jun 28 2015

Maple

map((t->(t-1,t)) @ combinat:-fibonacci,[1,$4..100]); # Robert Israel, Jun 29 2015

Mathematica

f = Fibonacci[Range[60]]; u = Union[f, f - 1]

PARI

a(n)=if(n<6,n-1,fibonacci((n+5)\2)-n%2) \\ Charles R Greathouse IV, Jun 28 2015

PARI

concat(0, Vec(-x^2*(x^6+x^5+x^4-x^2-2*x-1)/((x-1)*(x+1)*(x^4+x^2-1)) + O(x^100))) \\ Colin Barker, Feb 16 2017

Formula

From Colin Barker, Jun 28 2015: (Start)

a(n) = 2*a(n-2) - a(n-6) for n>8.

G.f.: -x^2*(x^6+x^5+x^4-x^2-2*x-1) / ((x-1)*(x+1)*(x^4+x^2-1)).

(End)

For n >= 3, a(n) = A000045(A004526(n+5)) - A000035(n). - Robert Israel, Jun 29 2015