A280523 a(n) = Fibonacci(2n + 1) - n.

1, 3, 10, 30, 84, 227, 603, 1589, 4172, 10936, 28646, 75013, 196405, 514215, 1346254, 3524562, 9227448, 24157799, 63245967, 165580121, 433494416, 1134903148, 2971215050, 7778742025, 20365011049, 53316291147, 139583862418

Offset

1,2

Comments

Old (and equivalent) definition: these are the indices of records in the Fibonachos sequence A280521: the least k such that A280521(k) = n.

Conjecture: a(n) = A215004(2n - 2). - Peter Kagey. The conjecture is true. See link for analysis of connections with A215004 and A280521. - Nathan Fox, Jan 27 2017

Define the n-th Fibonacci compositions CF(n) by CF(1)={(1)}, CF(2)={(2)}, and CF(n) is 1 adjoined at the end of each composition in CF(n-1) union 2 adjoined to the end of each composition in CF(n-2). The union is disjoint, so |CF(n)| is the n-th Fibonacci number. Define the weight of a composition c by 2^(number of singletons in c). For example, 2122 has 1 singleton and weight 2. Let s(n) be the sum of the weights of CF(n). Conjecture: a(n)= s(2n+4)-s(2n+3). - George Beck, Jan 31 2020

Links

Table of n, a(n) for n=1..27.

Nathan Fox, Proof that a(n) = Fibonacci(2n + 1) - n.

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2

Index entries for linear recurrences with constant coefficients, signature (5,-8,5,-1).

Formula

G.f.: -x*(-1+2*x-3*x^2+x^3) / ( (x^2-3*x+1)*(x-1)^2 ). - R. J. Mathar, Mar 11 2017

a(n) = 5*a(n-1)-8*a(n-2)+5*a(n-3)-a(n-4). - Wesley Ivan Hurt, Apr 26 2021

Example

a(1) = 1    because A280521(1)    = 1;
a(2) = 3    because A280521(3)    = 2;
a(3) = 10   because A280521(10)   = 3;
a(4) = 30   because A280521(30)   = 4;
a(5) = 84   because A280521(84)   = 5;
a(6) = 227  because A280521(227)  = 6;
a(7) = 603  because A280521(603)  = 7;
a(8) = 1589 because A280521(1589) = 8.

Mathematica

Table[Fibonacci[2n + 1] - n, {n, 30}] (* Alonso del Arte, Jan 29 2017 *)
LinearRecurrence[{5, -8, 5, -1}, {1, 3, 10, 30}, 30] (* Harvey P. Dale, Feb 06 2024 *)

Prog

(PARI) F=vector(64, n, fibonacci(n+2)-1); \\ Resize as needed
A280521(n)=my(s); while(n, s++; t=setsearch(F, n, 1); if(t, n-=F[t-1], return(s))); s
first(n)=my(v=vector(n), k, t, mn=1, gaps=n); while(gaps, t=A280521(k++); if(t>=mn && t<=n && v[t]==0, v[t]=k; while(mn<=n && v[mn], mn++); print("a("t") = "k); gaps--)); v \\ Charles R Greathouse IV, Jan 04 2017

Crossrefs

Cf. A001519, A215004, A280521.

Sequence in context: A027256 A332360 A115515 * A117869 A092756 A027205

Adjacent sequences: A280520 A280521 A280522 * A280524 A280525 A280526

Keyword

nonn,easy

Author

Peter Kagey, Jan 04 2017

Extensions

Corrected and extended by Charles R Greathouse IV, Jan 04 2017

a(26) from Charles R Greathouse IV, Jan 09 2017

a(27) from Charles R Greathouse IV, Jan 16 2017

Replaced old definition by simple formula established by Nathan Fox. - N. J. A. Sloane, Jan 30 2017

Status

approved