A372501 -id:A372501 - cp's OEIS frontend

A001911 a(n) = Fibonacci(n+3) - 2.

0, 1, 3, 6, 11, 19, 32, 53, 87, 142, 231, 375, 608, 985, 1595, 2582, 4179, 6763, 10944, 17709, 28655, 46366, 75023, 121391, 196416, 317809, 514227, 832038, 1346267, 2178307, 3524576, 5702885, 9227463, 14930350, 24157815, 39088167, 63245984

Offset: 0

N. J. A. Sloane

This is the sequence A(0,1;1,1;2) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 17 2010
Ternary words of length n - 1 with subwords (0, 1), (0, 2) and (2, 2) not allowed. - Olivier Gérard, Aug 28 2012
For subsets of (1, 2, 3, 5, 8, 13, ...) Fibonacci Maximal terms (Cf. A181631) equals the number of leading 1's per subset. For example, (7-11) in Fibonacci Maximal = (1010, 1011, 1101, 1110, 1111), numbers of leading 1's = (1 + 1 + 2 + 3 + 4) = 11 = a(4) = row 4 of triangle A181631. - Gary W. Adamson, Nov 02 2010
As in our 2009 paper, we use two types of Fibonacci trees: - Ta: Fibonacci analog of binomial trees; Tb: Binary Fibonacci trees. Let D(r(k)) be the sum over all distances of the form d(r, x), across all vertices x of the tree rooted at r of order k. Ignoring r, but overloading, let D(a(k)) and D(b(k)) be the distance sums for the Fibonacci trees Ta and Tb respectively of the order k. Using the sum-of-product form in Equations (18) and (21) in our paper it follows that F(k+4) - 2 = D(a(k+1)) - D(b(k-1)). - K.V.Iyer and P. Venkata Subba Reddy, Apr 30 2011
a(n) is the length of the n-th palindromic prefix of the infinite Fibonacci word A003849. - Dimitri Hendriks, May 19 2014
The first k terms of the infinite Fibonacci word A014675 are palindromic if and only if k is a positive term of this sequence. - Clark Kimberling, Jul 14 2014
Can be expressed in terms of a rule similar to Recamán's sequence (A005132). Instead of following the Recamán rule "subtract if possible, otherwise add", this sequence follows the rule "If subtraction is possible, do nothing; otherwise add." For example when at the fourth term, 6, it is possible to subtract 4 (giving 2 which is not in {0, 1, 3, 6}) so nothing is done with 4. It is not possible to subtract 5 (6-5=1, which is in {0, 1, 3, 6}), so it is added, resulting in 11. - Matthew Malone, Jan 03 2020
For n>=1, a(n) is the maximum number of vertices (Moore bound) of a (1,1)-regular mixed graph with diameter n-1. - Miquel A. Fiol, Jun 24 2024
Repunits in the lazy Fibonacci representation (A104326), and which is the first row of array A372501. - A.H.M. Smeets, Jun 25 2025

			G.f. = x + 3*x^2 + 6*x^3 + 11*x^4 + 19*x^5 + 32*x^6 + 53*x^7 + 87*x^8 + ...

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 233.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Charles R Greathouse IV, Table of n, a(n) for n = 0..4783 (next term has 1001 digits)
Stefano Bilotta, Variable-length Non-overlapping Codes, arXiv preprint arXiv:1605.03785 [cs.IT], 2016.
D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.
C. Dalfó, G. Erskine, G. Exoo, M. A. Fiol, N. López, A. Messegué, and J. Tuite, On large regular (1,1,k)-mixed graphs, Discrete Appl. Math. 356 (2024), 209-228.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
K. Viswanathan Iyer and K. R. Uday Kumar Reddy, Wiener index of binomial trees and Fibonacci trees, arXiv:0910.4432 [cs.DM], 2009. (Corrigendum: Eq.(23) to be corrected as follows on the right-side: in the fourth term F(k)-1 should be replaced by F(k); a term F(k)*F(K+1)-1 is to be included; pointed out by Emeric Deutsch).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
M. Rigo, P. Salimov, and E. Vandomme, Some Properties of Abelian Return Words, Journal of Integer Sequences, Vol. 16 (2013), #13.2.5.
D. G. Rogers, An application of renewal sequences to the dimer problem, pp. 142-153 of Combinatorial Mathematics VI (Armidale 1978), Lect. Notes Math. 748, 1979.
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences, 2010.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000032, A000045, A101220, A104326, A108617, A181631, A372501.
Cf. A001611, A000071, A157725, A001911, A157726, A006327, A157727, A157728, A157729, A167616. [Added by N. J. A. Sloane, Jun 25 2010 in response to a comment from Aviezri S. Fraenkel]
Partial sums of A000045(n+1).
Right-hand column 3 of triangle A011794.
See also A165910.
Subsequence of A226538.
Column k=3 of A261019.

a001911 n = a001911_list !! n
a001911_list = 0 : 1 : map (+ 2) (zipWith (+) a001911_list $ tail a001911_list)
-- Reinhard Zumkeller, Jun 18 2013

[(Fibonacci(n+3))-2: n in [0..85]]; // Vincenzo Librandi, Apr 23 2011

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+a[n-2]+2 od: seq(a[n],n=0..50); # Miklos Kristof, Mar 09 2005
A001911:=(1+z)/(z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation with another offset
a:= n-> (Matrix([[0,-1,1]]). Matrix([[1,1,0], [1,0,0], [2,0,1]])^n)[1,1]: seq(a(n), n=0..50); # Alois P. Heinz, Jul 24 2008

Table[Fibonacci[n+3] -2, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)
LinearRecurrence[{2,0,-1}, {0,1,3}, 40] (* Harvey P. Dale, Jun 06 2011 *)
Fibonacci[Range[3,40]]-2 (* Harvey P. Dale, Jun 28 2015 *)

a(n)=fibonacci(n+3)-2 \\ Charles R Greathouse IV, Mar 14 2012

[fibonacci(n+3)-2 for n in range(60)] # G. C. Greubel, Oct 21 2024

From Michael Somos, Jun 09 1999: (Start)
a(n) = A000045(n+3) - 2.
a(n) = a(n-1) + a(n-2) + 2, a(0) = 0, a(1) = 1. (End)
G.f.: x*(1+x)/((1-x)*(1-x-x^2)).
Sum of consecutive pairs of A000071 (partial sums of Fibonacci numbers). - Paul Barry, Apr 17 2004
a(n) = A101220(2, 1, n). - Ross La Haye, Jan 28 2005
a(n) = A108617(n+1, 2) = A108617(n+1, n-1) for n > 0. - Reinhard Zumkeller, Jun 12 2005
a(n) = term (1,1) in the 1 X 3 matrix [0,-1,1].[1,1,0; 1,0,0; 2,0,1]^n. - Alois P. Heinz, Jul 24 2008
a(0) = 0, a(1) = 1, a(2) = 3, a(n) = 2*a(n-1)-a(n-3). - Harvey P. Dale, Jun 06 2011
Eigensequence of an infinite lower triangular matrix with the natural numbers as the left border and (1, 0, 1, 0, ...) in all other columns. - Gary W. Adamson, Jan 30 2012
a(n) = (-2+(2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5))+(1+sqrt(5))^n*(2+sqrt(5))))/sqrt(5)). - Colin Barker, May 12 2016
a(n) = A000032(6+n)-1 mod A000045(6+n). - Mario C. Enriquez, Apr 01 2017
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 2*exp(x). - Stefano Spezia, May 08 2022

More terms and better description from Michael Somos

A385436 Tribonacci array of the second kind, read by upward antidiagonals.

Original entry on oeis.org

0, 2, 1, 4, 5, 3, 6, 8, 10, 7, 9, 12, 16, 20, 14, 11, 18, 23, 31, 38, 27, 13, 21, 34, 44, 58, 71, 51, 15, 25, 40, 64, 82, 108, 132, 95, 17, 29, 47, 75, 119, 152, 200, 244, 176, 19, 32, 54, 88, 139, 220, 281, 369, 450, 325, 22, 36, 60, 101, 163, 257, 406, 518, 680

Offset: 1

A.H.M. Smeets, Jun 28 2025

The array is, as a sequence, a permutation of the nonnegative integers; however it does not satisfy the conditions for interspersion and dispersion as given by Eric Weisstein's World of Mathematics. However, when all terms are increased by 1, it does satisfy the conditions for interspersion and dispersion!
Rows satisfy the recurrence: T(m,k) = 2*T(m,k-1) - T(m,k-4) for all k>4.
This array belongs to a family of Wythoff like arrays, based on binary number representations like the greedy and lazy Fibonacci number representations (see A035513 and A372501 for arrays), greedy and lazy Narayana number representations (A136189 for the array related to greedy representation).
The array is related to the lazy tribonacci number representation A352103. The first column lists the even numbers, i.e., for wich 0 suffix A352103(T(m,1)). The odd numbers are represented in the columns k > 1: A352103(T(m,k)) = A352103(T(m,1)) + 1^(k-1). Here + stands for concatenation and ^ stands for repeated concatenation.

			Array including some prepended columns (p = 1..4):
  p=4 p=3 p=2 p=1 | k=1 k=2 k=3  k=4  k=5  k=6  k=7   k=8   k=9  k=10
   -2  -1  -1  -1 |   0   1   3    7   14   27   51    95   176   325
   -2  -1   0   0 |   2   5  10   20   38   71  132   244   450   829
   -2   0   0   1 |   4   8  16   31   58  108  200   369   680
   -2   0   1   2 |   6  12  23   44   82  152  281   518
   -1   0   2   4 |   9  18  34   64  119  220  406   748
   -1   1   2   5 |  11  21  40   75  139  257  474   873
   -1   1   3   6 |  13  25  47   88  163  301  555  1022
   -1   1   4   7 |  15  29  54  101  187  345  636  1171
   -1   2   4   8 |  17  32  60  112  207  382  704  1296
   -1   2   5   9 |  19  36  67  125
    0   2   6  11 |  22  42  78  145
Each row of the array satisfies the recurrence relation T(m,k) = 2*T(m,k-1) - T(m,k-4); from this, the prepended columns are obtained by rowwise backward recursion.

A.H.M. Smeets, Table of n, a(n) for n = 1..20100 (first 200 antidiagonals).
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, No. 1 (February 2012), pp. 11-18.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Eric Weisstein's World of Mathematics, Interspersion.
Eric Weisstein's World of Mathematics, Sequence Dispersion.

Cf. A035513, A136189, A372501, A027084 (m=1), A351631 (k=1), A352103.

Prepended columns: A385455 (p=1), A385532 (p=2), A385533 (p=3).

def ToDual_111_Zeck(n):
    if n == 0:
        return "0"
    f0, f1, f2, sf = 1, 0, 0, 0
    while n > sf:
        f0, f1, f2 = f0+f1+f2, f0, f1
        sf += f0
    r, s = sf-n, "1"
    while f0 > 1:
        f0, f1, f2 = f1, f2, f0-f1-f2
        r, s = r%f0, s+str(1-r//f0)
    return s
def From_111_Zeck(s):
    f0, f1, f2, i, n = 1, 1, 0, len(s), 0
    while i > 0:
        i -= 1
        f0, f1, f2, n = f0+f1+f2, f0, f1, n+int(s[i])*f0
    return n
d, a, n, c1 = 0, 0, 0, []
while d < 11:
    s = ToDual_111_Zeck(a)
    if s[len(s)-1] == "0": # == even
        n, d = n+1, d+1
        print(a, end = ", ")
        i, c1, p1 = d-1, c1+[s], ""
        while i > 0:
            n, i, p1 = n+1, i-1, p1+"1"
            print(From_111_Zeck(c1[i]+p1), end = ", ")
    a += 1

cp's OEIS Frontend

A001911 a(n) = Fibonacci(n+3) - 2.

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