OFFSET
0,3
COMMENTS
2-ranks of difference sets constructed from Segre hyperovals.
Sometimes called Leonardo numbers. - George Pollard, Jan 02 2008
a(n) is the number of nodes in the Fibonacci tree of order n. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node (see the Knuth reference, p. 417). - Emeric Deutsch, Jun 14 2010
Also odd numbers whose index is a Fibonacci number: odd(Fib(k)). - Carmine Suriano, Oct 21 2010
This is the sequence A(1,1;1,1;1) of the family of sequences [a,b:c,d:k] considered by Gary Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 17 2010
In general, adding a constant to each successive term of a Horadam sequence with signature (c,d) will result in a third-order recurrence with signature (c+1, d-c,-d). - Gary Detlefs, Feb 01 2023
REFERENCES
E. W. Dijkstra, 'Fibonacci numbers and Leonardo numbers', circulated privately, July 1981.
E. W. Dijkstra, 'Smoothsort, an alternative for sorting in situ', Science of Computer Programming, 1(3): 223-233, 1982.
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
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).
J. Ziegenbalg, Algorithmen, Spektrum Akademischer Verlag, 1996, p. 172.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..500
Paula M. M. C. Catarino and Anabela Borges, On Leonardo numbers, Acta Mathematica Universitatis Comenianae (2019), 1-12.
Paula M. M. C. Catarino and Anabela Borges, A Note on Incomplete Leonardo Numbers, Integers 20A (2020) A43.
Fernando S. de Carvalho, First-order expansions of coupled Leonardo sequences, Rev. Tocantinense Mat. 1(1) 2026, 1-9. See p. 1.
E. W. Dijkstra, Smoothsort, an alternative for sorting in situ (EWD796a).
E. W. Dijkstra, Fibonacci numbers and Leonardo numbers (EWD797).
Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang, Gauss sums, Jacobi sums, and p-ranks of cyclic difference sets, J. Combin. Theory Ser. A, 87(1) (1999), 74-119.
Ronald Evans, Henk Hollmann, Christian Krattenthaler, and Qing Xiang, Supplement to "Gauss Sums, Jacobi Sums and p-Ranks ...".
Sergio Falcon, Sum of the Squares of the Extended (k, t)-Fibonacci Numbers, Preprints (2024), 2024081150. See p. 2.
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 2.
Yasuichi Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly 20(2) (1982), 168-178.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1019
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Kálmán Liptai, László Németh, Tamás Szakács, and László Szalay, On certain Fibonacci representations, arXiv:2403.15053 [math.NT], 2024. See p. 8.
Thor Martinsen, Non-Fisherian generalized Fibonacci numbers, Notes Num. Theor. Disc. Math. 31(2) (2025), 370-389. See p. 388.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Dept. Comp. Sci., Hunter College, CUNY, 2019.
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.
Kalika Prasad and Munesh Kumari, The generalized k-Leonardo numbers: a non-homogeneous generalization of the Fibonacci numbers, Palestine J. Math. (2025) Vol. 14, No. 3, 637-650. See p. 638.
Diana Savin and Elif Tan, On Companion sequences associated with Leonardo quaternions: Applications over finite fields, arXiv:2403.01592 [math.CO], 2024. See p. 2.
David Singmaster, Some counterexamples and problems on linear recurrences and part 2, Fib. Quart. 8 (1970), 264-267.
Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 24.
Hunar Sherzad Taher and Saroj Kumar Dash, Fibonacci Numbers that Are eta-concatenations of Leonardo and Lucas Numbers, Proc. Bulgar. Acad. Sci. 78(2) (2025), 171-180.
Elif Tan and Ho-Hon Leung, On Leonardo p-Numbers, Integers 23 (2023), Art. A7.
Bibhu Prasad Tripathy and Bijan Kumar Patel, Common Values of Generalized Fibonacci and Leonardo Sequences, J. Int. Seq. 26 (2023), Art. 23.6.2.
Qing Xiang, On Balanced Binary Sequences with Two-Level Autocorrelation Functions, IEEE Trans. Inform. Theory 44 (1998), 3153-3156.
Tülay Yaǧmur, On Gaussian Leonardo Hybrid Polynomials, Symmetry 15(7) (2023), 1422.
Tülay Yağmur, On dual and hyper-dual k-Leonardo numbers, Math. Bohem. (2025).
Feng-Zhen Zhao, The log-behavior of some sequences related to the generalized Leonardo numbers, Integers 24 (2024), Art. A82.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).
FORMULA
a(n) = 2*Fibonacci(n+1) - 1 = A006355(n+2) - 1. - Richard L. Ollerton, Mar 22 2002
G.f.: (1-x+x^2)/(1-2x+x^3) = 2/(1-x-x^2) - 1/(1-x). [Conjectured by Simon Plouffe in his 1992 dissertation; this is readily verified.]
a(n) = (2/sqrt(5))*((1+sqrt(5))/2)^(n+1) - 2/sqrt(5)*((1-sqrt(5))/2)^(n+1) - 1.
a(n+1)/a(n) is asymptotic to Phi = (1+sqrt(5))/2. - Jonathan Vos Post, May 26 2005
For n >= 2, a(n+1) = ceiling(Phi*a(n)). - Franklin T. Adams-Watters, Sep 30 2009
a(n) = Sum_{k=0..n+1} A109754(n-k+1,k) - Sum_{k=0..n} A109754(n-k,k) = Sum_{k=0..n+1} A101220(n-k+1,0,k) - Sum_{k=0..n} A101220(n-k,0,k). - Ross La Haye, May 31 2006
a(n) = Fibonacci(n-1) + Fibonacci(n+2) - 1. - Zerinvary Lajos, Jan 31 2008, corrected by R. J. Mathar, Dec 17 2010
a(n) = 2*a(n-1) - a(n-3); a(0)=1, a(1)=1, a(2)=3. - Harvey P. Dale, Aug 07 2012
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - exp(x). - Stefano Spezia, Jan 23 2024
EXAMPLE
a(7) = odd(F(7)) = odd(8) = 15. - Carmine Suriano, Oct 21 2010
MAPLE
L := 1, 3: for i from 3 to 40 do l := nops([ L ]): L := L, op(l, [ L ])+op(l-1, [ L ])+1: od: [ L ];
A001595:=(1-z+z**2)/(z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation
with(combinat): seq(fibonacci(n-1)+fibonacci(n+2)-1, n=0..40); # Zerinvary Lajos, Jan 31 2008
MATHEMATICA
Join[{1, 3}, Table[a[1]=1; a[2]=3; a[i]=a[i-1]+a[i-2]+1, {i, 3, 40} ] ]
(* Alternative: *)
Table[2*Fibonacci[n+1]-1, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009; modified by G. C. Greubel, Jul 10 2019 *)
(* Alternative: *)
RecurrenceTable[{a[0]==a[1]==1, a[n]==a[n-1]+a[n-2]+1}, a, {n, 40}] (* Harvey P. Dale, Aug 07 2012 *)
(* Alternative: *)
LinearRecurrence[{2, 0, -1}, {1, 1, 3}, 40] (* Harvey P. Dale, Aug 07 2012 *)
PROG
(PARI) a(n) = 2*fibonacci(n+1)-1 \\ Franklin T. Adams-Watters, Sep 30 2009
(Haskell)
a001595 n = a001595_list !! n
a001595_list =
1 : 1 : (map (+ 1) $ zipWith (+) a001595_list $ tail a001595_list)
-- Reinhard Zumkeller, Aug 14 2011
(Magma) [2*Fibonacci(n+1)-1: n in [0..40]]; // G. C. Greubel, Jul 10 2019
(SageMath) [2*fibonacci(n+1)-1 for n in (0..40)] # G. C. Greubel, Jul 10 2019
(GAP) List([0..40], n-> 2*Fibonacci(n+1) -1); # G. C. Greubel, Jul 10 2019
(Python)
from sympy import fibonacci
def A001595(n): return (fibonacci(n+1)<<1)-1 # Chai Wah Wu, Sep 10 2024
CROSSREFS
KEYWORD
nonn,easy,nice,changed
AUTHOR
EXTENSIONS
Additional comments from Christian Krattenthaler (kratt(AT)ap.univie.ac.at)
Further edits from Franklin T. Adams-Watters, Sep 30 2009, and N. J. A. Sloane, Oct 03 2009
STATUS
approved