A033538 -id:A033538 - cp's OEIS frontend

A001595 a(n) = a(n-1) + a(n-2) + 1, with a(0) = a(1) = 1.

1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049, 242785, 392835, 635621, 1028457, 1664079, 2692537, 4356617, 7049155, 11405773, 18454929, 29860703, 48315633, 78176337

Offset: 0

N. J. A. Sloane

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

			a(7) = odd(F(7)) = odd(8) = 15. - _Carmine Suriano_, Oct 21 2010

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.

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.
P. Catarino and A. Borges, A Note on Incomplete Leonardo Numbers, INTEGERS 20A (2020) A43.
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, No. 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. (2025) Vol. 31, No. 2, 370-389. See p. 388.
Saad Mneimneh, Simple Variations on the Tower of Hanoi to Guide the Study of Recurrences and Proofs by Induction, Department of Computer Science, 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.
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. (2025) Vol. 78, No. 2, 171-180.
Elif Tan and Ho-Hon Leung, On Leonardo p-Numbers, Integers (2023) Vol. 23, #A7.
Bibhu Prasad Tripathy and Bijan Kumar Patel, Common Values of Generalized Fibonacci and Leonardo Sequences, J. Int. Seq. (2023) Vol. 26, 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 (2023) Vol. 15, No. 7, 1422.
Feng-Zhen Zhao, The log-behavior of some sequences related to the generalized Leonardo numbers, Integers (2024) Vol. 24, Art. No. A82.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A000071, A006355, A033538, A049112, A049114, A101220, A109754, A128587.

List([0..40], n-> 2*Fibonacci(n+1) -1); # G. C. Greubel, Jul 10 2019

a001595 n = a001595_list !! n
a001595_list =
   1 : 1 : (map (+ 1) $ zipWith (+) a001595_list $ tail a001595_list)
-- Reinhard Zumkeller, Aug 14 2011

[2*Fibonacci(n+1)-1: n in [0..40]]; // G. C. Greubel, Jul 10 2019

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

Join[{1, 3}, Table[a[1]=1; a[2]=3; a[i]=a[i-1]+a[i-2]+1, {i, 3, 40} ] ]
Table[2*Fibonacci[n+1]-1, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009; modified by G. C. Greubel, Jul 10 2019 *)
RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1]+a[n-2]+1},a,{n,40}] (* or *) LinearRecurrence[{2,0,-1},{1,1,3},40] (* Harvey P. Dale, Aug 07 2012 *)

a(n) = 2*fibonacci(n+1)-1 \\ Franklin T. Adams-Watters, Sep 30 2009

from sympy import fibonacci
def A001595(n): return (fibonacci(n+1)<<1)-1 # Chai Wah Wu, Sep 10 2024

[2*fibonacci(n+1)-1 for n in (0..40)] # G. C. Greubel, Jul 10 2019

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) = A000071(n+3) - A000045(n). - Vladimir Joseph Stephan Orlovsky, Oct 13 2009
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

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

A033539 a(0)=1, a(1)=1, a(2)=1, a(n) = 2*a(n-1) + a(n-2) + 1.

Original entry on oeis.org

1, 1, 1, 4, 10, 25, 61, 148, 358, 865, 2089, 5044, 12178, 29401, 70981, 171364, 413710, 998785, 2411281, 5821348, 14053978, 33929305, 81912589, 197754484, 477421558, 1152597601, 2782616761, 6717831124, 16218279010, 39154389145

Offset: 0

Antti Karttunen

a(n) or a(n+1) gives the number of times certain simple recursive procedures are called to effect a reversal of a sequence of n elements (including both the top-level call and any subsequent recursive calls). See example and program lines.

			See the Python, Erlang (myrev), PARI (rev) and Forth definitions (REV3) given at Program section.
PARI, Python and Erlang functions are called a(n+1) times for the list of length n, while Forth word REV3 is called a(n) times if there are n elements in the parameter stack.

T. D. Noe, Table of n, a(n) for n = 0..300
Antti Karttunen, More information
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Cf. A002203, A033538.

# definition, demonstrating the reversal of the lists:
myrev([]) -> [];
myrev([A]) -> [A];
myrev([X|Y]) ->
   [A|B] = myrev(Y),
   [A|myrev([X|myrev(B)])].

# definition, demonstrating how to turn a parameter stack upside down:
: REV3 DEPTH 0= IF ELSE DEPTH 1 = IF ELSE DEPTH 2 = IF SWAP ELSE >R RECURSE R> SWAP >R >R RECURSE R> RECURSE R> THEN THEN THEN ;
-- Antti Karttunen, Mar 04 2013

Concatenation([1], List([1..30], n-> (3*Lucas(2,-1,n-1)[2] -2)/4 )); # G. C. Greubel, Oct 13 2019

a033539 n = a033539_list !! n
a033539_list =
   1 : 1 : 1 : (map (+ 1) $ zipWith (+) (tail a033539_list)
                                        (map (2 *) $ drop 2 a033539_list))
-- Reinhard Zumkeller, Aug 14 2011

I:=[1,1,4]; [1] cat [n le 3 select I[n] else 3*Self(n-1) - Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Oct 13 2019

seq(coeff(series((1 -2*x -x^2 +3*x^3)/((1-x)*(1-2*x-x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 13 2019

Join[{1},RecurrenceTable[{a[0]==a[1]==1,a[n]==2a[n-1]+a[n-2]+1},a,{n,30}]] (* or *) LinearRecurrence[{3,-1,-1},{1,1,1,4},30] (* Harvey P. Dale, Nov 20 2011 *)
Table[If[n==0, 1, (3*LucasL[n-1, 2] -2)/4], {n, 0, 30}] (* G. C. Greubel, Oct 13 2019 *)

/* needs version >= 2.5 */
/* function demonstrating the reversal of the lists and counting the function calls: */
rev( L )={ cnt++; if( #L>1, my(x,y); x=L[#L]; listpop(L); L=rev(L); y=L[#L]; listpop(L); L=rev(L); listput(L,x); L=rev(L); listput(L,y)); L }
for(n=0,50,cnt=0;print(n": rev(",L=List(vector(n,i,i)),")=",rev(L),",  cnt="cnt)) \\ Antti Karttunen, Mar 05 2013, partially based on previous PARI code from Michael Somos, 1999. Edited by M. F. Hasler, Mar 05 2013

concat([1], vector(30, n, (3*sum(k=0,(n-1)\2, binomial(n-1,2*k) * 2^k) - 1)/2 )) \\ G. C. Greubel, Oct 13 2019

rev([],[]).
   rev([A],[A]).
   rev([A|XB],[B|YA]) :-
rev(XB,[B|Y]), rev(Y,X), rev([A|X],YA). % Lewis Baxter, Jan 04 2021

# function, demonstrating the reversal of the lists:
def myrev(lista):
  '''Reverses a list, in cumbersome way.'''
  if(len(lista) < 2): return(lista)
  else:
    tr = myrev(lista[1:])
    return([tr[0]]+myrev([lista[0]]+myrev(tr[1:])))

[1]+[(3*lucas_number2(n-1,2,-1) -2)/4 for n in (1..30)] # G. C. Greubel, Oct 13 2019

a(n) = (3/4)*(1+sqrt(2))^(n-1) + 3/4*(1-sqrt(2))^(n-1) - 1/2 + 3*0^n, with n >= 0. - Jaume Oliver Lafont, Sep 10 2009
G.f.: (1 - 2*x - x^2 + 3*x^3)/((1-x)*(1-2*x-x^2)). - Jaume Oliver Lafont, Sep 09 2009
a(n) = 3*a(n-1) - a(n-2) - a(n-3), a(0)=1, a(1)=1, a(2)=1, a(3)=4. - Harvey P. Dale, Nov 20 2011
a(n) = (3*A001333(n-1) - 1)/2. - R. J. Mathar, Mar 04 2013
a(n) = -1/2 - (3/4)*(1+sqrt(2))^n - (3/4)*sqrt(2)*(1-sqrt(2))^n - (3/4)*(1-sqrt(2))^n + (3/4)*(1+sqrt(2))^n*sqrt(2) for n >= 1. - Alexander R. Povolotsky, Mar 05 2013
E.g.f.: 3 + (1/2)*exp(x)*(-1 - 3*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Oct 13 2019

cp's OEIS Frontend

A001595 a(n) = a(n-1) + a(n-2) + 1, with a(0) = a(1) = 1.

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A057508 Self-inverse permutation of natural numbers induced by the function 'reverse' (present in programming languages like Lisp, Scheme, Prolog and Haskell) when it acts on symbolless S-expressions encoded by A014486/A063171.

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A033539 a(0)=1, a(1)=1, a(2)=1, a(n) = 2*a(n-1) + a(n-2) + 1.

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