A033192 -id:A033192 - cp's OEIS frontend

A033193 Binomial transform of A033192.

1, 2, 6, 19, 62, 207, 704, 2430, 8486, 29903, 106098, 378391, 1354700, 4863834, 17499302, 63055947, 227465414, 821215295, 2966571096, 10721076118, 38757594758, 140143505031, 506827217210, 1833150646599, 6630915738212, 23986989146162, 86775559512774, 313930265564035

Offset: 0

N. J. A. Sloane

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 3, s(2n) = 3. - Herbert Kociemba, Jun 16 2004

Michael De Vlieger, Table of n, a(n) for n = 0..1791
N. J. A. Sloane, Transforms.
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A033192.

CoefficientList[Series[(x^4 - 7 x^3 + 11 x^2 - 6 x + 1)/((1 - 3 x + x^2) (1 - 5 x + 5 x^2)), {x, 0, 23}], x] (* Michael De Vlieger, Feb 12 2022 *)

Vec((x^4-7*x^3+11*x^2-6*x+1)/((1-3*x+x^2)*(1-5*x+5*x^2)) + O(x^24)) \\ Stefano Spezia, Aug 22 2025

G.f.: (x^4-7*x^3+11*x^2-6*x+1)/((1-3*x+x^2)*(1-5*x+5*x^2)).
a(n) = (1/5)*Sum_{r=1..9} sin(3*r*Pi/10)^2*(2*cos(r*Pi/10))^(2*n), n >= 1. - Herbert Kociemba, Jun 16 2004
For n > 0, a(n) = (phi^(2*n+1) + 1/phi^(2*n+1))/(2*sqrt(5)) + 5^(n/2-1)*(phi^(n+2) + 1/phi^(n+2))/2, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 22 2025

A003603 Fractal sequence obtained from Fibonacci numbers (or Wythoff array).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 5, 1, 6, 4, 3, 7, 2, 8, 5, 1, 9, 6, 4, 10, 3, 11, 7, 2, 12, 8, 5, 13, 1, 14, 9, 6, 15, 4, 16, 10, 3, 17, 11, 7, 18, 2, 19, 12, 8, 20, 5, 21, 13, 1, 22, 14, 9, 23, 6, 24, 15, 4, 25, 16, 10, 26, 3, 27, 17, 11, 28, 7, 29, 18, 2, 30, 19, 12, 31, 8, 32, 20, 5, 33

Offset: 1

N. J. A. Sloane, Mira Bernstein

Length of n-th row = A000045(n); last term of n-th row = A094967(n-1); sum of n-th row = A033192(n-1). - Reinhard Zumkeller, Jan 26 2012

			In the recurrence for making new rows, we get row 5 from row 4 thus: write row 4: 1,3,2, and then place 4 right after 1, and place 5 right after 2, getting 1,4,3,2,5. - _Clark Kimberling_, Oct 29 2009

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Rows n = 1..20 of triangle, flattened
J. H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
David Garth and Joseph Palmer, Self-Similar Sequences and Generalized Wythoff Arrays, Fibonacci Quart. 54 (2016), no. 1, 72-78.
Clark Kimberling, Fractal sequences.
Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See page 8.
N. J. A. Sloane, Classic Sequences.

Equals A019586(n) + 1. Cf. A003602, A000045, A033192, A035513, A035612, A094967, A265650.

-- according to Kimberling, see formula section.
a003603 n k = a003603_row n !! (k-1)
a003603_row n = a003603_tabl !! (n-1)
a003603_tabl = [1] : [1] : wythoff [2..] [1] [1] where
   wythoff is xs ys = f is xs ys [] where
      f js     []     []     ws = ws : wythoff js ys ws
      f js     []     [v]    ws = f js [] [] (ws ++ [v])
      f (j:js) (u:us) (v:vs) ws
        | u == v = f js us vs (ws ++ [v,j])
        | u /= v = f (j:js) (u:us) vs (ws ++ [v])
-- Reinhard Zumkeller, Jan 26 2012

A003603 := proc(n::posint)
    local r,c,W ;
    for r from 1 do
        for c from 1 do
            W := A035513(r,c) ;
            if W = n then
                return r ;
            elif W > n then
                break ;
            end if;
        end do:
    end do:
end proc:
seq(A003603(n),n=1..100) ; # R. J. Mathar, Aug 13 2021

num[n_, b_] := Last[NestWhile[{Mod[#[[1]], Last[#[[2]]]], Drop[#[[2]], -1], Append[#[[3]], Quotient[#[[1]], Last[#[[2]]]]]} &, {n, b, {}}, #[[2]] =!= {} &]];
left[n_, b_] := If[Last[num[n, b]] == 0, Dot[num[n, b], Rest[Append[Reverse[b], 0]]], n];
fractal[n_, b_] := # - Count[Last[num[Range[#], b]], 0] &@
   FixedPoint[left[#, b] &, n];
Table[fractal[n, Table[Fibonacci[i], {i, 2, 12}]], {n, 30}] (* Birkas Gyorgy, Apr 13 2011 *)
row[1] = row[2] = {1};
row[n_] := row[n] = Module[{ro, pos, lp, ins}, ro = row[n-1]; pos = Position[ro, Alternatives @@ Intersection[ro, row[n-2]]] // Flatten; lp = Length[pos]; ins = Range[lp] + Max[ro]; Do[ro = Insert[ro, ins[[i]], pos[[i]] + i], {i, 1, lp}]; ro];
Array[row, 9] // Flatten (* Jean-François Alcover, Jul 12 2016 *)

Vertical para-budding sequence: says which row of Wythoff array (starting row count at 1) contains n.
If one deletes the first occurrence of 1, 2, 3, ... the sequence is unchanged.
From Clark Kimberling, Oct 29 2009: (Start)
The fractal sequence of the Wythoff array can be constructed without reference to the Wythoff array or Fibonacci numbers. Write initial rows:
Row 1: .... 1
Row 2: .... 1
Row 3: .... 1..2
Row 4: .... 1..3..2
For n>4, to form row n+1, let k be the least positive integer not yet used; write row n, and right after the first number that is also in row n-1, place k; right after the next number that is also in row n-1, place k+1, and continue. A003603 is the concatenation of the rows. (End)
Conjecture: a(n) = abs(floor(n/phi) - floor(n*(1/phi + 1/(-phi)^(A035612(n) + 1)))) where phi = (1+sqrt(5))/2. - Alan Michael Gómez Calderón, Oct 27 2023

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

Keyword tabf added by Reinhard Zumkeller, Jan 26 2012

A005207 a(n) = (F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 127, 322, 826, 2135, 5545, 14445, 37701, 98514, 257608, 673933, 1763581, 4615823, 12082291, 31628466, 82798926, 216761547, 567474769, 1485645049, 3889431721, 10182603746, 26658304492, 69792188337, 182718064101, 478361686155, 1252366480135

Offset: 0

N. J. A. Sloane

Number of block fountains with exactly n coins in the base when mirror image fountains are identified. - Michael Woltermann (mwoltermann(AT)washjeff.edu), Oct 06 2010
a(n) = C(F(n+1)+1,2) + C(F(n)+1,2) = pairwise sums of A033192. - Ralf Stephan, Jul 06 2003
Number of (3412,54312)- and (3412,45321)-avoiding involutions in S_{n+1}. - Ralf Stephan, Jul 06 2003
Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1. - Herbert Kociemba, May 31 2004
The sequence 1,1,2,4,9,... has g.f. 1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x))))=(1-3*x+x^2+x^2)/(1-4*x+3*x^2+2*x^3-x^4), and general term (A001519(n)+A000045(n+1))/2. It is the binomial transform of A001519 aerated. - Paul Barry, Dec 17 2009
The Kn3 and Kn4 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 lead to this sequence. - Johannes W. Meijer, Jul 14 2011
Convolution of [1,1,1,2,5,...], which is A001519 with another leading 1, and A212804. - R. J. Mathar, Apr 14 2018
a(n) is the number of Motzkin n-paths of height <= 3. - Alois P. Heinz, Nov 24 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..300 from Vincenzo Librandi)
E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, sec. 8, arXiv:math/0307050 [math.CO], 2003.
S. Felsner, D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392.
M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392. (Annotated scanned copy)
Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, arXiv preprint arXiv:1105.3713 [math.CO], 2011.
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
Michael Woltermann, Problem 1826, Mathematics Magazine, 83 (2010), 304-305.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Cf. A000045, A001006, A001519, A131719.

A005207:=-(1-2*z-z^2+z^3)/(z^2-3*z+1)/(z^2+z-1); # Simon Plouffe in his 1992 dissertation with offset 0
a:= n-> (Matrix([[1,1,1,3]]). Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [4,-3,-2,1][i] else 0 fi)^n)[1,2]: seq(a(n), n=0..34); # Alois P. Heinz, Sep 06 2008

LinearRecurrence[{4, -3, -2, 1}, {1, 2, 4, 9}, 30] (* Jean-François Alcover, Jan 31 2016 *)

a(n)=(fibonacci(2*n-1)+fibonacci(n+1))/2

x='x+O('x^50); Vec(-x*(1-2*x-x^2+x^3)/((x^2+x-1)*(x^2-3*x+1))) \\ G. C. Greubel, Mar 05 2017

G.f.: 1-x*(1-2*x-x^2+x^3)/((x^2+x-1)*(x^2-3*x+1)).
a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4).
a(n) = (w^(2*n-1) + w^(1-2*n) + w^(n+1) - (-w)^(-1-n))/(4*w-2) where w = (1+sqrt(5))/2.
a(n) = (2/5)*Sum_{k=1..4} ( sin(Pi*k/5)^2*(1 + 2*cos(Pi*k/5))^n ). - Herbert Kociemba, May 31 2004
a(-1-2*n) = A027994(2*n); a(-2*n)=A059512(2*n+1).
Let M = an infinite tridiagonal matrix with all 1's in the super and main diagonals and [1,1,1,0,0,0,...] in the subdiagonal. Let V = vector [1,0,0,0,...]. The sequence is generated as leftmost column of M*V iterates. - Gary W. Adamson, Jun 07 2011
2*a(n) = A000045(n+1) + A001519(n). - R. J. Mathar, Apr 14 2018
a(n) mod 2 = A131719(n+3). - Alois P. Heinz, Nov 24 2023

a(0)=1 prepended by Alois P. Heinz, Nov 24 2023

A033191 Binomial transform of [ 1, 0, 1, 1, 3, 6, 15, 36, 91, 231, 595, ... ], which is essentially binomial(Fibonacci(k) + 1, 2).

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4861, 16778, 58598, 206516, 732825, 2613834, 9358677, 33602822, 120902914, 435668420, 1571649221, 5674201118, 20497829133, 74079051906, 267803779710, 968355724724, 3502058316337, 12666676646162, 45818284122149

Offset: 0

Simon P. Norton

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 1. - Herbert Kociemba, Jun 14 2004
The sequence 1,2,5,14,... has g.f. 1/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-2x)))) = (1-6x+10x^2-4x^3)/(1-8x+21x^2-20x^3+5x^4), and is the second binomial transform A001519 aerated. - Paul Barry, Dec 17 2009
Counts all paths of length (2*n), n>=0, starting and ending at the initial node on the path graph P_9, see the Maple program. - Johannes W. Meijer, May 29 2010

			1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + ...

Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243v1 [math.CO], 2012 (Corollary 3, case k=8, pages 10-11). [_Bruno Berselli_, May 12 2012]
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See p. 9.
Index entries for linear recurrences with constant coefficients, signature (8,-21,20,-5).

Cf. A033192.
Cf. A081567, A147748 and A178381.
Cf. A211216.

with(GraphTheory): G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=24; n2:=nmax*2: for n from 0 to n2 do B(n):=A^n; a(n):=B(n)[1,1]; od: seq(a(2*n),n=0..nmax); # Johannes W. Meijer, May 29 2010

CoefficientList[Series[(1-7x+15x^2-10x^3+x^4)/(1-8x+21x^2-20x^3+5x^4), {x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{8,-21,20,-5},{1,2,5,14}, 30]]  (* Harvey P. Dale, Apr 26 2011 *)

{a(n) = local(A); A = 1; for( i=1, 8, A = 1 / (1 - x*A)); polcoeff( A + x * O(x^n), n)} /* Michael Somos, May 12 2012 */

G.f.: (1-7x+15x^2-10x^3+x^4)/(1-8x+21x^2-20x^3+5x^4). - Ralf Stephan, May 13 2003
From Herbert Kociemba, Jun 14 2004: (Start)
a(n) = (1/5)*Sum_{r=1..9} sin(r*Pi/10)^2*(2*cos(r*Pi/10))^(2n), n >= 1;
a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4), n >= 5. (End)
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x )))))))). - Michael Somos, May 12 2012

A081667 a(n) = Fibonacci(binomial(n+2,2)).

Original entry on oeis.org

1, 2, 8, 55, 610, 10946, 317811, 14930352, 1134903170, 139583862445, 27777890035288, 8944394323791464, 4660046610375530309, 3928413764606871165730, 5358359254990966640871840, 11825896447871834976429068427, 42230279526998466217810220532898

Offset: 0

Paul Barry, Mar 26 2003

Diagonal of Fibonacci-Pascal triangle A045995.

Alois P. Heinz, Table of n, a(n) for n = 0..96
Peter M. Chema, Illustration of first 12 terms on a square spiral
T. Kotek, J. A. Makowsky, Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs, arXiv preprint arXiv:1309.4020 [math.CO], 2013.

Cf. A000045, A000217, A033192, A045995, A054783.

with(combinat): seq(fibonacci((n^2-n)/2),n=2..16); # Zerinvary Lajos, May 18 2008
# second Maple program:
a:= n-> (<<0|1>, <1|1>>^((n+1)*(n+2)/2))[1, 2]:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 20 2017

Table[Fibonacci[Binomial[n+2,2]],{n,0,20}] (* Harvey P. Dale, Dec 03 2014 *)

[fibonacci(binomial(n,2)) for n in range(2, 17)] # Zerinvary Lajos, Nov 30 2009

a(n) = sqrt(5)2^(-n(n+3)/2)(sqrt(5)+1)^((n^2+3n+2)/2)/10 + sqrt(5)2^(-n(n + 3)/2)(sqrt(5)-1)^((n^2+3n+ 2)/2)(-1)^(n(n+3)/2)/10.
a(n) = A045995(n+2,2).
a(n) = A000045(A000217(n+1)). - Peter M. Chema, Sep 18 2016. See the name.

Name edited by Michel Marcus, Sep 25 2016

A088858 Define a Fibonacci-type sequence to be one of the form s(0) = s_1 >= 1, s(1) = s_2 >= 1, s(n+2) = s(n+1) + s(n); then a(n) = maximal m such that n is the m-th term in some Fibonacci-type sequence.

Original entry on oeis.org

1, 2, 3, 3, 4, 3, 4, 5, 4, 4, 5, 4, 6, 5, 4, 5, 5, 6, 5, 5, 7, 5, 6, 5, 5, 6, 5, 6, 7, 5, 6, 5, 6, 8, 5, 6, 7, 6, 6, 5, 6, 7, 6, 6, 7, 6, 8, 6, 6, 7, 6, 6, 7, 6, 9, 6, 6, 7, 6, 8, 7, 6, 7, 6, 6, 7, 6, 8, 7, 6, 7, 6, 8, 7, 6, 9, 7, 6, 7, 6, 8, 7, 6, 7, 7, 8, 7, 6, 10, 7

Offset: 1

Don Reble, Nov 20 2003

A033192(k) is the number of integers m such that a(m) = k. - Michel Marcus, Aug 03 2017

Michel Marcus, Table of n, a(n) for n = 1..1000
J. H. E. Cohn, Recurrent Sequences Including N, Fib. Quart., 29-1, 1991.
T. Denes, Problem 413, Discrete Math. 272 (2003), 302 (but there are several errors in the table given there).
James P. Jones and Péter Kiss, Representation of integers as terms of a linear recurrence with maximal index, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae, 25. (1998) pp. 21-37.

Cf. A033192, A088527 (which is a(n)+1).

max = 12; s[n_] := (1/2)*((3*s1 - s2)*Fibonacci[n] + (s2 - s1)*LucasL[n]); a[n_] := Reap[Do[If[s[m] == n, Sow[m - 1]], {m, 1, max}, {s1, 1, max}, {s2, 1, max}]][[2, 1]] // Max; Table[a[n], {n, 1, 90}] (* Jean-François Alcover, Jan 15 2013 *)

a(n) = {if (n==1, return (1)); r = 2; while (ceil(((-1)^r*fibonacci(r-2)*n + 1)/fibonacci(r-1)) <= floor(((-1)^r*fibonacci(r-1)*n - 1)/fibonacci(r)), r++); r-1;} \\ Michel Marcus, Aug 02 2017

For n>1, a(n) is the largest integer r>1 such that ceiling(((-1)^r*Fibonacci(r-2)*n + 1)/Fibonacci(r-1)) <= floor(((-1)^r*Fibonacci(r-1)*n - 1)/Fibonacci(r)). See Theorem 2.12 in Jones & Kiss. - Michel Marcus, Aug 02 2017

A301809 Group the natural numbers such that the first group is (1) then (2),(3),(4,5),(6,7,8),... with the n-th group containing F(n) sequential terms where F(n) is the n-th Fibonacci number (A000045(n)). Sequence gives the sum of terms in the n-th group.

Original entry on oeis.org

1, 2, 3, 9, 21, 55, 140, 364, 945, 2465, 6435, 16821, 43992, 115102, 301223, 788425, 2063817, 5402651, 14143524, 37026936, 96935685, 253777537, 664392743, 1739393929, 4553778096, 11921922650, 31211961195, 81713914569, 213929707485, 560075086495, 1466295355580, 3838810662436, 10050136117497

Offset: 1

Frank M Jackson, Mar 27 2018

a(n) is the sum of all nodes at height n-1 within a binary tree structure recursively built from the Hofstadter G-sequence (see comments for A005206).

			a(7) = 14 + 15 + 16 + ... + 21 = (F(9)+1)*F(6)/2 = 140.

Colin Barker, Table of n, a(n) for n = 1..1000
Wikipedia, Hofstadter sequence.
Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1, 1).

Cf. A000045, A005206, A122931.

Cf. A000217, A033192.

[1] cat [(Fibonacci(n+2)+1)*Fibonacci(n-1) div 2 : n in [2..35] ]; // Vincenzo Librandi, Apr 18 2018

a[n_] := If[n==1, 1, (Fibonacci[n+2]+1)Fibonacci[n-1]/2]; Array[a, 50]
Join[{1}, LinearRecurrence[{3, 1, -5, -1, 1}, {2, 3, 9, 21, 55}, 40]] (* Vincenzo Librandi, Apr 18 2018 *)

a(n) = if (n==1, 1, (fibonacci(n+2)+1)*fibonacci(n-1)/2); \\ Michel Marcus, Apr 21 2018

Vec(x*(1 - x)*(1 - 4*x^2 - x^3 + x^4) / ((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)) + O(x^60)) \\ Colin Barker, May 11 2018

a(1) = 1 and for n > 1, a(n) = (F(n+2)+1)*F(n-1)/2, where F(n) is the n-th Fibonacci number (A000045).
From Colin Barker, Mar 27 2018: (Start)
G.f.: x*(1 - x)*(1 - 4*x^2 - x^3 + x^4) / ((1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)).
a(n) = 3*a(n-1) + a(n-2) - 5*a(n-3) - a(n-4) + a(n-5) for n>6. (End)
a(n) = A033192(n+1) - A033192(n) for n > 1. - J.S. Seneschal, Jul 07 2025

A032441 a(n) = Sum_{i=0..2} binomial(Fibonacci(n),i).

Original entry on oeis.org

1, 2, 2, 4, 7, 16, 37, 92, 232, 596, 1541, 4006, 10441, 27262, 71254, 186356, 487579, 1276004, 3339821, 8742472, 22885996, 59912932, 156848617, 410626154, 1075018897, 2814412826, 7368190922, 19290113572, 50502074767, 132215989336, 346145696821, 906220783316

Offset: 0

N. J. A. Sloane

nonn

Index entries for linear recurrences with constant coefficients, signature (4,-2,-6,4,2,-1).

Cf. A000045, A033192.

a:= n-> (f-> f*(f+1)/2+1)((<<0|1>, <1|1>>^n)[1, 2]):
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 01 2018

Table[Sum[Binomial[Fibonacci[n],i],{i,0,2}],{n,0,30}] (* or *) LinearRecurrence[ {4,-2,-6,4,2,-1},{1,2,2,4,7,16},30] (* Harvey P. Dale, Feb 02 2015 *)

a(0)=1, a(1)=2, a(2)=2, a(3)=4, a(4)=7, a(5)=16, a(n)=4*a(n-1)- 2*a(n-2)- 6*a(n-3)+4*a(n-4)+2*a(n-5)-a(n-6). - Harvey P. Dale, Feb 02 2015

a(n) = A033192(n) + 1. - Alois P. Heinz, Jul 01 2018

A296516 a(n) is the number of terms in expanded form of bivariate polynomial Q_n, where (P_n, Q_n) is defined by: P_0 = x, Q_0 = y, P_(n+1) = P_n + Q_n, Q_(n+1) = P_n * Q_n.

Original entry on oeis.org

1, 1, 2, 5, 14, 40, 111, 300, 797, 2098, 5499, 14389, 37634, 98435, 257516, 673827, 1763460, 4615686, 12082137, 31628294

Offset: 0

Luc Rousseau, Feb 27 2018

Programs based on the direct application of the definition quickly reach a limitation by combinatorial explosion, hence this short list of values in section Data. The first conjectured formula (see Formulas) obtained by the observation of a pattern in the 2D shape of Q_n (as drawn in the Examples) is more computationally efficient and makes it possible to produce a significantly longer list of (nonguaranteed) values: see attached a-file in b-file format, section Links.

A003686 and A064847 are values of P_n and Q_n at x=y=1 (i.e., sums of coefficients in these polynomials). At x=2, y=1 (or vice versa) P_n and Q_n seem to give same sequences but shifted. At x=y=-1, P_n seems to give A000058 negated interleaved with -1's, while Q_n seems to give A007018 interleaved with the same sequence negated. - Andrey Zabolotskiy, May 22 2018

			Q_0 = y -> one term -> a(0) = 1;
Q_1 = x*y -> one term -> a(1) = 1;
Q_2 = x^2*y + x*y^2 -> two terms -> a(2) = 2;
Q_3 = x^3*y + 2*x^2*y^2 + x^3*y^2 + x*y^3 + x^2*y^3 -> five terms -> a(3) = 5;
...
Locations of terms in 2D arrays indexed by the exponents of x and y:
   0: .   1: ..   2: ...   3: ....   4: ......   5: .........
      X      .X      ..X      ...X      ....X.      .....X...
                     .X.      ..XX      ...XXX      ....XXXX.
                              .XX.      ..XXXX      ...XXXXXX
                                        .XXXX.      ..XXXXXXX
                                        ..XX..      .XXXXXXXX
                                                    ..XXXXXX.
                                                    ..XXXXX..
                                                    ...XXX...

Luc Rousseau, 201 terms obtained with conjectured formula.
Rémy Sigrist, Colored scatterplot of the points (i, j) such that the coefficient of x^i*y^j in Q_14 is nonzero (where the color is function of the coefficient of x^i*y^j in Q_14).

Cf. A000045, A000217, A001622, A033192, A003686, A064847, A000058, A007018.

Cf. A005207 (which with +1 added appears to be to P_n as a(n) is to Q_n).

{p[0], q[0]} := {x, y};
{p[n_], q[n_]} := {p[n - 1] + q[n - 1], p[n - 1] q[n - 1]};
a[n_] := Length@CoefficientRules[q[n]];
Table[a[n], {n, 0, 10}] (* Andrey Zabolotskiy, Peter Luschny, May 30 2018 *)
From Luc Rousseau, Feb 27 2018: (Start)
(* conjectured: *)
T[n_] := n*(n + 1)/2
F[n_] := Fibonacci[n]
V[n_] := Sum[F[k]*(Sum[(n - 2 - l)*F[l], {l, k + 1, n - 3}]), {k, 1, n - 4}]
W[n_] := Sum[(n - 2 - l)*T[F[l]], {l, 1, n - 3}]
AA[n_] := (F[n + 1])^2 - T[n - 1] - T[F[n - 1]] - 2*V[n] - 2 W[n]
Table[AA[n], {n, 1, 50}]
(End)

def A296516(n):
    P, Q = {(1,0)}, {(0,1)}
    for _ in range(n): P, Q = P|Q, set((p[0]+q[0],p[1]+q[1]) for p in P for q in Q)
    return len(Q) # Chai Wah Wu, Oct 18 2021

Conjectured:
a(n) = F(n+1)^2 - T(n-1) - T(F(n-1)) - 2*V(n) - 2*W(n) for n > 0, where
  F(n) is the n-th Fibonacci number,
  T(n) is the n-th triangular number,
  V(n) = Sum_{i=1..n-4} F(i)*(Sum_{j=i+1..n-3} (n-2-j)*F(j)),
  W(n) = Sum_{i=1..n-3} (n-2-i)*T(F(i)).
Or:
a(n) = F(n+1)^2 - T(n-1) - T(F(n-1)) - [n>=4]*U(n) where [] is the Iverson bracket, and U(n) = F(n-1)*F(n-3) + F(n+1)*(2*F(n-4)-1) + 5 - 8*(n mod 2).
Conjectures from Colin Barker, Mar 28 2018: (Start)
G.f.: 1 + x*(1 - 5*x + 9*x^2 - 5*x^3 - 2*x^4 + x^5) / ((1 - x)^3*(1 - 3*x + x^2)*(1 - x - x^2)).
a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 6*a(n-4) - 6*a(n-5) + 5*a(n-6) - a(n-7) for n > 7.
(End)
From Luc Rousseau, Mar 30 2018: (Start)
Derived from the conjectured order-7 linear recurrence above, for n > 0,
a(n) = -(1/2)*n^2 + (1/2)*n - 1 + ((2+phi)/10)*(phi^2)^n + ((4-3*phi)/10)*(-phi^(-1))^n + ((1+3*phi)/10)*phi^n + ((3-phi)/10)*(phi^(-2))^n,
where phi denotes the golden ratio and lim_{n->oo} a(n+1)/a(n) = phi^2.
a(n) = (F(2*n+1) + F(n+2) - n^2 + n - 2) / 2.
(End)

a(15)-a(19) from Andrey Zabolotskiy, May 30 2018

A381503 Number of rectangles in a Fibonacci(n) X Fibonacci(n+1) grid.

Original entry on oeis.org

1, 3, 18, 90, 540, 3276, 21021, 137445, 916300, 6167700, 41812200, 284604840, 1942428033, 13278352815, 90862598190, 622150990734, 4261620339460, 29198279495220, 200080147593645, 1371167039301345, 9397260307853496, 64406143791454248, 441430873666787088, 3025546968019741200

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Darío Clavijo, Feb 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A000045, A033192, A096948, A377704.

Times @@@ Partition[PolygonalNumber[Fibonacci[Range[25]]], 2, 1] (* Paolo Xausa, Mar 13 2025 *)

from sympy import fibonacci
A096948 = lambda n,m: (m * (m + 1) * n * (n + 1)) >> 2
a = lambda n: A096948(fibonacci(n),fibonacci(n+1))
print([a(n) for n in range(1, 25)])

a(n) = A096948(Fibonacci(n),Fibonacci(n+1)).

a(n) = A033192(n) * A033192(n+1).

cp's OEIS Frontend

A033193 Binomial transform of A033192.

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A003603 Fractal sequence obtained from Fibonacci numbers (or Wythoff array).

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A005207 a(n) = (F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.

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A033191 Binomial transform of [ 1, 0, 1, 1, 3, 6, 15, 36, 91, 231, 595, ... ], which is essentially binomial(Fibonacci(k) + 1, 2).

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A081667 a(n) = Fibonacci(binomial(n+2,2)).

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A088858 Define a Fibonacci-type sequence to be one of the form s(0) = s_1 >= 1, s(1) = s_2 >= 1, s(n+2) = s(n+1) + s(n); then a(n) = maximal m such that n is the m-th term in some Fibonacci-type sequence.

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A301809 Group the natural numbers such that the first group is (1) then (2),(3),(4,5),(6,7,8),... with the n-th group containing F(n) sequential terms where F(n) is the n-th Fibonacci number (A000045(n)). Sequence gives the sum of terms in the n-th group.