A001519 -id:A001519 - cp's OEIS frontend

A059929 a(n) = Fibonacci(n)*Fibonacci(n+2).

0, 2, 3, 10, 24, 65, 168, 442, 1155, 3026, 7920, 20737, 54288, 142130, 372099, 974170, 2550408, 6677057, 17480760, 45765226, 119814915, 313679522, 821223648, 2149991425, 5628750624, 14736260450, 38580030723, 101003831722, 264431464440, 692290561601, 1812440220360

Offset: 0

Henry Bottomley, Feb 09 2001

Expansion of golden ratio (1+sqrt(5))/2 as an infinite product: phi = Product_{i>=0} (1+1/(Fibonacci(2*i+1) * Fibonacci(2*i+3)-1)) * (1-1/(Fibonacci(2*i+2) * Fibonacci(2i+4)+1)). - Thomas Baruchel, Nov 11 2003
Each of these is one short of or one over the square of a Fibonacci number (A007598). This means that a rectangle sized F(n) by F(n + 2) units can't be converted into a square with sides of length F(n + 1) units unless one square unit of material is added or removed. - Alonso del Arte, May 03 2011
These are the integer parts of the numerators of the numbers with continued fraction representations [1, 2, 2, 2, ...], [1, 1, 2, 2, 2, ...], [1, 1, 1, 2, 2, 2, ...], etc., that is, sqrt(2), (2+sqrt(2))/2, 3-sqrt(2), (10+sqrt(2))/7, (24-sqrt(2))/14, etc. - Geoffrey Caveney, May 03 2014
a(n) appears also as the third component of the square of [F(n), F(n+1), F(n+2), F(n+3)], for n >= 0, where F(n) = A000045(n), in the Clifford algebra Cl_2 over Euclidean 2-space. The whole quartet of sequences for this square is [-A248161(n), A079472(n+1), a(n), A121801(n+1)]. See the Oct 15 2014 comment in A147973 where also a reference is given. - Wolfdieter Lang, Nov 01 2014
Numbers with a continued fraction expansion with the repeating sequence of length n [1, 1, ..., 1, 2], n-1 ones followed by a single two, for n > = 1, appear to be equal to (F(n) + sqrt(a(n)))/F(n+1), where F(n) = A000045(n). - R. James Evans, Nov 21 2018
The preceding conjecture is true. Proof: For n >= 1 let c(n) := confrac(repeat(1^{n-1}, 2)) where 1^{k} denotes 1 taken k times. This can be computed, e.g. from [Perron, third and fourth eq. on p. 62], as c(n) = (F(n) + sqrt(F(n+1)^2 - (-1)^n)) / F(n+1), which is the conjectured formula because F(n+1)^2 - (-1)^n = a(n). - Wolfdieter Lang, Jan 05 2019

			G.f. = 2*x + 3*x^2 + 10*x^3 + 24*x^4 + 65*x^5 + 168*x^6 + ... - _Michael Somos_, Mar 18 2022

Oskar Perron, Die Lehre von den Kettenbrüchen, Band I, 3. Auflage, B. G. Teubner, Stuttgart, 1954, pp. 61-61.

Muniru A Asiru, Table of n, a(n) for n = 0..2374 (first 501 terms from Harry J. Smith)
Tim Jones (producer), Engineering Bits & Bytes: The Fibonacci Puzzle, Wayne State University College of Engineering (2011).
Eric H. Kuo, Applications of graphical condensation for enumerating matchings and tilings, arXiv:math/0304090 [math.CO], 2003.
Marc Renault, Properties of the Fibonacci Sequence Under Various Moduli, Master's Thesis, Wake Forest University, 1996.
Michel Waldschmidt, Open Diophantine problems, arXiv:math/0312440 [math.NT], 2003-2004.
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Bisection of A070550.
First differences of A059840.
Cf. A000045, A001519, A001654, A007598, A033999, A035513, A079472, A121801, A126358, A147973, A248161.

a:=List([0..30],n->Fibonacci(n)*Fibonacci(n+2));; Print(a); # Muniru A Asiru, Jan 05 2019

[Fibonacci(n)*Fibonacci(n+2): n in [0..30]]; // Vincenzo Librandi, Jul 02 2014

with(combinat): a:=n->fibonacci(n)*fibonacci(n+2): seq(a(n), n=0..26); # Zerinvary Lajos, Oct 07 2007

Table[Fibonacci[n]*Fibonacci[n+2],{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

a(n) = fibonacci(n)*fibonacci(n + 2) \\ Harry J. Smith, Jun 30 2009

from sympy import fibonacci
[fibonacci(n)*fibonacci(n+2) for n in range(30)] # Stefano Spezia, Jan 05 2019

[fibonacci(n)*fibonacci(n+2) for n in range(30)] # G. C. Greubel, Nov 21 2018

a(n) = Fibonacci(n+1)^2 - (-1)^n = A007598(n+1) + A033999(n+1) = A000045(n+1)^2 - A033999(n).
G.f.: (2*x-x^2) / ((1+x)*(1-3*x+x^2)).
Sum_{n>=1} 1/a(n) = 1.
Sum_{n>=1} (-1)^n/a(n) = 2 - sqrt(5).
Sum_{n>=1} 1/a(2n-1) = 1/phi = (sqrt(5) - 1)/2. - Franz Vrabec, Sep 15 2005
Sum_{n>=1} 1/a(2n) = (3 - sqrt(5))/2. - Franz Vrabec, Nov 30 2009
a(n) = ((7+3*sqrt(5))/10)*((3+sqrt(5))/2)^(n-1) + ((7-3*sqrt(5))/10)*((3-sqrt(5))/2)^(n-1) + (3/5)*(-1)^(n-1). - Tim Monahan, Aug 09 2011
a(n) = (Lucas(n+1)^2 - Fibonacci(n+1)^2)/4. - Vincenzo Librandi, Aug 02 2014
a(n) = F(n-2)*F(n) + F(n-1)*F(n) + F(n-2)*F(n+1) + F(n-1)*F(n+1), where F=A000045, F(-2)=-1, F(-1)=1. - Bruno Berselli, Nov 03 2015
a(n) = A035513(1,n-1)*A035513(3,n-1)/2 = A035513(1,n-1)*A035513(4,n-1)/3. - R. J. Mathar, Sep 04 2016
a(n)+a(n+1) = A001519(n+2). - R. J. Mathar, Oct 19 2021
a(n) = 2*A001654(n) - A001654(n-1). - R. J. Mathar, Oct 19 2021
a(n)+a(n+3) = 2*F(2n+5) = A126358(n+2). - Andrés Ventas, Oct 25 2021
Sum_{n>=1} Fibonacci(n+1)/a(n) = 2. - Amiram Eldar, Jan 11 2022
a(n) = a(-2-n) and a(n) + a(n+3) = 2*(a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Mar 18 2022

A055588 a(n) = 3*a(n-1) - a(n-2) - 1 with a(0) = 1 and a(1) = 2.

Original entry on oeis.org

1, 2, 4, 9, 22, 56, 145, 378, 988, 2585, 6766, 17712, 46369, 121394, 317812, 832041, 2178310, 5702888, 14930353, 39088170, 102334156, 267914297, 701408734, 1836311904, 4807526977, 12586269026, 32951280100, 86267571273

Offset: 0

Wolfdieter Lang, May 30 2000; Barry E. Williams, Jun 04 2000

Number of directed column-convex polyominoes with area n+2 and having two cells in the bottom row. - Emeric Deutsch, Jun 14 2001
a(n) is the length of the list generated by the substitution: 3->3, 4->(3,4,6), 6->(3,4,6,6): {3, 4}, {3, 3, 4, 6}, {3, 3, 3, 4, 6, 3, 4, 6, 6}, {3, 3, 3, 3, 4, 6, 3, 4, 6, 6, 3, 3, 4, 6, 3, 4, 6, 6, 3, 4, 6, 6}, etc. - Wouter Meeussen, Nov 23 2003
Equals row sums of triangle A144955. - Gary W. Adamson, Sep 27 2008
Equals the INVERT transform of A034943 and the INVERTi transform of A094790. - Gary W. Adamson, Apr 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 20.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
M. M. Mogbonju, I. A. Ogunleke, and O. A. Ojo, Graphical Representation Of Conjugacy Classes In The Order-Preserving Full Transformation Semigroup, International Journal of Scientific Research and Engineering Studies (IJSRES), Volume 1(5) (2014), ISSN: 2349-8862.
László Németh, Hyperbolic Pascal pyramid, arXiv:1511.02067 [math.CO], 2015 (1st line of Table 1 is 3*a(n-2)).
László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (1st line of Table 1 is a(n-2)).
Yan X Zhang, Four Variations on Graded Posets, arXiv:1508.00318 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A001906, A008346, A019827, A034943, A055587, A094790, A144955.
Partial sums of A001519.
Apart from the first term, same as A052925.

List([0..40], n-> Fibonacci(2*n)+1 ); # G. C. Greubel, Jun 06 2019

[Fibonacci(2*n)+1: n in [0..40]]; // Vincenzo Librandi, Sep 30 2017

g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)+1), n=0..27); # Zerinvary Lajos, Mar 22 2009

Table[Fibonacci[2n] +1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -4, 1}, {1, 2, 4}, 40] (* Vincenzo Librandi, Sep 30 2017 *)

vector(40, n, n--; fibonacci(2*n)+1) \\ G. C. Greubel, Jun 06 2019

[lucas_number1(n,3,1)+1 for n in range(40)] # Zerinvary Lajos, Jul 06 2008

a(n) = (((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n)/sqrt(5) + 1.
a(n) = Sum_{m=0..n} A055587(n, m) = 1 + A001906(n).
G.f.: (1 - 2*x)/((1 - 3*x + x^2)*(1-x)).
From Paul Barry, Oct 07 2004: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3);
a(n) = Sum_{k=0..floor(n/3)} binomial(n-k, 2*k)2^(n-3*k). (End)
From Paul Barry, Oct 26 2004: (Start)
a(n) = A001906(n) + 1.
a(n) = Sum_{k=0..n} Fibonacci(2*k+2)*(2*0^(n-k) - 1).
a(n) = A008346(2*n). (End)
a(n) = Sum_{k=0..2*n+1} ((-1)^(k+1))*Fibonacci(k). - Michel Lagneau, Feb 03 2014
E.g.f.: cosh(x) + sinh(x) + 2*exp(3*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Stefano Spezia, May 14 2024
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/10) (A019827). - Amiram Eldar, Nov 28 2024

A124302 Number of set partitions with at most 3 blocks; number of Dyck paths of height at most 4; dimension of space of symmetric polynomials in 3 noncommuting variables.

Original entry on oeis.org

1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165, 3812798742494, 11438396227481

Offset: 0

Mike Zabrocki, Oct 25 2006

Row sums of triangle in A056241. - Philippe Deléham, Oct 30 2006
Row sums of triangle in A147746. - Philippe Deléham, Dec 04 2008
Hankel transform is := [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008
Number of nonisomorphic graded posets with 0 and 1 and uniform Hasse graph of rank n with no 3-element antichain. (Uniform used in the sense of Retakh, Serconek and Wilson.  Graded used in Stanley's sense that every maximal chain has the same length n.) - David Nacin, Feb 26 2012
Number of Dyck paths of length 2n and height at most 4. - Ira M. Gessel, Aug 06 2012

			There are 15 set partitions of {1,2,3,4}, only {{1},{2},{3},{4}} has more than 3 blocks, so a(4) = 14.
G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 41*x^5 + 122*x^6 + 365*x^7 + ...

R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Per Alexandersson and Frether Getachew, An involution on derangements, arXiv:2105.08455 [math.CO], 2021.
N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
S. Felsner and 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. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 24 2012
V. Jelinek, T. Mansour, and M. Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292-326, - From _N. J. A. Sloane_, Jan 01 2013
Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243 [math.CO], 2012-2014 (Corollary 3, case k=4, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
M. Rosas and B. Sagan, Symmetric Functions in Noncommuting Variables, Transactions of the American Mathematical Society, 358 (2006), no. 1, 215-232.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A056272, A123183, A001519, A000110, A008277, A038754, A048328, A068911, A211216.

Essentially the same as A007051.

I:=[1, 1, 2]; [n le 3 select I[n] else  4*Self(n-1) - 3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 25 2012

a:= proc(n); if n<3 then [1,1,2][n+1]; else 4*a(n-1)-3*a(n-2); fi; end:
# Mike Zabrocki, Oct 25 2006
with(GraphTheory): G:=PathGraph(5): A:= AdjacencyMatrix(G): nmax:=27; for n from 0 to 2*nmax do B(n):=A^n; b(n):=B(n)[1,1]; od: for n from 0 to nmax do a(n):=b(2*n) od: seq(a(n),n=0..nmax);
# Johannes W. Meijer, May 29 2010

a=Exp[x]-1; Range[0, 20]! CoefficientList[Series[1+a+a^2/2+a^3/6, {x,0,20}],x]
Join[{1}, LinearRecurrence[{4, -3}, {1, 2}, 20]] (* David Nacin, Feb 26 2012 *)
CoefficientList[Series[1 / (1 - x / (1 - x / (1 - x / (1 - x)))), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 25 2012 *)
Table[Sum[StirlingS2[n,k],{k,0,3}],{n,0,30}] (* Robert A. Russell, Mar 29 2018 *)

{a(n) = if( n<1, n==0, (3^(n-1) + 1) / 2)}; /* Michael Somos, Apr 03 2014 */

def a(n, adict={0:1, 1:1, 2:2}):
    if n in adict:
        return adict[n]
    adict[n]=4*a(n-1) - 3*a(n-2)
    return adict[n] # David Nacin, Mar 04 2012

O.g.f.: (q^2 - 3*q + 1)/(3*q^2 - 4*q + 1) = Sum_{k=0..3} (q^k/Product_{i=1..k} (1-i*q)).
a(n) = 4*a(n-1) - 3*a(n-2); a(0) = 1, a(1) = 1, a(2) = 2, a(n) = Sum_{k=1..3} A008277(n,k).
Inverse binomial transform of A007581. - Philippe Deléham, Oct 30 2006
a(n) = Sum_{k=0..n} A056241(n,k), n >= 1. - Philippe Deléham, Oct 30 2006
a(0) = 1, a(n) = (3^(n-1) + 1)/2 for n >= 1, see A007051. - Philippe Deléham, Oct 30 2006
E.g.f.: (2 + 3*exp(x) + exp(3x))/6.
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x)))). - Michael Somos, May 03 2012
G.f.: 1 + x + 3*x^2*U(0)/2 where U(k) = 1 + 2/(3*3^k + 3*3^k/(1 - 18*x*3^k/ (9*x*3^k - 1/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 01 2012
G.f.: 1+x*G(0) where G(k) = 1 + 2*x/( 1-2*x - x*(1-2*x)/(x + (1-2*x)*2/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
a(n) = Sum_{k=0..3} Stirling2(n,k). - Robert A. Russell, Mar 29 2018
G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=3. - Robert A. Russell, Apr 25 2018

A075737 Prime Fibonacci numbers with prime indices.

Original entry on oeis.org

2, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241

Offset: 1

Jani Melik, Oct 07 2002

nonn

Same as A005478 except that F(4) = 3 has been omitted.

Sequence of primes in A001519. [James R. Buddenhagen, May 20 2010]

			5 is a prime and fibonacci(5)=5 is also a prime, 7 is a prime and fibonacci(7)=13 is also a prime, but 2 is a prime and fibonacci(2)=1 is not a prime.

T. D. Noe, Table of n, a(n) for n = 1..20

Subsequence of A030426.

Cf. A000045, A005478, A083668, A001519.

with(combinat, fibonacci): fib_supM_pra := proc(n); if (isprime(n)='true') then if (isprime(fibonacci(n))='true') then RETURN(fibonacci(n)); fi; fi; end: seq(fib_supM_pra(i), i=1..500);

Fibonacci[ Prime[ Select[ Range[50], PrimeQ[ Fibonacci[ Prime[ # ]]] & ]]]
Module[{nn=500,fibs},fibs=Fibonacci[Range[nn]];Select[Pick[fibs,Table[ If[ PrimeQ[n],1,0],{n,nn}],1],PrimeQ]] (* Harvey P. Dale, Sep 13 2018 *)

forprime(p=2,1e3,if(isprime(t=fibonacci(p)), print1(t", "))) \\ Charles R Greathouse IV, Feb 03 2014

A166444 a(0) = 0, a(1) = 1 and for n > 1, a(n) = sum of all previous terms.

Original entry on oeis.org

0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Robert G. Wilson v, Oct 13 2009

Essentially a duplicate of A000079. - N. J. A. Sloane, Oct 15 2009
a(n) is the number of compositions of n into an odd number of parts.
Also 0 together with A011782. - Omar E. Pol, Oct 28 2013
Inverse INVERT transform of A001519. - R. J. Mathar, Dec 08 2022

			G.f. = x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 16*x^6 + 32*x^7 + 64*x^8 + 128*x^9 + ...

Indranil Ghosh, Table of n, a(n) for n = 0..3317
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000045, A000079, A000213, A000288, A000322, A000383, A001519.
Cf. A011782, A034008, A060455, A123526, A127193, A127194, A127624.
Cf. A131577, A163551.

[n le 1 select n else 2^(n-2): n in [0..40]]; // G. C. Greubel, Jul 27 2024

a:= n-> `if`(n<2, n, 2^(n-2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 02 2021

a[0] = 0; a[1] = 1; a[n_] := a[n] = Plus @@ Array[a, n - 1]; Array[a, 35, 0]

[(2^n +2*int(n==1) -int(n==0))/4 for n in range(41)] # G. C. Greubel, Jul 27 2024

a(n) = A000079(n-1) for n > 0.
O.g.f.: x*(1 - x) / (1 - 2*x) = x / (1 - x / (1 - x)).
a(n) = (1-n) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
E.g.f.: (exp(2*x) + 2*x - 1)/4. - Stefano Spezia, Aug 07 2022

A080937 Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2*n steps with all values <= 5.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 131, 417, 1341, 4334, 14041, 45542, 147798, 479779, 1557649, 5057369, 16420730, 53317085, 173118414, 562110290, 1825158051, 5926246929, 19242396629, 62479659622, 202870165265, 658715265222, 2138834994142, 6944753544643, 22549473023585

Offset: 0

Henry Bottomley, Feb 25 2003

With interpolated zeros (1,0,1,0,2,...), counts closed walks of length n at start or end node of P_6. The sequence (0,1,0,2,...) counts walks of length n between the start and second node. - Paul Barry, Jan 26 2005
HANKEL transform of sequence and the sequence omitting a(0) is the sequence A130716. This is the unique sequence with that property. - Michael Somos, May 04 2012
From Wolfdieter Lang, Mar 30 2020: (Start)
a(n) is also the upper left entry of the n-th power of the 3 X 3 tridiagonal matrix M_3 = Matrix([1,1,0], [1,2,1], [0,1,2]) from A332602: a(n) = ((M_3)^n)[1,1].
Proof: (M_3)^n = b(n-2)*(M_3)^2 - (6*b(n-3) - b(n-4))*M_3 + b(n-3)*1_3, for n >= 0, with b(n) = A005021(n), for n >= -4. For the proof of this see a comment in A005021. Hence (M_3)^n[1,1] = 2*b(n-2) - 5*b(n-3) + b(n-4), for n >= 0. This proves the 3 X 3 part of the conjecture in A332602 by Gary W. Adamson.
The formula for a(n) given below in terms of r = rho(7) = A160389 proves that a(n)/a(n-1) converges to rho(7)^2 = A116425 = 3.2469796..., because r - 2/r = 0.6920... < 1, and r^2 - 3 = 0.2469... < 1. This limit was conjectured in A332602 by Gary W. Adamson.
(End)

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 131*x^6 + 417*x^7 + 1341*x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..600
Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, Generalized Narayana arrays, restricted Dyck paths, and related bijections, Univ. Bourgogne (France, 2025). See p. 27.
Jean-Luc Baril, Toufik Mansour, José L. Ramírez, and Mark Shattuck, Catalan words avoiding a pattern of length four, Univ. de Bourgogne (France, 2024). See p. 3.
Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
Wei Chen, Enumeration of Set Partitions Refined by Crossing and Nesting Numbers, MS Thesis, Department of Mathematics. Simon Fraser University, Fall 2014. Table 4.1, line k=2.
Johann Cigler, Number of bounded Dyck paths with "negative length", MathOverflow question, Sep 26 2020.
Michael Dairyko, Lara Pudwell, Samantha Tyner, and Casey Wynn, Non-contiguous pattern avoidance in binary trees, Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227.
Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
Paul Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641 [math.CO], 2011.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.
Stefan Felsner and Daniel 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.
Matthew Hyatt and Jeffrey Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 24 2012
Aleksandar Ilic and Andreja Ilic, On the number of restricted Dyck paths, Filomat 25:3 (2011), 191-201; DOI: 10.2298/FIL1103191I.
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=5, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
Erkko Lehtonen and Tamás Waldhauser, Associative spectra of graph algebras I. Foundations, undirected graphs, antiassociative graphs, arXiv:2011.07621 [math.CO], 2020. See also J. of Algebraic Combinatorics (2021) Vol. 53, 613-638.
Toufik Mansour and Mark Shattuck, Some enumerative results related to ascent sequences, arXiv preprint arXiv:1207.3755 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 22 2012
Sophie Morier-Genoud, Valentin Ovsienko, and Serge Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, arXiv:1008.3359 [math.AG], 2010-2011. - From _N. J. A. Sloane_, Dec 26 2012
Sophie Morier-Genoud, Valentin Ovsienko, and Serge Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Annales de l'institut Fourier, 62 no. 3 (2012), 937-987. - From _N. J. A. Sloane_, Dec 26 2012
David Nečas and Ivan Ohlídal, Consolidated series for efficient calculation of the reflection and transmission in rough multilayers, Optics Express, Vol. 22, 2014, No. 4; DOI:10.1364/OE.22.004499.
László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
Lara Pudwell, Pattern avoidance in trees, slides from a talk, mentions many sequences, 2012. - From _N. J. A. Sloane_, Jan 03 2013
Lara Pudwell and Andrew Baxter, Ascent sequences avoiding pairs of patterns, 2014.
Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Cf. A000007, A000012, A001519, A007051, A011782, A024175, A080937, A080938.
Cf. A033191 which essentially provide the same sequence for different limits and tend to A000108.
Cf. A005021, A094790, A094789.
Cf. A211216, A005021, A160389, A116425, A332602.

I:=[1,1,2]; [n le 3 select I[n] else 5*Self(n-1)-6*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jan 09 2016

a:= n-> (<<0|1|0>, <0|0|1>, <1|-6|5>>^n. <<1, 1, 2>>)[1, 1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 09 2012

nn=56;Select[CoefficientList[Series[(1-4x^2+3x^4)/(1-5x^2+6x^4-x^6), {x,0,nn}], x],#>0 &] (* Geoffrey Critzer, Jan 26 2014 *)
LinearRecurrence[{5,-6,1},{1,1,2},30] (* Jean-François Alcover, Jan 09 2016 *)

a=vector(99); a[1]=1; a[2]=2;a[3]=5; for(n=4,#a,a[n]=5*a[n-1]-6*a[n-2] +a[n-3]); a \\ Charles R Greathouse IV, Jun 10 2011

{a(n) = if( n<0, n = -n; polcoeff( (1 - 3*x + x^2) / (1 - 6*x + 5*x^2 - x^3) + x * O(x^n), n), polcoeff( (1 - 4*x + 3*x^2) / (1 - 5*x + 6*x^2 - x^3) + x * O(x^n), n))} /* Michael Somos, May 04 2012 */

a(n) = A080934(n,5).
G.f.: (1-4*x+3*x^2)/(1-5*x+6*x^2-x^3). - Ralf Stephan, May 13 2003
a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3). - Herbert Kociemba, Jun 11 2004
a(n) = A096976(2*n). - Floor van Lamoen, Nov 02 2005
a(n) = (4/7-4/7*cos(1/7*Pi)^2)*(4*(cos(Pi/7))^2)^n + (1/7-2/7*cos(1/7*Pi) + 4/7*cos(1/7*Pi)^2)*(4*(cos(2*Pi/7))^2)^n + (2/7+2/7*cos(1/7*Pi))*(4*(cos(3*Pi/7))^2)^n for n>=0. - Richard Choulet, Apr 19 2010
G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x / (1 - x))))). - Michael Somos, May 04 2012
a(-n) = A038213(n). a(n + 2) * a(n) - a(n + 1)^2 = a(1 - n). Convolution inverse is A123183 with A123183(0)=1. - Michael Somos, May 04 2012
From Wolfdieter Lang, Mar 30 2020: (Start)
In terms of the algebraic number r = rho(7) = A160389 of degree 3 the formula given by Richard Choulet becomes a(n) = (1/7)*(r)^(2*n)*(C1(r) + C2(r)*(r - 2/r)^(2*n) + C3(r)*(r^2 - 3)^(2*n)), with C1(r) = 4 - r^2, C2(r) = 1 - r + r^2, and C3 = 2 + r.
a(n) = ((M_3)^n)[1,1] = 2*b(n-2) - 5*b(n-3) + b(n-4), for n >= 0, with the 3 X 3 tridiagonal matrix M_3 = Matrix([1,1,0], [1,2,1], [0,1,2]) from A332602, and b(n) = A005021(n) (with offset n >= -4). (End)

A214992 Power ceiling-floor sequence of (golden ratio)^4.

Original entry on oeis.org

7, 47, 323, 2213, 15169, 103969, 712615, 4884335, 33477731, 229459781, 1572740737, 10779725377, 73885336903, 506417632943, 3471038093699, 23790849022949, 163064905066945, 1117663486445665, 7660579500052711

Offset: 0

Clark Kimberling, Nov 08 2012, Jan 24 2013

Let f = floor and c = ceiling.  For x > 1, define four sequences as functions of x, as follows:
p1(0) = f(x), p1(n) = f(x*p1(n-1));
p2(0) = f(x), p2(n) = c(x*p2(n-1)) if n is odd and p2(n) = f(x*p1(n-1)) if n is even;
p3(0) = c(x), p3(n) = f(x*p3(n-1)) if n is odd and p3(n) = c(x*p3(n-1)) if n is even;
p4(0) = c(x), p4(n) = c(x*p4(n-1)).
The present sequence is given by a(n) = p3(n).
Following the terminology at A214986, call the four sequences power floor, power floor-ceiling, power ceiling-floor, and power ceiling sequences.  In the table below, a sequence is identified with an A-numbered sequence if they appear to agree except possibly for initial terms.  Notation: S(t)=sqrt(t), r = (1+S(5))/2 = golden ratio, and Limit = limit of p3(n)/p2(n).
x ......p1..... p2..... p3..... p4.......Limit
r.......A000045 A000045 A000045 A000045..r
r^2.....A001519 A001654 A061646 A001906..-1+S(5)
r^3.....A024551 A001076 A015448 A049652..-1+S(5)
r^4.....A049685 A157335 A214992 A004187..-19+9*S(5)
r^5.....A214993 A049666 A015457 A214994...(-9+5*S(5))/2
r^6.....A007805 A156085 A214995 A049660..-151+68*S(5)
1+S(2)..A024537 A000129 A001333 A048739..S(2)
2+S(2)..A007052 A214996 A214997 A007070..(1+S(2))/2
1+S(3)..A057960 A002605 A028859 A077846..(1+S(3))/2
2+S(3)..A001835 A109437 A214998 A001353..-4+3*S(3)
S(5)....A214999 A215091 A218982 A218983..1.26879683...
2+S(5)..A024551 A001076 A015448 A049652..-1+S(5)
2+S(6)..A218984 A090017 A123347 A218985..S(3/2)
2+S(7)..A218986 A015530 A126473 A218987..(1+S(7))/3
2+S(8)..A218988 A057087 A086347 A218989..(1+S(2))/2
3+S(8)..A001653 A084158 A218990 A001109..-13+10*S(2)
3+S(10).A218991 A005668 A015451 A218992..-2+S(10)
...
Properties of p1, p2, p3, p4:
(1) If x > 2, the terms of p2 and p3 interlace: p2(0) < p3(0) < p2(1) < p3(1) < p2(2) < p3(2)... Also, p1(n) <= p2(n) <= p3(n) <= p4(n) <= p1(n+1) for all x>0 and n>=0.
(2) If x > 2, the limits L(x) = limit(p/x^n) exist for the four functions p(x), and L1(x) <= L2(x) <= L3(x) <= L4 (x). See the Mathematica programs for plots of the four functions; one of them also occurs in the Odlyzko and Wilf article, along with a discussion of the special case x = 3/2.
(3) Suppose that x = u + sqrt(v) where v is a nonsquare positive integer.  If u = f(x) or u = c(x), then p1, p2, p3, p4 are linear recurrence sequences.  Is this true for sequences p1, p2, p3, p4 obtained from x = (u + sqrt(v))^q for every positive integer q?
(4) Suppose that x is a Pisot-Vijayaraghavan number. Must p1, p2, p3, p4 then be linearly recurrent?  If x is also a quadratic irrational b + c*sqrt(d), must the four limits L(x) be in the field Q(sqrt(d))?
(5) The Odlyzko and Wilf article (page 239) raises three interesting questions about the power ceiling function; it appears that they remain open.

			a(0) = ceiling(r) = 7, where r = ((1+sqrt(5))/2)^4 = 6.8...; a(1) = floor(7*r) = 47; a(2) = ceiling(47) = 323.

Clark Kimberling, Table of n, a(n) for n = 0..250
A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem, Glasgow Math. J. 33, 235-240, 1991.
Index entries for linear recurrences with constant coefficients, signature (6,6,-1).
Index entries for sequences related to the Josephus Problem.

Cf. A001622, A214986, A049685, A157335, A004187.

(* Program 1.  A214992 and related sequences *)
x = GoldenRatio^4; z = 30; (* z = # terms in sequences *)
z1 = 100; (* z1 = # digits in approximations *)
f[x_] := Floor[x]; c[x_] := Ceiling[x];
p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x];
p1[n_] := f[x*p1[n - 1]]
p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]]
p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]]
p4[n_] := c[x*p4[n - 1]]
Table[p1[n], {n, 0, z}]  (* A049685 *)
Table[p2[n], {n, 0, z}]  (* A157335 *)
Table[p3[n], {n, 0, z}]  (* A214992 *)
Table[p4[n], {n, 0, z}]  (* A004187 *)
Table[p4[n] - p1[n], {n, 0, z}]  (* A004187 *)
Table[p3[n] - p2[n], {n, 0, z}]  (* A098305 *)
(* Program 2.  Plot of power floor and power ceiling functions, p1(x) and p4(x) *)
f[x_] := f[x] = Floor[x]; c[x_] := c[x] = Ceiling[x];
p1[x_, 0] := f[x]; p1[x_, n_] := f[x*p1[x, n - 1]];
p4[x_, 0] := c[x]; p4[x_, n_] := c[x*p4[x, n - 1]];
Plot[Evaluate[{p1[x, 10]/x^10, p4[x, 10]/x^10}], {x, 2, 3}, PlotRange -> {0, 4}]
(* Program 3. Plot of power floor-ceiling and power ceiling-floor functions, p2(x) and p3(x) *)
f[x_] := f[x] = Floor[x]; c[x_] := c[x] = Ceiling[x];
p2[x_, 0] := f[x]; p3[x_, 0] := c[x];
p2[x_, n_] := If[Mod[n, 2] == 1, c[x*p2[x, n - 1]], f[x*p2[x, n - 1]]]
p3[x_, n_] := If[Mod[n, 2] == 1, f[x*p3[x, n - 1]], c[x*p3[x, n - 1]]]
Plot[Evaluate[{p2[x, 10]/x^10, p3[x, 10]/x^10}], {x, 2, 3}, PlotRange -> {0, 4}]

a(n) = floor(r*a(n-1)) if n is odd and a(n) = ceiling(r*a(n-1)) if n is even, where a(0) = ceiling(r), r = (golden ratio)^4 = (7 + sqrt(5))/2.
a(n) = 6*a(n-1) + 6*a(n-2) - a(n-3).
G.f.: (7 + 5*x - x^2)/((1 + x)*(1 - 7*x + x^2)).
a(n) = (10*(-2)^n+(10+3*sqrt(5))*(7-3*sqrt(5))^(n+2)+(10-3*sqrt(5))*(7+3*sqrt(5))^(n+2))/(90*2^n). - Bruno Berselli, Nov 14 2012
a(n) = 7*A157335(n) + 5*A157335(n-1) - A157335(n-2). - R. J. Mathar, Feb 05 2020
E.g.f.: exp(-x)*(5 + 2*exp(9*x/2)*(155*cosh(3*sqrt(5)*x/2) + 69*sqrt(5)*sinh(3*sqrt(5)*x/2)))/45. - Stefano Spezia, Oct 28 2024

A326210 Number of labeled simple graphs with vertices {1..n} containing a nesting pair of edges, where two edges {a,b}, {c,d} are nesting if a < c and b > d or a > c and b < d.

Original entry on oeis.org

0, 0, 0, 0, 16, 672, 29888, 2071936, 268204288, 68717285888, 35184350796800, 36028796807919616, 73786976292712960000, 302231454903635611721728, 2475880078570760326175178752, 40564819207303340845566684397568, 1329227995784915872903782635437883392

Offset: 0

Gus Wiseman, Jun 19 2019

nonn

Also simple graphs containing a crossing pair of edges, where two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b.

Also simple graphs such that, if the edges are listed in lexicographic order, their maxima (seconds) are not weakly increasing.

			The a(4) = 16 nesting edge-sets:
  {14,23}
  {12,14,23}
  {13,14,23}
  {14,23,24}
  {14,23,34}
  {12,13,14,23}
  {12,14,23,24}
  {12,14,23,34}
  {13,14,23,24}
  {13,14,23,34}
  {14,23,24,34}
  {12,13,14,23,24}
  {12,13,14,23,34}
  {12,14,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}
The a(4) = 16 crossing edge-sets:
  {13,24}
  {12,13,24}
  {13,14,24}
  {13,23,24}
  {13,24,34}
  {12,13,14,24}
  {12,13,23,24}
  {12,13,24,34}
  {13,14,23,24}
  {13,14,24,34}
  {13,23,24,34}
  {12,13,14,23,24}
  {12,13,14,24,34}
  {12,13,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Non-nesting graphs are A054726.
Nesting digraphs are A326209.
Nesting (or crossing) set partitions are A016098.
MM-numbers of nesting multiset partitions are A326256.
Cf. A001519, A006125, A095661, A324170, A324172, A324326.
Cf. A326211, A326243, A326244, A326248, A326250.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!OrderedQ[Last/@#]&]],{n,0,5}]

seq(n)={my(p=1 + 3/2*x - x^2 - x/2*sqrt(1 - 12*x + 4*x^2 + O(x^n))); concat([0], vector(n, k, 2^binomial(k,2)-polcoef(p,k)))} \\ Andrew Howroyd, Aug 26 2019

A006125(n) = a(n) + A054726(n).

Terms a(7) and beyond from Andrew Howroyd, Aug 26 2019

A326243 Number of capturing set partitions of {1..n}.

Original entry on oeis.org

0, 0, 0, 0, 1, 11, 80, 503, 2993, 17609, 105017, 644528, 4107600, 27313805, 189866541, 1379728831, 10470032837, 82833202559, 681977545967, 5832430910181, 51723181525978, 474866750479993, 4506706112772881, 44151975623559477, 445958774322599940, 4638590033810841345

Offset: 0

Gus Wiseman, Jun 19 2019

nonn

A set partition is capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t. This is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.

			The a(5) = 11 capturing set partitions:
  {{1,2,5},{3,4}}
  {{1,3,4},{2,5}}
  {{1,3,5},{2,4}}
  {{1,4},{2,3,5}}
  {{1,4,5},{2,3}}
  {{1,5},{2,3,4}}
  {{1},{2,5},{3,4}}
  {{1,4},{2,3},{5}}
  {{1,5},{2},{3,4}}
  {{1,5},{2,3},{4}}
  {{1,5},{2,4},{3}}

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.

Non-capturing set partitions are A054391.
Crossing and nesting set partitions are (both) A016098.
Cf. A000108, A000110, A001519, A054391, A058681, A122880, A324170.
Cf. A326209, A326211, A326237, A326246, A326249, A326255, A326259.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
capXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;xt||x>z&&y

a(n) = A000110(n) - A054391(n).

a(12) and beyond from Christian Sievers, Aug 23 2024

A001870 Expansion of (1-x)/(1 - 3*x + x^2)^2.

Original entry on oeis.org

1, 5, 19, 65, 210, 654, 1985, 5911, 17345, 50305, 144516, 411900, 1166209, 3283145, 9197455, 25655489, 71293590, 197452746, 545222465, 1501460635, 4124739581, 11306252545, 30928921224, 84451726200, 230204999425

Offset: 0

N. J. A. Sloane

a(n) = ((n+1)*F(2*n+3)+(2*n+3)*F(2*(n+1)))/5 with F(n)=A000045(n) (Fibonacci numbers). One half of odd-indexed A001629(n), n >= 2 (Fibonacci convolution).
Convolution of F(2n+1) (A001519) and F(2n+2) (A001906(n+1)). - Graeme McRae, Jun 07 2006
Number of reentrant corners along the lower contours of all directed column-convex polyominoes of area n+3 (a reentrant corner along the lower contour is a vertical step that is followed by a horizontal step). a(n) = Sum_{k=0..ceiling((n+1)/2)} k*A121466(n+3,k). - Emeric Deutsch, Aug 02 2006
From Wolfdieter Lang, Jan 02 2012: (Start)
a(n) = A024458(2*n), n >= 1 (bisection, even arguments).
a(n) is also the odd part of the bisection of the half-convolution of the sequence A000045(n+1), n >= 0, with itself. See a comment on A201204 for the definition of the half-convolution of a sequence with itself. There one also finds the rule for the o.g.f. which in this case is Chato(x)/2 with the o.g.f. Chato(x) = 2*(1-x)/(1-3*x+x^2)^2 of A001629(2*n+3), n >= 0.
(End)

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Elena Barcucci, Renzo Pinzani, and Renzo Sprugnoli , Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Descent distribution on Catalan words avoiding a pattern of length at most three, arXiv:1803.06706 [math.CO], 2018.
Éva Czabarka, Rigoberto Flórez, and Leandro Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
Éva Czabarka, Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math. 341 (10) (2018), 2789-2807. See Cor. 6.
Emeric Deutsch and Helmut Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 5-6, 14-15, 17, 19.
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.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 15.
Pieter Moree, Convoluted Convolved Fibonacci Numbers, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.
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
John Riordan, Notes to N. J. A. Sloane, Jul. 1968
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

a(n) = A060921(n+1, 1)/2.
Partial sums of A030267. First differences of A001871.
Cf. A121466.
Cf. A023610.

F:=Fibonacci;; List([0..30], n-> ((n+1)*F(2*n+3)+(2*n+3)*F(2*(n+1)))/5); # G. C. Greubel, Jul 15 2019

a001870 n = a001870_list !! n
a001870_list = uncurry c $ splitAt 1 $ tail a000045_list where
   c us vs'@(v:vs) = (sum $ zipWith (*) us vs') : c (v:us) vs
-- Reinhard Zumkeller, Oct 31 2013

I:=[1, 5, 19, 65]; [n le 4 select I[n] else 6*Self(n-1) -11*Self(n-2)+6*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 10 2012

A001870:=-(-1+z)/(z**2-3*z+1)**2; # Simon Plouffe in his 1992 dissertation.

CoefficientList[Series[(1-x)/(1-3*x+x^2)^2,{x,0,40}],x] (* Vincenzo Librandi, Jun 10 2012 *)
LinearRecurrence[{6,-11,6,-1},{1,5,19,65},30] (* Harvey P. Dale, Aug 17 2013 *)
With[{F=Fibonacci}, Table[((n+1)*F[2*n+3]+(2*n+3)*F[2*n+2])/5, {n,0,30}]] (* G. C. Greubel, Jul 15 2019 *)

Vec((1-x)/(1-3*x+x^2)^2+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

f=fibonacci; [((n+1)*f(2*n+3)+(2*n+3)*f(2*n+2))/5 for n in (0..30)] # G. C. Greubel, Jul 15 2019

a(n) = Sum_{k=1..n+1} k*binomial(n+k+1, 2k). - Emeric Deutsch, Jun 11 2003
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 10 2012
a(n) = (A238846(n) + A001871(n))/2. - Philippe Deléham, Mar 06 2014
a(n) = ((2*n-1)*Fibonacci(2*n) - n*Fibonacci(2*n-1))/5 [Czabarka et al.]. - N. J. A. Sloane, Sep 18 2018
E.g.f.: exp(3*x/2)*(5*(5 + 11*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(13 + 25*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025

More terms from Christian G. Bower

cp's OEIS Frontend

A059929 a(n) = Fibonacci(n)*Fibonacci(n+2).

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A055588 a(n) = 3*a(n-1) - a(n-2) - 1 with a(0) = 1 and a(1) = 2.

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A124302 Number of set partitions with at most 3 blocks; number of Dyck paths of height at most 4; dimension of space of symmetric polynomials in 3 noncommuting variables.

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A075737 Prime Fibonacci numbers with prime indices.

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A166444 a(0) = 0, a(1) = 1 and for n > 1, a(n) = sum of all previous terms.

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A080937 Number of Catalan paths (nonnegative, starting and ending at 0, step +/-1) of 2*n steps with all values <= 5.

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