A108617 -id:A108617 - 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

A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3a(n-1) - a(n-2) - 2a(n-3).

Original entry on oeis.org

0, 1, 2, 5, 11, 24, 51, 107, 222, 457, 935, 1904, 3863, 7815, 15774, 31781, 63939, 128488, 257963, 517523, 1037630, 2079441, 4165647, 8342240, 16702191, 33433039, 66912446, 133899917, 267921227, 536038872, 1072395555, 2145305339

Offset: 0

Clark Kimberling

Number of compositions of n with at least one even part (offset 2). - Vladeta Jovovic, Dec 29 2004
First differences of A008466. a(n) = A008466(n+2) - A008466(n+1). - Alexander Adamchuk, Apr 06 2006
Starting with "1" = eigensequence of a triangle with the Fibonacci series as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
An elephant sequence, see A175654. For the corner squares 24 A[5] vectors, with decimal values between 11 and 416, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A099036 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
a(n) = Sum_{k=1..n} A108617(n,k) / 2. - Reinhard Zumkeller, Oct 07 2012
a(n) is the number of binary strings that contain the substring 11 or end in 1. a(3) = 5 because we have: 001, 011, 101, 110, 111. - Geoffrey Critzer, Jan 04 2014
a(n-1), n >= 1, is the number of nonexisting (due to the maturation delay) "[male-female] pairs of Fibonacci rabbits" at the beginning of the n-th month. - Daniel Forgues, May 06 2015
a(n-1) is the number of subsets of {1,2,..,n} that contain n that have at least one pair of consecutive integers. For example, for n=5, a(4) = 11 and the 11 subsets are {4,5}, {1,2,5}, {1,4,5}, {2,3,5}, {2,4,5}, {3,4,5}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, {1,2,3,4,5}.  Note that A008466(n) is the number of all subsets of {1,2,..,n} that have at least one pair of consecutive integers. - Enrique Navarrete, Aug 15 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Patrick Letendre, Polynomials with integer roots, arXiv:1911.00480 [math.NT], 2019. See p. 4.
Toufik Mansour and Mark Shattuck, Pattern avoidance in flattened derangements, Disc. Math. Lett. (2025) Vol. 15, 67-74. See p. 74.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
OEIS Wiki, Fibonacci rabbits
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Row sums of triangle A131767. - Gary W. Adamson, Jul 13 2007
a(n) = A101220(1, 2, n+1).
T(n, n) + T(n, n+1) + ... + T(n, 2n), T given by A027926.
Diagonal sums of A055248.
Cf. A000045, A000079, A008466, A059570, A099036, A047967, A000032.

List([0..35], n-> 2^n - Fibonacci(n+1) ); # G. C. Greubel, Sep 27 2019

a027934 n = a027934_list !! n
a027934_list = 0 : 1 : 2 : zipWith3 (\x y z -> 3 * x - y - 2 * z)
               (drop 2 a027934_list) (tail a027934_list) a027934_list
-- Reinhard Zumkeller, Oct 07 2012

[2^n - Fibonacci(n+1): n in [0..35]]; // G. C. Greubel, Sep 27 2019

A027934:= proc(n) local K; K:= Matrix ([[2,0,0], [0,1,1], [0,1,0]])^n; K[1,1]-K[2,2] end: seq (A027934(n), n=0..31); # Alois P. Heinz, Jul 28 2008
a := n -> 2^n - combinat:-fibonacci(n+1): seq(a(n),n=0..31); # Peter Luschny, May 09 2015

nn=31; a:=1/(1-x-x^2); b:=1/(1-2x); CoefficientList[Series[a*x*(1+x*b), {x,0,nn}], x] (* Geoffrey Critzer, Jan 04 2014 *)
LinearRecurrence[{3,-1,-2}, {0,1,2}, 32] (* Jean-François Alcover, Jan 09 2016 *)
nxt[{a_,b_,c_}]:={b,c,3c-b-2a}; NestList[nxt,{0,1,2},40][[;;,1]] (* Harvey P. Dale, Feb 02 2025 *)

a(n)=2^n-fibonacci(n+1) \\ Charles R Greathouse IV, Jun 11 2015

[2^n - fibonacci(n+1) for n in (0..35)] # G. C. Greubel, Sep 27 2019

a(n) = Sum_{j=0..floor(n/2)} Sum_{k=0..n-2*j} binomial(n-j, n-2*j-k). - Paul Barry, Feb 07 2003
From Paul Barry, Jan 23 2004: (Start)
Row sums of A105809.
G.f.: x*(1-x)/((1-2*x)*(1-x-x^2)).
a(n) = 2^n - Fibonacci(n+1). (End) - corrected Apr 06 2006 and Oct 05 2012
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n-k, k+j). - Paul Barry, Aug 29 2004
a(n) = (Sum of (n+1)-th row of the triangle in A108617) / 2. - Reinhard Zumkeller, Jun 12 2005
a(n) = term (1,1) - term (2,2) in the 3 X 3 matrix [2,0,0; 0,1,1; 0,1,0]^n. - Alois P. Heinz, Jul 28 2008
a(n) = 2^n - A000045(n+1). - Geoffrey Critzer, Jan 04 2014
a(n) ~ 2^n. - Daniel Forgues, May 06 2015
From Bob Selcoe, Mar 29 2016: (Start)
a(n) = 2*a(n-1) + A000045(n-2).
a(n) = 4*a(n-2) + A000032(n-2). (End)
a(n) = 2^(n-1) - ( ((1+sqrt(5))/2)^n - ((1-sqrt(5))/2)^n)/sqrt(5). - Haider Ali Abdel-Abbas, Aug 17 2019

Simpler definition from Miklos Kristof, Nov 24 2003
Initial zero added by N. J. A. Sloane, Feb 13 2008
Definition fixed by Reinhard Zumkeller, Oct 07 2012

A074829 Triangle formed by Pascal's rule, except that the n-th row begins and ends with the n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 7, 8, 7, 5, 8, 12, 15, 15, 12, 8, 13, 20, 27, 30, 27, 20, 13, 21, 33, 47, 57, 57, 47, 33, 21, 34, 54, 80, 104, 114, 104, 80, 54, 34, 55, 88, 134, 184, 218, 218, 184, 134, 88, 55, 89, 143, 222, 318, 402, 436, 402, 318, 222, 143, 89

Offset: 1

Joseph L. Pe, Sep 30 2002

			The first and second Fibonacci numbers are 1, 1, so the first and second rows of the triangle are 1; 1 1; respectively. The third row of the triangle begins and ends with the third Fibonacci number, 2 and the middle term is the sum of the contiguous two terms in the second row, i.e., 1 + 1 = 2, so the third row is 2 2 2.
Triangle begins:
   1;
   1,  1;
   2,  2,  2;
   3,  4,  4,   3;
   5,  7,  8,   7,   5;
   8, 12, 15,  15,  12,   8;
  13, 20, 27,  30,  27,  20, 13;
  21, 33, 47,  57,  57,  47, 33, 21;
  34, 54, 80, 104, 114, 104, 80, 54, 34;
  ...
Formatted as a symmetric triangle:
                           1;
                        1,    1;
                     2,    2,    2;
                  3,    4,    4,    3;
               5,    7,    8,    7,    5;
            8,   12,   15,   15,   12,    8;
        13,   20,   27,   30,   27,   20,   13;
     21,   33,   47,   57,   57,   47,   33,   21;
  34,   54,   80,  104,  114,  104,   80,   54,   34;

Reinhard Zumkeller, Rows n = 1..120 of table, flattened
Hebert Pérez-Rosés, Asymptotic Analysis of Central Binomiacci Numbers, arXiv:2503.17462 [math.CO], 2025.
Index entries for triangles and arrays related to Pascal's triangle

Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A105809, A108617, A109906, A111006, A114197, A162741, A228074.

Cf. A074878 (row sums).

T:= function(n,k)
    if k=1 then return Fibonacci(n);
    elif k=n then return Fibonacci(n);
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([1..15], n-> List([1..n], k-> T(n,k) ))); # G. C. Greubel, Jul 12 2019

a074829 n k = a074829_tabl !! (n-1) !! (k-1)
a074829_row n = a074829_tabl !! (n-1)
a074829_tabl = map fst $ iterate
   (\(u:_, vs) -> (vs, zipWith (+) ([u] ++ vs) (vs ++ [u]))) ([1], [1,1])
-- Reinhard Zumkeller, Aug 15 2013

A074829 := proc(n,k)
    option remember ;
    if k=1 or k=n then
        combinat[fibonacci](n) ;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc:
seq(seq(A074829(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Mar 31 2025

T[n_, 1]:= Fibonacci[n]; T[n_, n_]:= Fibonacci[n]; T[n_, k_]:= T[n-1, k-1] + T[n-1, k]; Table[T[n, k], {n, 1, 12}, {k, 1, n}]//Flatten (* G. C. Greubel, Jul 12 2019 *)

T(n,k) = if(k==1 || k==n, fibonacci(n), T(n-1,k-1) + T(n-1,k));
for(n=1,12, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 12 2019

def T(n, k):
    if (k==1 or k==n): return fibonacci(n)
    else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 12 2019

More terms from Philippe Deléham, Sep 20 2006

Data error in 7th row fixed by Reinhard Zumkeller, Aug 15 2013

A227550 A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 6, 4, 4, 6, 24, 10, 8, 10, 24, 120, 34, 18, 18, 34, 120, 720, 154, 52, 36, 52, 154, 720, 5040, 874, 206, 88, 88, 206, 874, 5040, 40320, 5914, 1080, 294, 176, 294, 1080, 5914, 40320, 362880, 46234, 6994, 1374, 470, 470, 1374, 6994, 46234, 362880, 3628800

Offset: 0

Vincenzo Librandi, Aug 04 2013

A003422 gives the second column (after 0).

			Triangle begins:
       1;
       1,     1;
       2,     2,    2;
       6,     4,    4,    6;
      24,    10,    8,   10,  24;
     120,    34,   18,   18,  34, 120;
     720,   154,   52,   36,  52, 154,  720;
    5040,   874,  206,   88,  88, 206,  874, 5040;
   40320,  5914, 1080,  294, 176, 294, 1080, 5914, 40320;
  362880, 46234, 6994, 1374, 470, 470, 1374, 6994, 46234, 362880;

Vincenzo Librandi, Rows n = 0..70, flattened

Cf. similar triangles with t on the borders: A007318 (t = 1), A028326 (t = 2), A051599 (t = prime(n)), A051601 (t = n), A051666 (t = n^2), A108617 (t = fibonacci(n)), A134636 (t = 2n+1), A137688 (t = 2^n), A227075 (t = 3^n).
Cf. A003422.
Cf. A227791 (central terms), A001563, A074911.

a227550 n k = a227550_tabl !! n !! k
a227550_row n = a227550_tabl !! n
a227550_tabl = map fst $ iterate
   (\(vs, w:ws) -> (zipWith (+) ([w] ++ vs) (vs ++ [w]), ws))
   ([1], a001563_list)
-- Reinhard Zumkeller, Aug 05 2013

function T(n,k)
  if k eq 0 or k eq n then return Factorial(n);
  else return T(n-1,k-1) + T(n-1,k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 02 2021

t = {}; Do[r = {}; Do[If[k == 0||k == n, m = n!, m = t[[n, k]] + t[[n, k + 1]]]; r = AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t = Flatten[t]

def T(n,k): return factorial(n) if (k==0 or k==n) else T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 02 2021

From G. C. Greubel, May 02 2021: (Start)
T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = T(n, n) = n!.
Sum_{k=0..n} T(n, k) = 2^n * (1 +Sum_{j=1..n-1} j*j!/2^j) = A140710(n). (End)

A167821 a(n) is the number of n-tosses having a run of 3 or more heads or a run of 3 or more tails for a fair coin (i.e., probability is a(n)/2^n).

Original entry on oeis.org

0, 0, 2, 6, 16, 38, 86, 188, 402, 846, 1760, 3630, 7438, 15164, 30794, 62342, 125904, 253782, 510758, 1026684, 2061730, 4136990, 8295872, 16627166, 33311646, 66716028, 133582106, 267406998, 535206832, 1071049286, 2143127030, 4287918140, 8578528818

Offset: 1

V.J. Pohjola, Nov 13 2009

A167821(n) is the difference between A000918(n), the number of branches of a complete binary tree of n levels, and the number of recursive calls needed to compute the (n+1)-th Fibonacci number F(n+1) as defined in A019274: A167821(n) = A000918(n) - A019274(n+1). - Denis Lorrain, Jan 14 2012

Partial sums of A027934 multiplied term by term by 2 (as shown by the second formula), i.e., partial sums of row sums of A108617. - J. M. Bergot, Oct 02 2012, clarified by R. J. Mathar, Oct 05 2012

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3, -1, -2).

Cf. A008466, A050231.

[2^n-2*Fibonacci(n+1): n in [1..40]]; // Vincenzo Librandi, Jun 28 2016

CoefficientList[Series[(2 x^2)/(1 - 3 x + x^2 + 2 x^3), {x, 0, 30}], x]
Table[2^n - 2*Fibonacci[n + 1], {n, 1, 31}]
LinearRecurrence[{3, -1, -2}, {0, 0, 2}, 50] (* G. C. Greubel, Jun 27 2016 *)

G.f.: (2 x^2)/(1 - 3 x + x^2 + 2 x^3);
a(n) = 2^n - 2*Fibonacci(n+1).
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3). - G. C. Greubel, Jun 27 2016

A316938 Triangle read by rows formed using Pascal's rule except that n-th row begins and ends with Fibonacci(n+4).

Original entry on oeis.org

3, 5, 5, 8, 10, 8, 13, 18, 18, 13, 21, 31, 36, 31, 21, 34, 52, 67, 67, 52, 34, 55, 86, 119, 134, 119, 86, 55, 89, 141, 205, 253, 253, 205, 141, 89, 144, 230, 346, 458, 506, 458, 346, 230, 144, 233, 374, 576, 804, 964, 964, 804, 576, 374, 233, 377, 607, 950, 1380, 1768, 1928, 1768, 1380, 950, 607, 377

Offset: 0

Vincenzo Librandi, Jul 27 2018

Row sums give A316937.

			Triangle begins:
    3;
    5,   5;
    8,  10,   8;
   13,  18,  18,  13;
   21,  31,  36,  31,  21;
   34,  52,  67,  67,  52,  34;
   55,  86, 119, 134, 119,  86,  55;
   89, 141, 205, 253, 253, 205, 141,  89;
  144, 230, 346, 458, 506, 458, 346, 230, 144;
...

Cf. A000045 (Fibonacci numbers), A316937 (row sums).

Other Fibonacci borders: A074829, A108617, A316939.

t={}; Do[r={}; Do[If[k==0||k==n, m=Fibonacci[n+4], m=t[[n,k]]+t[[n,k+1]]]; r=AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t

A316939 Triangle read by rows formed using Pascal's rule except that n-th row begins and ends with Fibonacci(n+2).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 7, 7, 5, 8, 12, 14, 12, 8, 13, 20, 26, 26, 20, 13, 21, 33, 46, 52, 46, 33, 21, 34, 54, 79, 98, 98, 79, 54, 34, 55, 88, 133, 177, 196, 177, 133, 88, 55, 89, 143, 221, 310, 373, 373, 310, 221, 143, 89, 144, 232, 364, 531, 683, 746, 683, 531, 364, 232, 144, 233, 376, 596, 895, 1214, 1429

Offset: 0

Vincenzo Librandi, Jul 28 2018

			Triangle begins:
   1;
   2,  2;
   3,  4,   3;
   5,  7,   7,   5;
   8, 12,  14,  12,   8;
  13, 20,  26,  26,  20,  13;
  21, 33,  46,  52,  46,  33,  21;
  34, 54,  79,  98,  98,  79,  54, 34;
  55, 88, 133, 177, 196, 177, 133, 88, 55;
  ...

Cf. A316528 (row sums).
Columns k=0..2: A000045, A000071, A001924.
Other Fibonacci borders: A074829, A108617, A316938.

f:= proc(n,k) option remember;
  if k=0 or k=n then combinat:-fibonacci(n+2) else procname(n-1,k)+procname(n-1,k-1) fi
end proc:
for n from 0 to 10 do
  seq(f(n,k),k=0..n)
od; # Robert Israel, Sep 20 2018

t={}; Do[r={}; Do[If[k==0||k==n, m=Fibonacci[n + 2], m=t[[n, k]] + t[[n, k + 1]]]; r=AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t // Flatten

Incorrect g.f. removed by Georg Fischer, Feb 18 2020

A176085 a(n) = A136431(n,n).

Original entry on oeis.org

0, 1, 3, 11, 41, 155, 591, 2267, 8735, 33775, 130965, 509015, 1982269, 7732659, 30208749, 118167055, 462760369, 1814091011, 7118044023, 27952660883, 109853552255, 432021606103, 1700093447847, 6694137523051, 26372544576331, 103950885100775, 409928481296331

Offset: 0

Paul Curtz, Apr 08 2010

a(n+1) is also the number of sequences of length 2n obeying the regular expression "0^* (1 or 2)^* 3^*" and having sum 3n. For example, a(3)=11 because of the sequences 0033, 0123, 0213, 0222, 1113, 1122, 1212, 1221, 2112, 2121, 2211. - Don Knuth, May 11 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A000045, A054441, A108617.

List([0..30], n-> Sum([0..n], k-> Binomial(2*n-k-1, n-k)*Fibonacci(k) )); # G. C. Greubel, Nov 28 2019

[(&+[Binomial(2*n-k-1, n-k)*Fibonacci(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 28 2019

with(combinat); seq( add(binomial(2*n-k-1, n-k)*fibonacci(k), k=0..n), n=0..30); # G. C. Greubel, Nov 28 2019
1/(sqrt(1 - 4*x) + 1/x - 4): series(%, x, 27):
seq(coeff(%, x, k), k=0..26); # Peter Luschny, May 29 2021

t[n_, k_]:= CoefficientList[ Series[x/(1-x-x^2)/(1-x)^k, {x,0,k}], x][[k+1]]; Array[ t[#, #] &, 20]
Table[Sum[Binomial[2*n-k-1, n-k]*Fibonacci[k], {k,0,n}], {n,0,30}] (* G. C. Greubel, Nov 28 2019 *)

a(n):=sum(fib(k)*binomial(2*n-k-1,n-k),k,1,n); /*  Vladimir Kruchinin, Mar 17 2016 */

a(n) = sum(k=1, n, fibonacci(k)*binomial(2*n-k-1, n-k)) \\ Michel Marcus, Mar 17 2016

[sum(binomial(2*n-k-1, n-k)*fibonacci(k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 28 2019

a(n+1) - 4*a(n) = -A081696(n-1).
From Vaclav Kotesovec, Oct 21 2012: (Start)
G.f.: x*(x-sqrt(1-4*x))/(sqrt(1-4*x)*(x^2+4*x-1)).
Recurrence: (n-2)*a(n) = 2*(4*n-9)*a(n-1) - (15*n-38)*a(n-2) - 2*(2*n-5)* a(n-3).
a(n) ~ 4^n/sqrt(Pi*n). (End)
a(n) = Sum_{k=1..n} (F(k)*binomial(2*n-k-1,n-k)), where F(k) = A000045(k). - Vladimir Kruchinin, Mar 17 2016
Simpler g.f.: x/sqrt(1-4*x)/(x+sqrt(1-4*x)). - Don Knuth, May 11 2016
a(n) = A000045(3*n) - A054441(n). - Hrishikesh Venkataraman, May 27 2021
a(n) = 4*a(n-1) + a(n-2) - binomial(2*n-4,n-2) for n>=2. - Hrishikesh Venkataraman, Jul 02 2021
a(n) = A108617(2n,n)/2. - Alois P. Heinz, Jan 26 2025

A347584 Triangle formed by Pascal's rule, except that the n-th row begins and ends with the n-th Lucas number.

Original entry on oeis.org

2, 1, 1, 3, 2, 3, 4, 5, 5, 4, 7, 9, 10, 9, 7, 11, 16, 19, 19, 16, 11, 18, 27, 35, 38, 35, 27, 18, 29, 45, 62, 73, 73, 62, 45, 29, 47, 74, 107, 135, 146, 135, 107, 74, 47, 76, 121, 181, 242, 281, 281, 242, 181, 121, 76, 123, 197, 302, 423, 523, 562, 523, 423, 302, 197, 123

Offset: 0

Noah Carey and Greg Dresden, Sep 07 2021

nonn

Similar in spirit to the Fibonacci-Pascal triangle A074829, which uses Fibonacci numbers instead of Lucas numbers at the ends of each row.

If we consider the top of the triangle to be the 0th row, then the sum of terms in n-th row is 2*(2^(n+1) - Lucas(n+1)). This sum also equals 2*A027973(n-1) for n>0.

			The first two Lucas numbers (for n=0 and n=1) are 2 and 1, so the first two rows (again, for n=0 and n=1) of the triangle are 2 and 1, 1 respectively.
Triangle begins:
               2;
             1,  1;
           3,  2,  3;
         4,  5,  5,  4;
       7,  9, 10,  9,  7;
    11, 16, 19, 19, 16, 11;
  18, 27, 35, 38, 35, 27, 18;

Index entries for triangles and arrays related to Pascal's triangle

Cf. A000032, A027973.
Cf. A227550, A228196 (general formula).
Fibonacci borders: A074829, A108617, A316938, A316939.

T[n_, 0] := LucasL[n]; T[n_, n_] := LucasL[n];
T[n_, k_] := T[n - 1, k - 1] + T[n - 1, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten

a(n) = 2*A074829(n+1) - A108617(n).

cp's OEIS Frontend

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

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A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3).

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A074829 Triangle formed by Pascal's rule, except that the n-th row begins and ends with the n-th Fibonacci number.

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A227550 A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.

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A167821 a(n) is the number of n-tosses having a run of 3 or more heads or a run of 3 or more tails for a fair coin (i.e., probability is a(n)/2^n).

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A316938 Triangle read by rows formed using Pascal's rule except that n-th row begins and ends with Fibonacci(n+4).

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A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3a(n-1) - a(n-2) - 2a(n-3).