A004798 -id:A004798 - cp's OEIS frontend

A023610 Convolution of Fibonacci numbers and {F(2), F(3), F(4), ...}.

1, 3, 7, 15, 30, 58, 109, 201, 365, 655, 1164, 2052, 3593, 6255, 10835, 18687, 32106, 54974, 93845, 159765, 271321, 459743, 777432, 1312200, 2211025, 3719643, 6248479, 10482351, 17562870, 29391490, 49132669, 82048737, 136884293, 228160495, 379975140, 632293452

Offset: 0

Clark Kimberling

a(n-2) + 1 is the number of (3412,1243)-, (3412,2134)- and (3412,1324)-avoiding involutions in S_n, n>1. - Ralf Stephan, Jul 06 2003
The number of terms in all ordered partitions of (n+1) using only ones and twos. For example, a(3)=15 because there are 15 terms in 1+1+1+1;2+1+1;1+2+1;1+1+2;2+2 - Geoffrey Critzer, Apr 07 2008
a(n) is the number of n-matchings in the graph obtained by a zig-zag triangulation of a convex (2n+1)-gon. Example: a(2)=7 because in the triangulation of the convex pentagon ABCDEA with diagonals AD and AC we have 7 2-matchings: {AB,CD},{AB,DE},{BC,AD},{BC,DE},{BC,EA},{CD,EA} and {DE,AC}. - Emeric Deutsch, Dec 25 2004
Partial sums of A029907. First differences of A002940. - Peter Bala, Oct 24 2007
Equals row sums of triangle A144154. - Gary W. Adamson, Sep 12 2008
Equals the number of 1's in Fibonacci Maximal notation for subsets of
(1, 2, 3, 5, 8, 13, ...) terms. For example (cf. A181630): 4, 5, and 6 are the 3 terms 101, 110, and 111 in Fibonacci Maximal. Total number of 1's for those terms = 7 = a(2). - Gary W. Adamson, Nov 02 2010
a(n) is half the number of strokes needed to draw all the domino tilings of a 2 X (n+2) rectangle. - Roberto Tauraso, Mar 15 2014
a(n) is the total number of 1's in all (n+1)-bit dual Zeckendorf representations of integers (A104326). For example, a(2) = 7 counts the 1's in 101, 110, 111. - Shenghui Yang, Feb 09 2025

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci, Hypercubes and Isometric Words based on Swap and Mismatch Distance, arXiv:2303.09898 [math.CO], 2023.
Kassie Archer and Noel Bourne, Pattern avoidance in compositions and powers of permutations, arXiv:2505.05218 [math.CO], 2025. See p. 6.
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Jean-Luc Baril and José L. Ramírez, Fibonacci and Catalan paths in a wall, 2023.
Giuseppa Castiglione and Marinella Sciortino, Standard Sturmian words and automata minimization algorithms, Theoretical Computer Science, Volume 601, 11 October 2015, Pages 58-66 ("WORDS 2013").
Tomislav Došlic and Luka Podrug, Tilings of a Honeycomb Strip and Higher Order Fibonacci Numbers, arXiv:2203.11761 [math.CO], 2022.
Dmitry Efimov, Hafnian of two-parameter matrices, arXiv:2101.09722 [math.CO], 2021.
Eric S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, arXiv:math/0307050 [math.CO], 2003. Sec. 8.
Neven Elezović, Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers, J. Int. Seq. 17 (2014) # 14.2.1.
Martin Griffiths, A Restricted Random Walk defined via a Fibonacci Process, Journal of Integer Sequences, Vol. 14 (2011), #11.5.4.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045 (Fibonacci numbers).
Column 1 of triangle A063967.
Cf. A001629, A002940, A004798, A010049, A029907, A104326, A144154, A181630.

a023610 n = a023610_list !! n
a023610_list = f [1] $ drop 3 a000045_list where
   f us (v:vs) = (sum $ zipWith (*) us $ tail a000045_list) : f (v:us) vs
-- Reinhard Zumkeller, Jan 18 2014

Table[Sum[Binomial[n - i, i]*(n - i), {i, 0, n}], {n, 1, 33}] (* Geoffrey Critzer, May 04 2009 *)

a(n)=(n+2)*fibonacci(n+4)/5+(n-1)*fibonacci(n+2)/5 \\ Charles R Greathouse IV, Jun 11 2015

def A023610():
    a, b, c, d = 1, 3, 7, 15
    while True:
        yield a
        a, b, c, d = b, c, d, 2*(d-b)+c-a
a = A023610(); [next(a) for i in range(33)]  # Peter Luschny, Nov 20 2013

O.g.f.: (x+1)/(1-x-x^2)^2. - Len Smiley, Dec 11 2001
a(n) = (1/5)*((n+2)*F(n+4) + (n-1)*F(n+2)), with F(n)=A000045(n). - Ralf Stephan, Jul 06 2003
a(n) = Sum_{k=0..n+1} (n-k+1)*binomial(n-k+1, k). - Paul Barry, Nov 05 2005
Recurrence: a(n+2) = a(n+1) + a(n) + Fib(n+4), n >= 0. For n >= 2, a(n-2) = (-1)^n*((-2n+3)*Fib(-n) - (-n)*Fib(-n-1))/5 = (-1)^n*A010049(-n), the second-order Fibonacci numbers of negative index, where Fib(-n) = (-1)^(n+1)*Fib(n). - Peter Bala, Oct 24 2007
a(n) = (n+1)*F(n+2) - A001629(n+1) where F(n) is the n-th Fibonacci number. - Geoffrey Critzer, Apr 07 2008
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4), n >= 4. - L. Edson Jeffery, Mar 29 2013
a(n+1) = A004798(n) + A000045(n+2) for n >= 0. - John Molokach, Jul 04 2013
a(n) = A001629(n+1) + A001629(n+2). - Philippe Deléham, Oct 30 2013
E.g.f.: exp(x/2)*(5*(5 + 7*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(11 + 15*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 04 2023

A058071 A Fibonacci triangle: triangle T(n,k) = Fibonacci(k+1)*Fibonacci(n-k+1), for n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 5, 3, 4, 3, 5, 8, 5, 6, 6, 5, 8, 13, 8, 10, 9, 10, 8, 13, 21, 13, 16, 15, 15, 16, 13, 21, 34, 21, 26, 24, 25, 24, 26, 21, 34, 55, 34, 42, 39, 40, 40, 39, 42, 34, 55, 89, 55, 68, 63, 65, 64, 65, 63, 68, 55, 89, 144, 89, 110, 102, 105, 104, 104, 105, 102, 110, 89, 144

Offset: 0

N. J. A. Sloane, Nov 24 2000

Or, multiplication table of the positive Fibonacci numbers read by antidiagonals.
Or, triangle of products of nonzero Fibonacci numbers.
Or, a two-dimensional square Fibonacci array read by antidiagonals, with offset 1: T(1,1) = T(1,2) = T(2,1) = T(2,2) = 1; thereafter T(m,n) = max {T(m,n-2) + T(m,n-1), T(m-2,n) + T(m-1,n), T(m-2,n-2) + T(m-1,n-1)}. If "max" is changed to "min" we get A283845. - N. J. A. Sloane, Mar 31 2017
Row sums are A001629 (Fibonacci numbers convolved with themselves.). The main diagonal and first subdiagonal are Fibonacci numbers, for other entries T(n,k) = T(n-1,k) + T(n-2,k). The central numbers form A006498. - Gerald McGarvey, Jun 02 2005
Alternating row sums = (1, 0, 3, 0, 8, ...), given by Fibonacci(2n) if n even, else zero.
Row n = edge-counting vector for the Fibonacci cube F(n+1) embedded in the natural way in the hypercube Q(n+1). - Emanuele Munarini, Apr 01 2008
The augmentation of A058071 is the triangle A193595. To fit the definition of augmented triangle at A103091, it is helpful to represent A058071 using p(n,k)=F(k+1)*F(n+1-k) for 0<=k<=n. - Clark Kimberling, Jul 31 2011
T(n,k) = number of appearances of a(k) in p(n) in the n-th convergent p(n)/q(n) of the formal infinite continued fraction [a(0), a(1), ...]; e.g., p(3) = a(0)*a(1)*a(2)*a(3) + a(0)*a(1) + a(0)*a(3) + a(2)*a(3) + 1. Also, T(n,k) = number of appearances of a(k+1) in q(n+1); e.g., q(3) = a(1)*a(2)*a(3) + a(1) + a(3). - Clark Kimberling, Dec 21 2015
Each row is a palindrome, and the central term of row 2n is the square of the F(n+1), where F = A000045 (Fibonacci numbers). - Clark Kimberling, Dec 21 2015
Also called Hosoya's triangle, after the Japanese chemist Haruo Hosoya (b. 1936). - Amiram Eldar, Jun 10 2021

			Triangle begins as:
   1;
   1,  1;
   2,  1,  2;
   3,  2,  2,  3;
   5,  3,  4,  3,  5;
   8,  5,  6,  6,  5,  8;
  13,  8, 10,  9, 10,  8, 13;
  21, 13, 16, 15, 15, 16, 13, 21;
  34, 21, 26, 24, 25, 24, 26, 21, 34;
  ...
As a square array:
   1,  1,  2,  3,  5,  8, 13, 21, ...
   1,  1,  2,  3,  5,  8, 13, 21, ...
   2,  2,  4,  6, 10, 16, 26, ...
   3,  3,  6,  9, 15, 24, ...
   5,  5, 10, 15, 25, ...
   8,  8, 16, 24, ...
  13, 13, 26, ...
  21, 21, ...

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
Thomas Koshy, "Fibonacci and Lucas Numbers and Applications", Chap. 15, Hosoya's Triangle, Wiley, New York, 2001.

Emanuele Munarini and Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Arthur T. Benjamin and Daniela Elizondo, Counting on Hosoya's Triangle, Fibonacci Quart. 60 (2022), no. 5, 47-55.
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Matrices in the Hosoya triangle, arXiv:1808.05278 [math.CO], 2018.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 27.
Hsin-Yun Ching, Rigoberto Flórez, and Antara Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
Cristian Cobeli and Alexandru Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56 (104), No. 1 (2013), pp. 73-98. - From _N. J. A. Sloane_, Feb 16 2013
Rigoberto Flórez, Robinson A. Higuita and Leandro Junes, GCD Property of the Generalized Star of David in the Generalized Hosoya Triangle, J. Int. Seq., Vol. 17 (2014), Article 14.3.6.
Rigoberto Florez, Robinson A. Higuita, Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, arXiv:1706.04247 [math.CO], 2017.
Rigoberto Flórez, Robinson A. Higuita, and Antara Mukherjee, The Geometry of some Fibonacci Identities in the Hosoya Triangle, arXiv:1804.02481 [math.NT], 2018.
Martin Griffiths, Digit Proportions in Zeckendorf Representations, Fibonacci Quart., Vol. 48, No. 2 (2010), pp. 168-174.
Haruo Hosoya, Fibonacci Triangle, The Fibonacci Quarterly, Vol. 14, No. 2 (1976), pp. 173-178.
Sandi Klavžar and Iztok Peterin, Edge-counting vectors, Fibonacci cubes and Fibonacci triangle, 2005 preprint of Publ. Math. Debrecen, Vol. 71, No. 3-4 (2007), pp. 267-278.
Tiberiu V. Trif, Solution to Problem 10706 proposed by J. G. Propp, Amer. Math. Monthly, Vol. 107, No. 9 (Nov. 2000), pp. 866-867.

Cf. A000045, A003991, A001629, A098356, A283845.

Cf. A001654, A004798, A007598, A024458, A250111

a058071 n k = a058071_tabl !! n !! k
a058071_row n = a058071_tabl !! n
a058071_tabl = map (\fs -> zipWith (*) fs $ reverse fs) a104763_tabl
-- Reinhard Zumkeller, Aug 15 2013

[Fibonacci(k+1)*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 06 2022

row[n_] := Table[Fibonacci[k]*Fibonacci[n-k+1], {k, 1, n}]; Table[row[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

T(n,k)=fibonacci(k)*fibonacci(n+2-k) \\ Charles R Greathouse IV, Feb 07 2017

flatten([[fibonacci(k+1)*fibonacci(n-k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 06 2022

Row n: F(1)*F(n), F(2)*F(n-1), ..., F(n)*F(1).
G.f.: T(x,y) = 1/((1-x-x^2)(1-xy-x^2y^2)). Recurrence: T(n+4,k+2) = T(n+3,k+2) + T(n+3,k+1) + T(n+2,k+2) - T(n+2,k+1) + T(n+2,k) - T(n+1,k+1) - T(n+1,k) - T(n,k). - Emanuele Munarini, Apr 01 2008
T(n,k) = A104763(n+1,k+1) * A104763(n+1,n+1-k). - Reinhard Zumkeller, Aug 15 2013
Column k is the (generalized) Fibonacci sequence having first two terms F(k+1), F(k+1). - Clark Kimberling, Dec 21 2015
From G. C. Greubel, Apr 06 2022: (Start)
T(n,k) = Fibonacci(k+1)*Fibonacci(n-k+1).
Sum_{k=0..n} T(n, k) = A001629(n+2).
Sum_{k=0..floor(n/2)} T(n, k) = A024458(n+1).
Sum_{k=1..n-1} T(n, k) = A004798(n-1), n >= 2.
Sum_{k=0..floor(n/2)} T(n-k, k) = A250111(n+2).
T(n, 0) = A000045(n+1).
T(2*n, n) = A007598(n+1).
T(2*n+1, n) = A001654(n+1).
T(n, n-k) = T(n, k). (End)

More terms from James Sellers, Nov 27 2000
Edited by N. J. A. Sloane, Sep 15 2008 at the suggestion of R. J. Mathar
Name edited by G. C. Greubel, Apr 06 2022

A006367 Number of binary vectors of length n+1 beginning with 0 and containing just 1 singleton.

Original entry on oeis.org

1, 0, 2, 2, 5, 8, 15, 26, 46, 80, 139, 240, 413, 708, 1210, 2062, 3505, 5944, 10059, 16990, 28646, 48220, 81047, 136032, 228025, 381768, 638450, 1066586, 1780061, 2968040, 4944519, 8230370, 13689118, 22751528, 37786915, 62716752, 104028245

Offset: 0

David M. Bloom

Number of compositions of n+1 containing exactly one 1. - Emeric Deutsch, Mar 08 2002
Number of permutations with one fixed point avoiding 231 and 321.
A singleton is a run of length 1. - Michael Somos, Nov 29 2014
Second column of A105422. - Michael Somos, Nov 29 2014
Number of weak compositions of n with one 0 and no 1's. Example: Combine one 0 with the compositions of 5 without 1 to get a(5) = 8 weak compositions: 0,5; 5,0; 0,2,3; 0,3,2; 2,0,3; 3,0,2; 2,3,0; 3,2,0. - Gregory L. Simay, Mar 21 2018

			a(4) = 5 because among the 2^4 compositions of 5 only 4+1,1+4,2+2+1,2+1+2,1+2+2 contain exactly one 1.
a(4) = 5 because the binary vectors of length 4+1 beginning with 0 and with exactly one singleton are: 00001, 00100, 00110, 01100, 01111. - _Michael Somos_, Nov 29 2014
G.f. = 1 + 2*x^2 + 2*x^3 + 5*x^4 + 8*x^5 + 15*x^6 + 26*x^7 + 46*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 11.
Mengmeng Liu and Andrew Yezhou Wang, The Number of Designated Parts in Compositions with Restricted Parts, J. Int. Seq., Vol. 23 (2020), Article 20.1.8.
J. J. Madden, A generating function for the distribution of runs in binary words, arXiv:1707.04351 [math.CO], 2017, Theorem 1.1, r=k=1.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, arXiv:math/0204005 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000032, A000045, A004798, A006355, A010049, A105422, A139821, A212804.

I:=[1,0]; [n le 2 select I[n] else Self(n-1)+Self(n-2)+Fibonacci(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2014

nn=36; CoefficientList[Series[1/(1 -x/(1-x) +x)^2, {x, 0, nn}], x] (* Geoffrey Critzer, Feb 18 2014 *)
a[n_]:= If[ n<0, SeriesCoefficient[((1-x)/(1+x-x^2))^2, {x, 0, -2-n}], SeriesCoefficient[((1-x)/(1-x-x^2))^2, {x, 0, n}]]; (* Michael Somos, Nov 29 2014 *)

Vec( (1-x)^2/(1-x-x^2)^2 + O(x^66) ) \\ Joerg Arndt, Feb 20 2014

{a(n) = if( n<0, n = -2-n; polcoeff( (1 - x)^2 / (1 + x - x^2)^2 + x * O(x^n), n), polcoeff( (1 - x)^2 / (1 - x - x^2)^2 + x * O(x^n), n))}; /* Michael Somos, Nov 29 2014 */

from sympy import fibonacci
from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n==0 else 0 if n==1 else a(n - 1) + a(n - 2) + fibonacci(n - 3)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 20 2017

def A006367(n): return (1/5)*(n*lucas_number2(n-2, 1, -1) + fibonacci(n+1) + 4*fibonacci(n-1))
[A006367(n) for n in (0..40)] # G. C. Greubel, Apr 06 2022

a(n) = a(n-1) + a(n-2) + Fibonacci(n-3).
G.f.: (1-x)^2/(1-x-x^2)^2. - Emeric Deutsch, Mar 08 2002
a(n) = A010049(n+1) - A010049(n). - R. J. Mathar, May 30 2014
Convolution square of A212804. - Michael Somos, Nov 29 2014
a(n) = -(-1)^n * A004798(-1-n) for all n in Z. - Michael Somos, Nov 29 2014
0 = a(n)*(-2*a(n) - 7*a(n+1) + 2*a(n+2) + a(n+3)) + a(n+1)*(-4*a(n+1) + 10*a(n+2) - 2*a(n+3)) + a(n+2)*(+4*a(n+2) - 7*a(n+3)) + a(n+3)*(+2*a(n+3)) for all n in Z. - Michael Somos, Nov 29 2014
a(n) = (n*Lucas(n-2) + Fibonacci(n))/5 + Fibonacci(n-1). - Ehren Metcalfe, Jul 29 2017

A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

Original entry on oeis.org

7, 2, 3, 6, 0, 6, 7, 9, 7, 7, 4, 9, 9, 7, 8, 9, 6, 9, 6, 4, 0, 9, 1, 7, 3, 6, 6, 8, 7, 3, 1, 2, 7, 6, 2, 3, 5, 4, 4, 0, 6, 1, 8, 3, 5, 9, 6, 1, 1, 5, 2, 5, 7, 2, 4, 2, 7, 0, 8, 9, 7, 2, 4, 5, 4, 1, 0, 5, 2, 0, 9, 2, 5, 6, 3, 7, 8, 0, 4, 8, 9, 9, 4, 1, 4, 4, 1, 4, 4, 0, 8, 3, 7, 8, 7, 8, 2, 2, 7, 4

Offset: 0

Jean-François Alcover, May 20 2014

Quoting Steven Finch: "The slopes of the 'critical parallelogram' are (1+sqrt(5))/2 [phi] and (1-sqrt(5))/2 [-1/phi]."
Essentially the same as A229780, A134972, A134945, A098317 and A002163. - R. J. Mathar, May 23 2014
Let W_n be the collection of all binary words of length n that do not contain two consecutive 0's. Let r_n be the ratio of the total number of 1's in W_n divided by the total number of letters in W_n. Then lim_{n->oo} r_n = 0.723606... Equivalently, lim_{n->oo} A004798(n)/(n*A000045(n+2)) = 0.723606... - Geoffrey Critzer, Feb 04 2022
The limiting frequency of the digit 0 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - Amiram Eldar, Mar 18 2025

			k2 = 0.723606797749978969640917366873127623544...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.23, p. 176.

Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
Index entries for algebraic numbers, degree 2.

Cf. A000032, A000045, A001622, A002163, A004798, A020762, A081007, A094214, A094874, A098317, A134972, A229780, A244847, A296184, A300074, A344212.

RealDigits[(1+1/Sqrt[5])/2, 10, 100] // First

(1 + 1/sqrt(5))/2 \\ Stefano Spezia, Dec 07 2024

Equals (1 + 1/sqrt(5))/2.
Equals 1/A094874. - Michel Marcus, Dec 01 2018
From Amiram Eldar, Feb 11 2022: (Start)
Equals phi/sqrt(5), where phi is the golden ratio (A001622).
Equals lim_{k->oo} Fibonacci(k+1)/Lucas(k). (End)
From Amiram Eldar, Nov 28 2024: (Start)
Equals A344212/2 = A296184/5 = A300074^2 = sqrt(A229780).
Equals Product_{k>=1} (1 - 1/A081007(k)). (End)
Equals 1 - A244847. - Amiram Eldar, Mar 18 2025

A213587 Rectangular array: (row n) = bc, where b(h) = F(h+1), c(h) = F(n+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

Original entry on oeis.org

1, 4, 2, 10, 7, 3, 22, 17, 11, 5, 45, 37, 27, 18, 8, 88, 75, 59, 44, 29, 13, 167, 146, 120, 96, 71, 47, 21, 310, 276, 234, 195, 155, 115, 76, 34, 566, 511, 443, 380, 315, 251, 186, 123, 55, 1020, 931, 821, 719, 614, 510, 406, 301, 199, 89, 1819, 1675, 1497, 1332, 1162, 994, 825, 657, 487, 322, 144

Offset: 1

Clark Kimberling, Jun 19 2012

Principal diagonal: A213588.
Antidiagonal sums: A213589.
Row 1,  (1,2,3,5,...)**(1,2,3,5,...): A004798.
Row 2,  (1,2,3,5,...)**(2,3,5,8,...)
Row 3,  (1,2,3,5,...)**(3,5,8,13,...)
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
  1....4....10....22....45....88....167
  2....7....17....37....75....146...276
  3....11...27....59....120...234...443
  5....18...44....96....195...380...719
  8....29...71....155...315...614...1162
  13...47...115...251...510...994...1881

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A213500, A213588, A213589, A004798.

Flat( List([1..12], n-> List([1..n], k-> ((n-k+1)*Lucas(1,-1, n+3)[2] - Fibonacci(n-k+1)*Lucas(1,-1,k-1)[2])/5 ))); # G. C. Greubel, Jul 08 2019

[[((n-k+1)*Lucas(n+3) - Fibonacci(n-k+1)*Lucas(k-1))/5: k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 08 2019

(* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= Fibonacci[n+1];
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213587 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213588 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213589 *)
(* Second program *)
Table[((n-k+1)*LucasL[n+3] - Fibonacci[n-k+1]*LucasL[k-1])/5, {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 08 2019 *)

lucas(n) = fibonacci(n+1) + fibonacci(n-1);
t(n,k) = ((n-k+1)*lucas(n+3) - fibonacci(n-k+1)*lucas(k-1))/5;
for(n=1,12, for(k=1,n, print1(t(n,k), ", "))) \\ G. C. Greubel, Jul 08 2019

[[((n-k+1)*lucas_number2(n+3,1,-1) - fibonacci(n-k+1)* lucas_number2(k-1, 1,-1))/5 for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 08 2019

Rows: T(n,k) = 2*T(n,k-1) + T(n,k-2) - 2*T(n,k-3) - T(n,k-4).
Columns: T(n,k) = T(n-1,k) + T(n-2,k).
G.f. for row n: f(x)/g(x), where f(x) = F(n+1) + F(n+2)*x + F(n)*x^2 and g(x) = (1 - x - x^2)^2.
T(n, k) = (k*Lucas(n+k+2) - Fibonacci(k)*Lucas(n-1))/5. - G. C. Greubel, Jul 08 2019

A129717 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 101's (n >= 0, 0 <= k <= floor((n-1)/2)). A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

1, 2, 3, 4, 1, 4, 4, 4, 8, 1, 4, 12, 5, 4, 16, 13, 1, 4, 20, 25, 6, 4, 24, 41, 19, 1, 4, 28, 61, 44, 7, 4, 32, 85, 85, 26, 1, 4, 36, 113, 146, 70, 8, 4, 40, 145, 231, 155, 34, 1, 4, 44, 181, 344, 301, 104, 9, 4, 48, 221, 489, 532, 259, 43, 1, 4, 52, 265, 670, 876, 560, 147, 10, 4, 56

Offset: 0

Emeric Deutsch, May 12 2007

Row n has 1+floor((n-1)/2) terms for n >= 1.
Row sums are the Fibonacci numbers (A000045).
T(n,1) = A008574(n-3).
T(n,2) = A001844(n-5).
T(n,3) = A005900(n-6).
T(n,4) = A006325(n-7).
T(n,5) = A033455(n-10).
T(n,k) = A129718(n,k+1) (since in each word: 1 + the number of 101's = number of runs of 1's).
Sum_{k>=0} k*T(n,k) = A004798(n-2).

			T(6,2)=5 because we have 110101, 101101, 101010, 101011 and 010101.
Triangle starts:
  1;
  2;
  3;
  4,  1;
  4,  4;
  4,  8,  1;
  4, 12,  5;

Michael De Vlieger, Table of n, a(n) for n = 0..10100 (rows 0 <= n <= 200, flattened.)

Cf. A000045, A008574, A001844, A005900, A006325, A033455, A129718, A004798.

T:=proc(n,k) if n=0 and k=0 then 1 elif n=1 and k=0 then 2 elif n=2 and k=0 then 3 elif n=3 and k=1 then 1 elif k

MapAt[{0, 1} + # &, #, 4] /. {} -> {1} &@ Table[If[n < 3, n + 1, Binomial[n - k - 1, k] + 2 Binomial[n - k - 2, k] + Binomial[n - k - 3, k]], {n, 0, 17}, {k, 0, Floor[(n - 1)/2]}] // Flatten (* Michael De Vlieger, Nov 15 2019 *)

G.f.: G(t,z) = (1+z)*(1 + z^2 - t*z^2)/(1 - z - t*z^2).
G.f. of col 0: (1+z)(1+z^2)/(1-z), leading to the partial sums of 1,1,1,1,0,0,0,...
G.f. of col k: z^(2k+1)*(1+z)^2/(1-z)^(k+1) (k >= 1).
T(n,k) = binomial(n-k-1, k) + 2*binomial(n-k-2, k) + binomial(n-k-3, k) for n >= 4 and 0 <= k < n/2.

A171855 Triangle read by rows: T(n,k) is the number of binary words of length n that have no pair of adjacent 1's and have k subwords 0000 (n>=0; k=0 for n=0,1,2; 0<=k<=n-3 for n>=3).

Original entry on oeis.org

1, 2, 3, 5, 7, 1, 10, 2, 1, 15, 3, 2, 1, 22, 6, 3, 2, 1, 32, 11, 6, 3, 2, 1, 47, 18, 12, 6, 3, 2, 1, 69, 30, 20, 13, 6, 3, 2, 1, 101, 50, 34, 22, 14, 6, 3, 2, 1, 148, 81, 59, 38, 24, 15, 6, 3, 2, 1, 217, 130, 99, 68, 42, 26, 16, 6, 3, 2, 1, 318, 208, 163, 118, 77, 46, 28, 17, 6, 3, 2, 1, 466

Offset: 0

Emeric Deutsch, Feb 15 2010

Row n has n-2 entries.
Sum of entries in row n is the Fibonacci number A000045(n+2).
T(n,0)=A003410(n). Sum(k*T(n,k),k>=0)=A004798(n-3) for n>=4.

			T(5,1)=2 because we have 00001 and 10000; T(7,2)=3 because we have 0000010, 0100000, and 1000001.
Triangle starts:
1;
2;
3;
5;
7,1;
10,2,1;
15,3,2,1;
22,6,3,2,1;

Cf. A000045, A003410, A004798.

G := (1+z)*(1+z-t*z+z^2-t*z^2+z^3-t*z^3)/(1-t*z-z^2+t*z^3-z^3+t*z^4-z^4): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 2; 3; for n from 3 to 15 do seq(coeff(P[n], t, k), k = 0 .. n-3) end do; # yields sequence in triangular form

G.f.: G(t,z) = (1+z)(1+z-tz+z^2-tz^2+z^3-tz^3)/(1-tz-z^2+tz^3-z^3+tz^4-z^4).

A185081 Triangle T(n,k), read by rows, given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 4, 3, 0, 3, 9, 10, 5, 0, 5, 18, 28, 22, 8, 0, 8, 35, 68, 74, 45, 13, 0, 13, 66, 154, 210, 177, 88, 21, 0, 21, 122, 331, 541, 574, 397, 167, 34, 0, 34, 222, 686, 1302, 1656, 1446, 850, 310, 55

Offset: 0

Philippe Deléham, Jan 22 2012

Row sums: A133494.

			Triangle begins:
  1;
  0,  1;
  0,  1,  2;
  0,  2,  4,  3;
  0,  3,  9, 10,  5;
  0,  5, 18, 28, 22,  8;
  0,  8, 35, 68, 74, 45, 13;
From _Philippe Deléham_, Apr 11 2012: (Start)
Triangle in A209138 begins:
  1;
  1,  2;
  2,  4,  3;
  3,  9, 10,  5;
  5, 18, 28, 22,  8;
  8, 35, 68, 74, 45, 13; (End)

Cf. A000007, A000045, A000244, A004798, A033999, A084938, A133494, A209138.

nmax = 9; T[n_, n_] := Fibonacci[n+1]; T[, 0] = 0; T[n, 1] := Fibonacci[n]; T[n_, k_] /; 1 < k < n := T[n, k] = T[n - 1, k] + T[n - 1, k - 1] + T[n - 2, k] + T[n - 2, k - 1] + T[n - 2, k - 2]; T[, ] = 0;
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)

Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000007(n), A133494(n) for x = -1, 0, 1 respectively.
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), for n > 2, T(0,0) = T(1,1) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(2,2) = 2.
T(n+1,n) = A004798(n), T(n,n) = T(n+1,1) = A000045(n+1).
T(n,k) = A209138(n,k-1) for k >= 1. - Philippe Deléham, Apr 11 2012
G.f.: (-1 + x^2*y + x + x^2)/(-1 + x^2*y + x + x^2 + x*y + x^2*y^2). - R. J. Mathar, Aug 11 2015

Corrected by Jean-François Alcover, Jun 20 2017

A379039 G.f. A(x) satisfies A(x) = ( (1 + x) * (1 + x*A(x)^2) )^2.

Original entry on oeis.org

1, 4, 22, 172, 1513, 14356, 143228, 1480956, 15728516, 170558634, 1880568650, 21019304814, 237615558790, 2712066792304, 31210387143556, 361738488066632, 4218907281330372, 49476183230651216, 583066018329260673, 6901459436855306662, 82011678696864842013

Offset: 0

Seiichi Manyama, Dec 14 2024

nonn

Cf. A004798, A073155, A379037.

Cf. A364337, A379038.

a(n) = sum(k=0, n, binomial(4*k+2, k)*binomial(4*k+2, n-k)/(2*k+1));

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A364337.

a(n) = Sum_{k=0..n} binomial(4*k+2,k) * binomial(4*k+2,n-k)/(2*k+1).

A106408 Triangle, read by rows, where T(1,1) = 1; T(2,1) = T(2,2) = 2; for n > 2, T(n,n) = T(n-1,n-1) + T(n-2,n-2); T(n+1,n) = 2 * T(n,n); for all other entries, T(n,k) = T(n-1,k) + T(n-2,k).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 5, 6, 6, 5, 8, 10, 9, 10, 8, 13, 16, 15, 15, 16, 13, 21, 26, 24, 25, 24, 26, 21, 34, 42, 39, 40, 40, 39, 42, 34, 55, 68, 63, 65, 64, 65, 63, 68, 55, 89, 110, 102, 105, 104, 104, 105, 102, 110, 89, 144, 178, 165, 170, 168, 169, 168, 170, 165, 178, 144

Offset: 1

Gerald McGarvey, May 28 2005

Row sums are A004798 (convolution of Fibonacci numbers 1,2,3,5,... with themselves). Central numbers of the rows are A006498 (a(n) = a(n-1)+a(n-3)+a(n-4)). First column and main diagonal are Fibonacci numbers 1,2,3,5,... First subdiagonal are 2*Fibonacci numbers. T(n,k) = F(n-k+2)*F(k+1) where F(m) is the m-th Fibonacci number. For the antidiagonal sums b(n): b(1) = 1, b(2) = 2, then b(n) = b(n-1) + b(n-2) + F(floor((n+3)/2)).

T(n,k) is the number of Boolean intervals of the form [s_k,w] in the weak order on S_n, for a fixed simple reflection s_k. - Bridget Tenner, Jan 16 2020

			Triangle begins
   1;
   2,  2;
   3,  4,  3;
   5,  6,  6,  5;
   8, 10,  9, 10,  8;

B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.

Cf. A000045, A004798, A006498.

G.f.: (1+x+y+x*y)/((1-x-x^2)*(1-y-y^2)) [U coordinates] - N. J. A. Sloane, Jun 01 2005

cp's OEIS Frontend

A023610 Convolution of Fibonacci numbers and {F(2), F(3), F(4), ...}.

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A058071 A Fibonacci triangle: triangle T(n,k) = Fibonacci(k+1)*Fibonacci(n-k+1), for n >= 0, 0 <= k <= n.

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A006367 Number of binary vectors of length n+1 beginning with 0 and containing just 1 singleton.

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A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

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A213587 Rectangular array: (row n) = b**c, where b(h) = F(h+1), c(h) = F(n+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A129717 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 101's (n >= 0, 0 <= k <= floor((n-1)/2)). A Fibonacci binary word is a binary word having no 00 subword.

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A213587 Rectangular array: (row n) = bc, where b(h) = F(h+1), c(h) = F(n+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.