A104763 -id:A104763 - cp's OEIS frontend

A180662 The Golden Triangle: T(n,k) = A001654(k) for n>=0 and 0<=k<=n.

0, 0, 1, 0, 1, 2, 0, 1, 2, 6, 0, 1, 2, 6, 15, 0, 1, 2, 6, 15, 40, 0, 1, 2, 6, 15, 40, 104, 0, 1, 2, 6, 15, 40, 104, 273, 0, 1, 2, 6, 15, 40, 104, 273, 714, 0, 1, 2, 6, 15, 40, 104, 273, 714, 1870, 0, 1, 2, 6, 15, 40, 104, 273, 714, 1870, 4895

Offset: 0

Johannes W. Meijer, Sep 21 2010

The terms in the n-th row of the Golden Triangle are the first (n+1) golden rectangle numbers. The golden rectangle numbers are A001654(n)=F(n)*F(n+1), with F(n) the Fibonacci numbers. The mirror image of the Golden Triangle is A180663.
We define below 24 mostly new triangle sums. The Row1 and Row2 sums are the ordinary and alternating row sums respectively and the Kn11 and Kn12 sums are commonly known as antidiagonal sums. Each of the names of these sums, except for the row sums, comes from a (fairy) chess piece that moves in its own peculiar way over a chessboard, see Hooper and Whyld. All pieces are leapers: knight (sqrt(5) or 1,2), fil (sqrt(8) or 2,2), camel (sqrt(10) or 3,1), giraffe (sqrt(17) or 4,1) and zebra (sqrt(13) or 3,2). Information about the origin of these chess sums can be found in "Famous numbers on a chessboard", see Meijer.
Each triangle or chess sum formula adds up numbers on a chessboard using the moves of its namesake. Converting a number triangle to a square array of numbers shows this most clearly (use the table button!). The formulas given below are for number triangles.
The chess sums of the Golden Triangle lead to six different sequences, see the crossrefs. As could be expected all these sums are related to the golden rectangle numbers.
Some triangles with complete sets of triangle sums are: A002260 (Natural Numbers), A007318 (Pascal), A008288 (Delannoy) A013609 (Pell-Jacobsthal), A036561 (Nicomachus), A104763 (Fibonacci(n)), A158405 (Odd Numbers) and of course A180662 (Golden Triangle).
   #..Name....Type..Code....Definition of triangle sums.
Row......1....Row1..  a(n) = Sum_{k=0..n} T(n, k).
Row Alt..2....Row2..  a(n) = Sum_{k=0..n} (-1)^(n+k)*T(n, k).
Knight...1....Kn11..  a(n) = Sum_{k=0..floor(n/2)} T(n-k, k).
Knight...1....Kn12..  a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, k+1).
Knight...1....Kn13..  a(n) = Sum_{k=0..floor(n/2)} T(n-k+2, k+2).
Knight...2....Kn21..  a(n) = Sum_{k=0..floor(n/2)} T(n-k, n-2*k).
Knight...2....Kn22..  a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, n-2*k).
Knight...2....Kn23..  a(n) = Sum_{k=0..floor(n/2)} T(n-k+2, n-2*k).
Knight...3....Kn3...  a(n) = Sum_{k=0..n} T(n+k, 2*k).
Knight...4....Kn4...  a(n) = Sum_{k=0..n} T(n+k, n-k).
Fil......1....Fi1...  a(n) = Sum_{k=0..floor(n/2)} T(n, 2*k).
Fil......2....Fi2...  a(n) = Sum_{k=0..floor(n/2)} T(n, n-2*k).
Camel....1....Ca1...  a(n) = Sum_{k=0..floor(n/3)} T(n-2*k, k).
Camel....2....Ca2...  a(n) = Sum_{k=0..floor(n/3)} T(n-2*k, n-3*k).
Camel....3....Ca3...  a(n) = Sum_{k=0..n} T(n+2*k, 3*k).
Camel....4....Ca4...  a(n) = Sum_{k=0..n} T(n+2*k, n-k).
Giraffe..1....Gi1...  a(n) = Sum_{k=0..floor(n/4)} T(n-3*k, k).
Giraffe..2....Gi2...  a(n) = Sum_{k=0..floor(n/4)} T(n-3*k, n-4*k).
Giraffe..3....Gi3...  a(n) = Sum_{k=0..n} T(n+3*k, 4*k).
Giraffe..4....Gi4...  a(n) = Sum_{k=0..n} T(n+3*k, n-k).
Zebra....1....Ze1...  a(n) = Sum_{k=0..floor(n/2)} T(n+k, 3*k).
Zebra....2....Ze2...  a(n) = Sum_{k=0..floor(n/2)} T(n+k, n-2*k).
Zebra....3....Ze3...  a(n) = Sum_{k=0..floor(n/3)} T(n-k, 2*k).
Zebra....4....Ze4...  a(n) = Sum_{k=0..floor(n/3)} T(n-k, n-3*k).

			The first few rows of the Golden Triangle are:
  0;
  0, 1;
  0, 1, 2;
  0, 1, 2, 6;
  0, 1, 2, 6, 15;
  0, 1, 2, 6, 15, 40;

David Hooper and Kenneth Whyld, The Oxford Companion to Chess, p. 221, 1992.

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Verner E. Hoggatt, Jr., A new angle on Pascal’s triangle, The Fibonacci Quarterly, Vol. 6, Number 4, pp. 228-230, Oct. 1968.
Edouard Lucas, Recherches sur plusieurs ouvrages de Léonard de Pise, Ch. 1, pp. 12-14, 1877.
Johannes W. Meijer, Famous numbers on a chessboard, Acta Nova, Volume 4, No.4, December 2010; pp. 589-598.
Johannes W. Meijer, Illustrations of the triangle sums, Mar 07 2013.
S. Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.

Cf. A180663 (Mirror), A001654 (Golden Rectangle), A000045 (F(n)).

Triangle sums: A064831 (Row1, Kn11, Kn12, Kn4, Ca1, Ca4, Gi1, Gi4), A077916 (Row2), A180664 (Kn13), A180665 (Kn21, Kn22, Kn23, Fi2, Ze2), A180665(2*n) (Kn3, Fi1, Ze3), A115730(n+1) (Ca2, Ze4), A115730(3*n+1) (Ca3, Ze1), A180666 (Gi2), A180666(4*n) (Gi3).

import Data.List (inits)
a180662 n k = a180662_tabl !! n !! k
a180662_row n = a180662_tabl !! n
a180662_tabl = tail $ inits a001654_list
-- Reinhard Zumkeller, Jun 08 2013

[Fibonacci(k)*Fibonacci(k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, May 25 2021

F:= combinat[fibonacci]:
T:= (n, k)-> F(k)*F(k+1):
seq(seq(T(n, k), k=0..n), n=0..10); # revised Johannes W. Meijer, Sep 13 2012

Table[Times @@ Fibonacci@ {k, k + 1}, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 18 2016 *)
Module[{nn=20,f},f=Times@@@Partition[Fibonacci[Range[0,nn]],2,1];Table[Take[f,n],{n,nn}]]//Flatten (* Harvey P. Dale, Nov 26 2022 *)

T(n,k)=fibonacci(k)*fibonacci(k+1) \\ Charles R Greathouse IV, Nov 07 2016

flatten([[fibonacci(k)*fibonacci(k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 25 2021

T(n, k) = F(k)*F(k+1) with F(n) = A000045(n), for n>=0 and 0<=k<=n.
From Johannes W. Meijer, Jun 22 2015: (Start)
Kn1p(n) = Sum_{k=0..floor(n/2)} T(n-k+p-1, k+p-1), p >= 1.
Kn1p(n) = Kn11(n+2*p-2) - Sum_{k=0..p-2} T(n-k+2*p-2, k), p >= 2.
Kn2p(n) = Sum_{k=0..floor(n/2)} T(n-k+p-1, n-2*k), p >= 1.
Kn2p(n) = Kn21(n+2*p-2) - Sum_{k=0..p-2} T(n+k+p, n+2*k+2), p >= 2. (End)
G.f. as triangle: xy/((1-x)(1+xy)(1-3xy+x^2 y^2)). - Robert Israel, Sep 06 2015

A027926 Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 3, 4, 3, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 1, 1, 2, 3, 5, 8, 13, 20, 26, 25, 16, 6, 1, 1, 1, 2, 3, 5, 8, 13, 21, 33, 46, 51, 41, 22, 7, 1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 54, 79, 97, 92, 63, 29, 8, 1

Offset: 0

Clark Kimberling

T(n,k) = number of strings s(0),...,s(n) such that s(0)=0, s(n)=n-k and for 1<=i<=n, s(i)=s(i-1)+d, with d in {0,1,2} if i=0, in {0,2} if s(i)=2i, in {0,1,2} if s(i)=2i-1, in {0,1} if 0<=s(i)<=2i-2.

Can be seen as concatenation of triangles A104763 and A105809, with identifying column of Fibonacci numbers, see example. - Reinhard Zumkeller, Aug 15 2013

			.   0:                           1
.   1:                        1  1   1
.   2:                     1  1  2   2   1
.   3:                  1  1  2  3   4   3   1
.   4:               1  1  2  3  5   7   7   4   1
.   5:            1  1  2  3  5  8  12  14  11   5   1
.   6:          1 1  2  3  5  8 13  20  26  25  16   6   1
.   7:        1 1 2  3  5  8 13 21  33  46  51  41  22   7   1
.   8:      1 1 2 3  5  8 13 21 34  54  79  97  92  63  29   8  1
.   9:    1 1 2 3 5  8 13 21 34 55  88 133 176 189 155  92  37  9  1
.  10:  1 1 2 3 5 8 13 21 34 55 89 143 221 309 365 344 247 129 46 10  1
.
.   1:                           1
.   2:                        1  1
.   3:                     1  1  2
.   4:                  1  1  2  3
.   5:               1  1  2  3  5      columns = A000045, > 0
.   6:            1  1  2  3  5  8     +---------+
.   7:          1 1  2  3  5  8 13     | A104763 |
.   8:        1 1 2  3  5  8 13 21     +---------+
.   9:      1 1 2 3  5  8 13 21 34
.  10:    1 1 2 3 5  8 13 21 34 55
.  11:  1 1 2 3 5 8 13 21 34 55 89
.
.   0:                           1
.   1:                           1   1                +---------+
.   2:                           2   2   1            | A105809 |
.   3:                           3   4   3   1        +---------+
.   4:                           5   7   7   4   1
.   5:                           8  12  14  11   5   1
.   6:                          13  20  26  25  16   6   1
.   7:                          21  33  46  51  41  22   7   1
.   8:                          34  54  79  97  92  63  29   8  1
.   9:                          55  88 133 176 189 155  92  37  9  1
.  10:                          89 143 221 309 365 344 247 129 46 10  1

Reinhard Zumkeller, Rows n = 0..100 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle

Many columns of T are A000045 (Fibonacci sequence), also in T: A001924, A004006, A000071, A000124, A014162, A014166, A027927-A027933.

Some other Fibonacci-Pascal triangles: A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074.

Flat(List([0..10], n-> List([0..2*n], k-> Sum([0..Int((2*n-k+1)/2) ], j-> Binomial(n-j, 2*n-k-2*j) )))); # G. C. Greubel, Sep 05 2019

a027926 n k = a027926_tabf !! n !! k
a027926_row n = a027926_tabf !! n
a027926_tabf = iterate (\xs -> zipWith (+)
                               ([0] ++ xs ++ [0]) ([1,0] ++ xs)) [1]
-- Variant, cf. example:
a027926_tabf' = zipWith (++) a104763_tabl (map tail a105809_tabl)
-- Reinhard Zumkeller, Aug 15 2013

[&+[Binomial(n-j, 2*n-k-2*j): j in [0..Floor((2*n-k+1)/2)]]: k in [0..2*n], n in [0..10]]; // G. C. Greubel, Sep 05 2019

A027926 := proc(n,k)
    add(binomial(n-j,2*n-k-2*j),j=0..(2*n-k+1)/2) ;
end proc: # R. J. Mathar, Apr 11 2016

z = 15; t[n_, 0] := 1; t[n_, k_] := 1 /; k == 2 n; t[n_, 1] := 1;
t[n_, k_] := t[n, k] = t[n - 1, k - 2] + t[n - 1, k - 1];
u = Table[t[n, k], {n, 0, z}, {k, 0, 2 n}];
TableForm[u] (* A027926 array *)
v = Flatten[u] (* A027926 sequence *)
(* Clark Kimberling, Aug 31 2014 *)
Table[Sum[Binomial[n-j, 2*n-k-2*j], {j, 0, Floor[(2*n-k+1)/2]}], {n, 0, 10}, {k, 0, 2*n}]//Flatten (* G. C. Greubel, Sep 05 2019 *)

{T(n, k) = if( k<0 || k>2*n, 0, if( k<=1 || k==2*n, 1, T(n-1, k-2) + T(n-1, k-1)))}; /* _Michael Somos, Feb 26 1999 */

{T(n, k) = if( k<0 || k>2*n, 0, sum( j=max(0, k-n), k\2, binomial(k-j, j)))}; /* Michael Somos */

[[sum(binomial(n-j, 2*n-k-2*j) for j in (0..floor((2*n-k+1)/2))) for k in (0..2*n)] for n in (0..10)] # G. C. Greubel, Sep 05 2019

T(n, k) = Sum_{j=0..floor((2*n-k+1)/2)} binomial(n-j, 2*n-k-2*j). - Len Smiley, Oct 21 2001

Incorporates comments from Michael Somos.

Example extended by Reinhard Zumkeller, Aug 15 2013

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

A104762 Triangle read by rows: row n contains first n nonzero Fibonacci numbers in decreasing order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 3, 2, 1, 1, 8, 5, 3, 2, 1, 1, 13, 8, 5, 3, 2, 1, 1, 21, 13, 8, 5, 3, 2, 1, 1, 34, 21, 13, 8, 5, 3, 2, 1, 1, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 144, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1, 1, 233, 144, 89, 55, 34, 21, 13, 8, 5, 3, 2, 1

Offset: 1

Gary W. Adamson, Mar 23 2005, Mar 05 2007

Sum of n-th row = F(n+2) - 1; sequence A000071 starting (1, 2, 4, 7, 12, 20, ...).
Riordan array (1/(1-x-x^2),x). - Philippe Deléham, Apr 23 2009 [with offset 0]
Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A104762 is the reverse reluctant sequence of Fibonacci numbers (A000045), except 0. - Boris Putievskiy, Dec 13 2012

			First few rows of the triangle:
  1;
  1, 1;
  2, 1, 1;
  3, 2, 1, 1;
  5, 3, 2, 1, 1;
  8, 5, 3, 2, 1, 1;
  ...
From _Philippe Deléham_, Oct 07 2014: (Start)
Production matrix begins:
  1, 1;
  1, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 1;
  ... (End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A000045, A000071, A271355 (analogous Lucas triangle).

Companion triangle A104763, Fibonacci sequence in each row starting from the left. A121461.

seq(seq(combinat:-fibonacci(n-i),i=0..n-1),n=1..20); # Robert Israel, May 01 2016

r = N[(1 + Sqrt[5])/2, 100]; s = N[(1 - Sqrt[5])/2, 100];
t = Table[Abs[Round[(r^n)*(s^k)/Sqrt[5]]], {n, 2, 15}, {k, 1, n - 1}]
Flatten[t]
TableForm[t]
(* Clark Kimberling, May 01 2016 *)
Table[Reverse[Fibonacci[Range[n]]],{n,15}]//Flatten (* Harvey P. Dale, Jan 28 2019 *)

In every column, (1, 1, 2, 3, 5, ...); the nonzero Fibonacci numbers, A000045.
a(n,k) = A000045(n-k+1). - R. J. Mathar, Jun 23 2006
a(n) = A000045(m), where m = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012
Let P denote Pascal's triangle. Then P*A104762*P^(-1) = A121461. - Peter Bala, Apr 11 2013
a(n,k) = |round((r^n)*(s^k)/sqrt(5))|, where r = golden ratio = (1 + sqrt(5))/2, s = (1 - sqrt(5))/2, 1 <= k <= n-1, n >= 2. - Clark Kimberling, May 01 2016
G.f. of triangle: G(x,y) = x*y/((1-x-x^2)*(1-x*y)). - Robert Israel, May 01 2016

Edited by N. J. A. Sloane at the suggestion of Philippe Deléham, Jun 11 2007

More terms from Philippe Deléham, Apr 21 2009

A104765 Triangle T(n,k) read by rows: row n contains the first n Lucas numbers A000204.

Original entry on oeis.org

1, 1, 3, 1, 3, 4, 1, 3, 4, 7, 1, 3, 4, 7, 11, 1, 3, 4, 7, 11, 18, 1, 3, 4, 7, 11, 18, 29, 1, 3, 4, 7, 11, 18, 29, 47, 1, 3, 4, 7, 11, 18, 29, 47, 76, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 1, 3, 4, 7, 11

Offset: 1

Gary W. Adamson, Mar 24 2005

Reading rows from the right to the left yields A104764.

Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. Sequence A104765 is the reluctant sequence of A000204. - Boris Putievskiy, Dec 14 2012

			First few rows of the triangle are:
  1;
  1, 3;
  1, 3, 4;
  1, 3, 4, 7;
  1, 3, 4, 7, 11;
  1, 3, 4, 7, 11, 18;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A000204, A104765, A104762, A104763.

Cf. A027961 (row sums).

Table[LucasL[k], {n, 1, 10}, {k, 1, n}] // Flatten (* G. C. Greubel, Dec 21 2017 *)
Module[{nn=20,luc},luc=LucasL[Range[nn]];Table[Take[luc,n],{n,nn}]]//Flatten (* Harvey P. Dale, Jul 10 2024 *)

for(n=1,10, for(k=1,n, print1(fibonacci(k+1) + fibonacci(k-1), ", "))) \\ G. C. Greubel, Dec 21 2017

T(n,k) = A000204(k), 1<=k<=n.
T(n,k) = A104764(n,n-k+1).
a(n) = A000204(m), where m = n-t(t+1)/2, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 14 2012
G.f.: (x*y*(2*x*y+1))/((x-1)*(x^2*y^2+x*y-1)). - Vladimir Kruchinin, Jun 21 2025

Edited and extended by R. J. Mathar, Jul 23 2008

A180671 a(n) = Fibonacci(n+6) - Fibonacci(6).

Original entry on oeis.org

0, 5, 13, 26, 47, 81, 136, 225, 369, 602, 979, 1589, 2576, 4173, 6757, 10938, 17703, 28649, 46360, 75017, 121385, 196410, 317803, 514221, 832032, 1346261, 2178301, 3524570, 5702879, 9227457, 14930344, 24157809, 39088161, 63245978, 102334147, 165580133

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn15 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Vincenzo Librandi, Table of n, a(n) for n = 0..280
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045.

Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

List([0..40], n-> Fibonacci(n+6)-8); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+6)-Fibonacci(6): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+6)-fibonacci(6) od: seq(a(n),n=0..nmax);

f[n_]:= Fibonacci[n+6] - Fibonacci[6]; Array[f, 40, 0] (* or *)
LinearRecurrence[{2,0,-1}, {0,5,13}, 41] (* or *)
CoefficientList[Series[x(3x+5)/(x^3-2x+1), {x,0,40}], x] (* Robert G. Wilson v, Apr 11 2017 *)

for(n=1,40,print(fibonacci(n+6)-fibonacci(6))); \\ Anton Mosunov, Mar 02 2017

concat(0, Vec(x*(5+3*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Apr 20 2017

[fibonacci(n+6)-8 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+6) - F(6) with F = A000045.
a(n) = a(n-1) + a(n-2) + 8 for n>1, a(0)=0, a(1)=5, and where 8 = F(6).
From Colin Barker, Apr 13 2012: (Start)
G.f.: x*(5 + 3*x)/((1 - x)*(1 - x - x^2)).
a(n) = 2*a(n-1) - a(n-3). (End)
a(n) = (-8 + (2^(-n)*((1-sqrt(5))^n*(-9+4*sqrt(5)) + (1+sqrt(5))^n*(9+4*sqrt(5)))) / sqrt(5)). - Colin Barker, Apr 20 2017

A180672 a(n) = Fibonacci(n+7) - Fibonacci(7).

Original entry on oeis.org

0, 8, 21, 42, 76, 131, 220, 364, 597, 974, 1584, 2571, 4168, 6752, 10933, 17698, 28644, 46355, 75012, 121380, 196405, 317798, 514216, 832027, 1346256, 2178296, 3524565, 5702874, 9227452, 14930339, 24157804, 39088156, 63245973

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn16 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Vincenzo Librandi, Table of n, a(n) for n = 0..280
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A000071.

Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

List([0..40], n-> Fibonacci(n+7)-13 ); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+7) - Fibonacci(7): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+7)-fibonacci(7) od: seq(a(n),n=0..nmax);

Fibonacci[7 +Range[0, 40]] -13 (* G. C. Greubel, Jul 13 2019 *)

concat(0, Vec(x*(8+5*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017

a(n)=fibonacci(n+7)-fibonacci(7) \\ Charles R Greathouse IV, Feb 24 2017

[fibonacci(n+7)-13 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+7) - F(7) with F = A000045.
a(n) = a(n-1) + a(n-2) + 13 for n>1, a(0)=0, a(1)=8, and where 13 = F(7).
G.f.: x*(8 + 5*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-13 + (2^(-1-n)*((1-sqrt(5))^n*(-29+13*sqrt(5)) + (1+sqrt(5))^n*(29+13*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)
a(n) = 8*A000071(n+2) + 5*A000071(n+1). - Bruno Berselli, Feb 24 2017

A180673 a(n) = Fibonacci(n+8) - Fibonacci(8).

Original entry on oeis.org

0, 13, 34, 68, 123, 212, 356, 589, 966, 1576, 2563, 4160, 6744, 10925, 17690, 28636, 46347, 75004, 121372, 196397, 317790, 514208, 832019, 1346248, 2178288, 3524557, 5702866, 9227444, 14930331, 24157796, 39088148, 63245965, 102334134

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn17 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Vincenzo Librandi, Table of n, a(n) for n = 0..275
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A000071.

Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

List([0..40], n-> Fibonacci(n+8)-21); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+8) - Fibonacci(8): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=40: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+8)-fibonacci(8) od: seq(a(n),n=0..nmax);

Fibonacci[8 +Range[0, 40]] -21 (* G. C. Greubel, Jul 13 2019 *)

concat(0, Vec(x*(13+8*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017

a(n)=fibonacci(n+8)-21 \\ Charles R Greathouse IV, Feb 24 2017

[fibonacci(n+8)-21 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+8) - F(8) with F(n) the Fibonacci numbers A000045.
a(n) = a(n-1) + a(n-2) + 21 for n>1, a(0)=0, a(1)=13, and where 21 = F(8).
G.f.: x*(13 + 8*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
a(n) = 13*A000071(n+2) + 8*A000071(n+1). - Bruno Berselli, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-21 + (2^(-1-n)*((1-sqrt(5))^n*(-47+21*sqrt(5)) + (1+sqrt(5))^n*(47+21*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)

A180674 a(n) = Fibonacci(n+9) - Fibonacci(9).

Original entry on oeis.org

0, 21, 55, 110, 199, 343, 576, 953, 1563, 2550, 4147, 6731, 10912, 17677, 28623, 46334, 74991, 121359, 196384, 317777, 514195, 832006, 1346235, 2178275, 3524544, 5702853, 9227431, 14930318, 24157783, 39088135, 63245952, 102334121

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn18 sums of the Fibonacci(n) triangle A104763. See A180662 for information about these knight and other chess sums.

Vincenzo Librandi, Table of n, a(n) for n = 0..275
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000045, A000071.

Cf. A131524 (Kn11), A001911 (Kn12), A006327 (Kn13), A167616 (Kn14), A180671 (Kn15), A180672 (Kn16), A180673 (Kn17), A180674 (Kn18).

List([0..40], n-> Fibonacci(n+9)-34); # G. C. Greubel, Jul 13 2019

[Fibonacci(n+9) - Fibonacci(9): n in [0..40]]; // Vincenzo Librandi, Apr 24 2011

nmax:=31: with(combinat): for n from 0 to nmax do a(n):=fibonacci(n+9)-fibonacci(9) od: seq(a(n),n=0..nmax);

Fibonacci[9 +Range[0, 40]] -34 (* G. C. Greubel, Jul 13 2019 *)
LinearRecurrence[{2,0,-1},{0,21,55},40] (* Harvey P. Dale, Aug 24 2024 *)

concat(0, Vec(x*(21+13*x)/((1-x)*(1-x-x^2)) + O(x^40))) \\ Colin Barker, Feb 24 2017

a(n) = fibonacci(n+9) - fibonacci(9) \\ Charles R Greathouse IV, Feb 24 2017

[fibonacci(n+9)-34 for n in (0..40)] # G. C. Greubel, Jul 13 2019

a(n) = F(n+9) - F(9) with F = A000045.
a(n) = a(n-1) + a(n-2) + 34 for n>1, a(0)=0, a(1)=21, and where 34 = F(9).
G.f.: x*(21 + 13*x)/((1 - x)*(1 - x - x^2)). - Ilya Gutkovskiy, Feb 24 2017
a(n) = 21*A000071(n+2) + 13*A000071(n+1). - Bruno Berselli, Feb 24 2017
From Colin Barker, Feb 24 2017: (Start)
a(n) = (-34 + (2^(-n)*((1-sqrt(5))^n*(-38+17*sqrt(5)) + (1+sqrt(5))^n*(38+17*sqrt(5)))) / sqrt(5)).
a(n) = 2*a(n-1) - a(n-3) for n>2. (End)

A131410 A127647 * A000012.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144, 144

Offset: 1

Gary W. Adamson, Jul 08 2007

Row sums = A045925, n*Fib(n): (1, 2, 6, 12, 25, 48,...).

A104763 = A000012 * A127647.

			First few rows of the triangle are:
1;
1, 1;
2, 2, 2;
3, 3, 3, 3;
5, 5, 5, 5, 5;
8, 8, 8, 8, 8, 8;
...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A127647, A045925, A104763, A000045.

a131410 n k = a131410_tabl !! (n-1) !! (n-1)
a131410_row n = a131410_tabl !! (n-1)
a131410_tabl = zipWith replicate [1..] $ tail a000045_list
-- Reinhard Zumkeller, Oct 07 2012

Table[Fibonacci[n], {n, 15}, {n}] // Flatten (* Vincenzo Librandi, Jan 28 2017 *)

A127647 * A000012 as infinite lower triangular matrices.
Partial sums of A127647 starting from the right, read by rows.
By rows, F(n) occurs n times.

cp's OEIS Frontend

A180662 The Golden Triangle: T(n,k) = A001654(k) for n>=0 and 0<=k<=n.

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A027926 Triangular array T read by rows: T(n,0) = T(n,2n) = 1 for n >= 0; T(n,1) = 1 for n >= 1; T(n,k) = T(n-1,k-2) + T(n-1,k-1) for k = 2..2n-1, n >= 2.

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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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A104762 Triangle read by rows: row n contains first n nonzero Fibonacci numbers in decreasing order.

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A104765 Triangle T(n,k) read by rows: row n contains the first n Lucas numbers A000204.

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