A104762 -id:A104762 - cp's OEIS frontend

A155002 Triangle read by rows, A104762 * (A000129 * 0^(n-k)).

1, 1, 1, 2, 1, 2, 3, 2, 2, 5, 5, 3, 4, 5, 12, 8, 5, 6, 10, 12, 29, 13, 8, 10, 15, 24, 29, 70, 21, 13, 16, 25, 36, 58, 70, 169, 34, 21, 26, 40, 60, 87, 140, 169, 408, 55, 34, 42, 65, 96, 145, 210, 338, 408, 985

Offset: 1

Gary W. Adamson & Roger L. Bagula, Jan 18 2009

Eigentriangle, row sums = rightmost term of next row.

Row sums = the Pell series starting with offset 1: (1, 2, 5, 12, 29, ...).

			First ten rows of the triangle T(n, m):
n \ m   1  2  3  4  5   6   7   8   9  10 ...
1:      1
2:      1  1
3:      2  1  2
4:      3  2  2  5
5:      5  3  4  5 12
6:      8  5  6 10 12  29
7:     13  8 10 15 24  29  70
8:     21 13 16 25 36  58  70 169
9:     34 21 26 40 60  87 140 169 408
10:    55 34 42 65 96 145 210 338 408 985
... reformatted by - _Wolfdieter Lang_, Apr 13 2021
Row 4 = (3, 2, 2, 5) = termwise products of (3, 2, 1, 1) and (1, 1, 2, 5).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Cf. A104762, A000045, A000129, A215928.

Triangle read by rows, A104762 * (A000129 * 0^(n-k)).
A104762 = Fibonacci numbers "decrescendo", (1, 1, 2, 3, 5, ...) in every column.
(A000129 * 0^(n-k)) ) = the Pell series prefaced with a 1:
(1, 1, 2, 5, 12, ...) as the main diagonal and the rest zeros
From Wolfdieter Lang, Apr 13 2021: (Start)
T(n, m) = F(n+1-m)*A215928(m), with F = A000045, for n >= m >= 1, and 0 otherwise.
The lower triangular (infinite) matrix t with elements t(n, m) = T(n+1, m+1), for n >= m >= 0, and 0 otherwise, has row polynomials R(n, x) = Sum_{m=0..n} t(n, m)*x^m with o.g.f. G(z, x) = A(z)/(1 - x*z*A(x*z)) =
(1 - x*z - (x*z)^2)/((1 - z - z^2)*(1 - 2*x*z - (x*z)^2)), with the o.g.f. A(x) of (F_{n+1})_{n>=0}, where F = A000045.
The infinite dimensional lower triangular Riordan matrix TB := (1/(1 - x - x^2), x) (a Toeplitz matrix) with nonzero elements A104762(n+1, m+1) has sequence (A215928(m)){m >=0} as 'L-eigen-sequence' (cf. the Bernstein-Sloane link for 'eigen-sequence'). This means that (TB - L)*vec(B) = 0-matrix, where L has elements L(i, j) = delta{i, j-1} (first upper diagonal with 1s, otherwise 0), and the infinite vector vec(B) has the elements of A215928.
Thanks to Gary W. Adamson for motivating me to look at such triangles and sequences. (End)

A104763 Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 23 2005

Triangle of A104762, Fibonacci sequence in each row starts from the right.
The triangle or chess sums, see A180662 for their definitions, link the Fibonacci(n) triangle to sixteen different sequences, see the crossrefs. The knight sums Kn14 - Kn18 have been added. As could be expected all sums are related to the Fibonacci numbers. - Johannes W. Meijer, Sep 22 2010
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 A104763 is reluctant sequence of Fibonacci numbers (A000045), except 0. - Boris Putievskiy, Dec 13 2012

			First few rows of the triangle are:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3;
  1, 1, 2, 3, 5;
  1, 1, 2, 3, 5, 8;
  1, 1, 2, 3, 5, 8, 13; ...

Reinhard Zumkeller, Rows n = 1..100 of table, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A104762, A000045.
Cf. A000071 (row sums). - R. J. Mathar, Jul 22 2009
Triangle sums (see the comments): A000071 (Row1; Kn4 & Ca1 & Ca4 & Gi1 & Gi4); A008346 (Row2); A131524 (Kn11); A001911 (Kn12); A006327 (Kn13); A167616 (Kn14); A180671 (Kn15); A180672 (Kn16); A180673 (Kn17); A180674 (Kn18); A052952 (Kn21 & Kn22 & Kn23 & Fi2 & Ze2); A001906 (Kn3 &Fi1 & Ze3); A004695 (Ca2 & Ze4); A001076 (Ca3 & Ze1); A080239 (Gi2); A081016 (Gi3). - Johannes W. Meijer, Sep 22 2010

Flat(List([1..15], n -> List([1..n], k -> Fibonacci(k)))); # G. C. Greubel, Jul 13 2019

a104763 n k = a104763_tabl !! (n-1) !! (k-1)
a104763_row n = a104763_tabl !! (n-1)
a104763_tabl = map (flip take $ tail a000045_list) [1..]
-- Reinhard Zumkeller, Aug 15 2013

[Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 13 2019

Table[Fibonacci[k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jul 13 2019 *)

for(n=1,15, for(k=1,n, print1(fibonacci(k), ", "))) \\ G. C. Greubel, Jul 13 2019

[[fibonacci(k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 13 2019

F(1) through F(n) starting from the left in n-th row.
T(n,k) = A000045(k), 1<=k<=n. - R. J. Mathar, May 02 2008
a(n) = A000045(m), where m= n-t(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 13 2012
G.f.: (x*y)/((x-1)*(x^2*y^2+x*y-1)). - Vladimir Kruchinin, Jun 21 2025

Edited by R. J. Mathar, May 02 2008

Extended by R. J. Mathar, Aug 27 2008

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

A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 5, 2, 1, 8, 3, 1, 13, 5, 2, 21, 8, 3, 1, 34, 13, 5, 2, 1, 55, 21, 8, 3, 1, 89, 34, 13, 5, 2, 1, 144, 55, 21, 8, 3, 1, 233, 89, 34, 13, 5, 2, 1, 377, 144, 55, 21, 8, 3, 1, 610, 233, 89, 34, 13, 5, 2, 1

Offset: 0

Gary W. Adamson, Feb 14 2010

The row sums equal A052952.
Let the triangle = M. Then lim_{n->infinity} M^n = A173285 as a left-shifted vector.
A173284 * [1, 2, 3, ...] = A054451: (1, 1, 4, 5, 12, 17, 33, ...). - Gary W. Adamson, Mar 03 2010
From Johannes W. Meijer, Sep 05 2013: (Start)
Triangle read by rows formed from antidiagonals of triangle A104762.
The diagonal sums lead to A004695. (End)

			First few rows of the triangle:
    1;
    1;
    2,   1;
    3,   1;
    5,   2,  1;
    8,   3,  1;
   13,   5,  2,  1;
   21,   8,  3,  1;
   34,  13,  5,  2,  1;
   55,  21,  8,  3,  1;
   89,  34, 13,  5,  2, 1;
  144,  55, 21,  8,  3, 1;
  233,  89, 34, 13,  5, 2, 1;
  377, 144, 55, 21,  8, 3, 1;
  610, 233, 89, 34, 13, 5, 2, 1;
  ...

Cf. A000045, A004695, A052952, A054451, A173285.

Cf. (Similar triangles) A008315 (Catalan), A011973 (Pascal), A102541 (Losanitsch), A122196 (Fractal), A122197 (Fractal), A128099 (Pell-Jacobsthal), A152198, A152204, A207538, A209634.

T := proc(n, k): if n<0 then return(0) elif k < 0 or k > floor(n/2) then return(0) else combinat[fibonacci](n-2*k+1) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..14); # Johannes W. Meijer, Sep 05 2013

Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.
From Johannes W. Meijer, Sep 05 2013: (Start)
T(n,k) = A000045(n-2*k+1), n >= 0 and 0 <= k <= floor(n/2).
T(n,k) = A104762(n-k, k). (End)

Term a(15) corrected by Johannes W. Meijer, Sep 05 2013

A104796 Triangle read by rows: T(n,k) = (n+1-k)*Fibonacci(n+2-k), for n>=1, 1<=k<=n.

Original entry on oeis.org

1, 4, 1, 9, 4, 1, 20, 9, 4, 1, 40, 20, 9, 4, 1, 78, 40, 20, 9, 4, 1, 147, 78, 40, 20, 9, 4, 1, 272, 147, 78, 40, 20, 9, 4, 1, 495, 272, 147, 78, 40, 20, 9, 4, 1, 890, 495, 272, 147, 78, 40, 20, 9, 4, 1, 1584, 890, 495, 272, 147, 78, 40, 20, 9, 4, 1, 2796, 1584, 890, 495, 272

Offset: 1

Gary W. Adamson, Mar 26 2005

The first column is A023607 (without the leading zero).

			Rows 1,2,3,4,5,6 and columns 1,2,3,4,5,6 of the triangle are:
1;
4, 1;
9, 4, 1;
20, 9, 4, 1;
40, 20, 9, 4, 1;
78, 40, 20, 9, 4, 1;
...
Row 3 for example is 3*F(4), 2*F(3), 1*F(2) = 3*3, 2*2, 1*1 = 9, 4, 1.
Row 4 is 4*F(5), 3*F(4), 2*F(3), 1*F(2) = 4*5, 3*3, 2*2, 1*1 = 20, 9, 4, 1.
Reading the rows backwards gives an initial segment of the terms of A023607 (but without the initial zero).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Row sums are in A094584.

Cf. A023607, A104762, A104765, A000045.

Table[(n+1-k)Fibonacci[n+2-k],{n,20},{k,n}]//Flatten (* Harvey P. Dale, Sep 24 2020 *)
Module[{nn=20,c},c=LinearRecurrence[{2,1,-2,-1},{1,4,9,20},nn];Table[ Reverse[ Take[c,n]],{n,nn}]]//Flatten (* Harvey P. Dale, Sep 25 2020 *)

Edited by Ralf Stephan, Apr 05 2009

Entry revised by N. J. A. Sloane, Sep 23 2020

A121461 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having last ascent of length k (1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 8, 3, 1, 1, 21, 8, 3, 1, 1, 55, 21, 8, 3, 1, 1, 144, 55, 21, 8, 3, 1, 1, 377, 144, 55, 21, 8, 3, 1, 1, 987, 377, 144, 55, 21, 8, 3, 1, 1, 2584, 987, 377, 144, 55, 21, 8, 3, 1, 1, 6765, 2584, 987, 377, 144, 55, 21, 8, 3, 1, 1, 17711, 6765, 2584, 987, 377, 144, 55, 21

Offset: 1

Emeric Deutsch, Jul 31 2006

Also the number of directed column-convex polyominoes of area n, having k cells in the last column. Row sums are the odd-subscripted Fibonacci numbers (A001519). Sum_{k=1..n} k*T(n,k) = Fibonacci(2n) = A001906(n).
Riordan array ((1-2*x+x^2)/(1-3*x+x^2), x). - Philippe Deléham, Oct 04 2014
Antidiagonal sums are in A007598. - Philippe Deléham, May 22 2015

			T(4,2)=3 because we have UUDD(UU)DD, UUD(UU)DDD and UDUD(UU)DD, where U=(1,1) and D=(1,-1) (the last ascents are shown between parentheses).
Triangle starts:
   1;
   1,  1;
   3,  1, 1;
   8,  3, 1, 1;
  21,  8, 3, 1, 1;
  55, 21, 8, 3, 1, 1;
  ...

E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
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.
A. M. Baxter, L. K. Pudwell, Ascent sequences avoiding pairs of patterns, 2014.
E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.

Cf. A001519, A001906, A104762, A088305.

Cf. A000045, A007598.

Maple
```
with(combinat): T:=proc(n,k) if k
				
```

T(n,k) = Fibonacci(2(n-k)) if k < n; T(n,n)=1.
G.f.: G = G(t,z) = t*z*(1-z)^2/((1-3z+z^2)*(1-tz)).
From Gary W. Adamson, Jul 07 2011: (Start)
Let M be the production matrix:
  1, 1, 0, 0, 0, 0, ...
  2, 0, 1, 0, 0, 0, ...
  3, 0, 0, 1, 0, 0, ...
  4, 0, 0, 0, 1, 0, ...
  5, 0, 0, 0, 0, 1, ...
  ...
n-th row of triangle A121461 = top row terms of (n-1)-th power of M. (End)
Let P denote Pascal's triangle. Then P^(-1)*A121461*P = A104762. - Peter Bala, Apr 11 2013

A271355 Triangular array: T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 7, 4, 3, 2, 1, 11, 7, 4, 3, 2, 1, 18, 11, 7, 4, 3, 2, 1, 29, 18, 11, 7, 4, 3, 2, 1, 47, 29, 18, 11, 7, 4, 3, 2, 1, 76, 47, 29, 18, 11, 7, 4, 3, 2, 1, 123, 76, 47, 29, 18, 11, 7, 4, 3, 2, 1, 199, 123, 76, 47, 29, 18, 11, 7, 4, 3

Offset: 1

Clark Kimberling, May 01 2016

Row n consists of the first n numbers of A169985 = (1,2,3,4,7,... ) in reverse order; these are the Lucas numbers, A000032, with order of initial two terms reversed. Every column of the triangle is A169985.

			First six rows:
  1
  2   1
  3   2   1
  4   3   2   1
  7   4   3   2   1
  11  7   4   3   2   1

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A169985, A000032, A000045, A104762

r = N[(1 + Sqrt[5])/2, 100]; s = N[(1 - Sqrt[5])/2, 100];
t = Table[Abs[Round[(r^n)*(s^k)]], {n, 0, 15}, {k, 1, n}];
Flatten[t]  (* A271355, sequence *)
TableForm[t]  (* A271355, array *)

T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.

T(k+j-1,j) = A000032(k) = k-th Lucas number, for k >= 2.

A104733 Triangle T(n,k) = sum_{j=k..n} Fibonacci(n-j+1)*Fibonacci(k+1), read by rows, 0<=k<=n.

Original entry on oeis.org

1, 2, 1, 4, 2, 2, 7, 4, 4, 3, 12, 7, 8, 6, 5, 20, 12, 14, 12, 10, 8, 33, 20, 24, 21, 20, 16, 13, 54, 33, 40, 36, 35, 32, 26, 21, 88, 54, 66, 60, 60, 56, 52, 42, 34, 143, 88, 108, 99, 100, 96, 91, 84, 68, 55, 232, 143, 176, 162, 165, 160, 156, 147, 136, 110, 89

Offset: 0

Gary W. Adamson, Mar 20 2005

			The first few rows of the triangle are:
1;
2, 1;
4, 2, 2;
7, 4, 4, 3;
12, 7, 8, 6, 5;
20, 12, 14, 12, 10, 8

Cf. A000071 (1st and 2nd column), A019274 (3rd column)

Matrix product of T(n,k) = sum_j A104762(n+1,j)*A104763(j+1,k), both interpreted as lower triangular square arrays.

Incorrect conjecture on row sums removed. R. J. Mathar, Sep 17 2013

A104766 Triangle T(n,k) = A001629(n-k+2) read by rows, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 10, 5, 2, 1, 20, 10, 5, 2, 1, 38, 20, 10, 5, 2, 1, 71, 38, 20, 10, 5, 2, 1, 130, 71, 38, 20, 10, 5, 2, 1

Offset: 1

Gary W. Adamson, Mar 24 2005

The triangle is the matrix square of the triangle A104762: T(n,k) = sum_{j= k..n} A104762(n,j)*A104762(j,k).

			First few rows of the triangle:
1;
2, 1;
5, 2, 1;
10, 5, 2, 1;
20, 10, 5, 2, 1;
38, 20, 10, 5, 2, 1;
71, 38, 20, 10, 5, 2, 1;
...

Cf. A001629, A104762, A104763, A006478 (row sums).

cp's OEIS Frontend

A155002 Triangle read by rows, A104762 * (A000129 * 0^(n-k)).

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A104763 Triangle read by rows: Fibonacci(1), Fibonacci(2), ..., Fibonacci(n) in row n.

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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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A173284 Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.

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A104796 Triangle read by rows: T(n,k) = (n+1-k)*Fibonacci(n+2-k), for n>=1, 1<=k<=n.

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A121461 Triangle read by rows: T(n,k) is the number of nondecreasing Dyck paths of semilength n, having last ascent of length k (1 <= k <= n).

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A271355 Triangular array: T(n,k) = |round((r^n)*(s^k))|, where r = golden ratio = (1+sqrt(5))/2, s = (1-sqrt(5))/2, 1 <= k <= n, n >= 1.

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