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

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

Author

Gary W. Adamson, Mar 24 2005

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

			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;
  ...
		

Links

• 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.

Crossrefs

Cf. A000204, A104765, A104762, A104763.

Cf. A027961 (row sums).

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

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