A292030 Table read by ascending antidiagonals: T(n,k) = A000045(k+1)*n + A000045(k).

0, 1, 1, 2, 2, 1, 3, 3, 3, 2, 4, 4, 5, 5, 3, 5, 5, 7, 8, 8, 5, 6, 6, 9, 11, 13, 13, 8, 7, 7, 11, 14, 18, 21, 21, 13, 8, 8, 13, 17, 23, 29, 34, 34, 21, 9, 9, 15, 20, 28, 37, 47, 55, 55, 34, 10, 10, 17, 23, 33, 45, 60, 76, 89, 89, 55, 11, 11, 19, 26, 38, 53, 73, 97, 123, 144, 144, 89

Offset: 0

Author

Ely Golden, Sep 07 2017

Keywords

• nonn
• tabl

Comments

T(n,k) is entry k in the Fibonacci-like sequence which starts n, n+1, i.e., T(n,0) = n, T(n,1) = n+1, T(n,k) = T(n,k-1) + T(n,k-2).

Therefore, T(0,k), T(1,k), and T(2,k) are equivalent to A000045(k), A000045(k+2), and A000045(k+3) respectively.

Any two rows T(x,.) and T(y,.) contain infinitely many differing terms for any values of x and y, provided that max(x,y) >= 3.

Column k is the list of all positive integers congruent to A000045(k) mod A000045(k+1), with the exception of row 0 which is the same as row 1 with 0 included.

Examples

			T(4,2) = 9 because 9 is element 2 in the Fibonacci-like sequence starting with 4 and 5.
The array begins:
  0, 1,  1,  2,  3,  5,  8, ...
  1, 2,  3,  5,  8, 13, 21, ...
  2, 3,  5,  8, 13, 21, 34, ...
  3, 4,  7, 11, 18, 29, 47, ...
  4, 5,  9, 14, 23, 37, 60, ...
  5, 6, 11, 17, 28, 45, 73, ...
  6, 7, 13, 20, 33, 53, 86, ...
		

Links

• Ely Golden, Table of n, a(n) for n = 0..10000(contains 140 antidiagonals, flattened).
• Ely Golden, List of ordered pairs (j,k) such that T(j,k)=n for n = 0..10000

Crossrefs

Cf. A000045, A292031, A292032.

A023548 gives the antidiagonal sums.

Programs

Mathematica

T[n_,k_]:=SeriesCoefficient[(x+y-x*y)/((1-x)^2(1-y-y^2)),{x,0,n},{y,0,k}]; Table[T[n-k,k],{n,0,11},{k,0,n}]//Flatten (* Stefano Spezia, May 21 2025 *)

PARI

T(n, k) = fibonacci(k+1)*n + fibonacci(k);
tabl(nn) = matrix(nn+1,nn+1, i,j, T(i-1,j-1)); \\ Michel Marcus, Sep 27 2017

Python

def nextAntidiagonal(d):
  if(d[0]==0): return [d[1]+1,0]
  return [d[0]-1,d[1]+1]
def fibonacciM(m,n):
  f0,f1=0,1
  if(n<0):
    for _ in range(-n): f1,f0=f0,f1-f0
  else:
    for _ in range(n): f0,f1=f1,f0+f1
  return f1*m+f0
d=[0,0]
for i in range(10001):
  print(str(i)+" "+str(fibonacciM(d[0],d[1])))
  d=nextAntidiagonal(d)

Formula

G.f.: (x + y - x*y)/((1 - x)^2*(1 - y - y^2)). - Stefano Spezia, May 21 2025