A117501 Triangle generated from an array of generalized Fibonacci-like terms.

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

Offset: 1

Author

Gary W. Adamson, Mar 23 2006

Keywords

• nonn
• tabl

Comments

Difference terms of the array columns in triangle format becomes A117502.

Row sums of the triangle are A104161: (1, 2, 5, 10, 19, 34, 59, ...), generated by a(k) = a(k-1) + a(k-2) + n.

This is the lower triangular version of A109754 (without a row and column 0). - Ross La Haye, Apr 12 2006

Examples

			First few rows of the array T(n,k) are:
       k=1 k=2 k=3 k=4 k=5 k=6
  n=1:  1,  1,  2,  3,  5,  8, ...
  n=2:  1,  2,  3,  5,  8, 13, ...
  n=3:  1,  3,  4,  7, 11, 18, ...
  n=4:  1,  4,  5,  9, 14, 23, ...
  n=5:  1,  5,  6, 11, 17, 28, ...
First few rows of the triangle are:
  1;
  1, 1;
  1, 2, 2;
  1, 3, 3,  3;
  1, 4, 4,  5,  5;
  1, 5, 5,  7,  8,  8;
  1, 6, 6,  9, 11, 13, 13;
  1, 7, 7, 11, 14, 18, 21, 21; ...
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A000045, A001595, A104161 (diagonal sums), A109754 (with column of 0's), A117502.

Programs

GAP

F:=Fibonacci;; Flat(List([1..15], n-> List([1..n], k-> (n-k+1)*F(k-1) + F(k-2) ))); # G. C. Greubel, Jul 13 2019

Magma

F:=Fibonacci; [(n-k+1)*F(k-1) + F(k-2): k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 13 2019

Mathematica

a[n_, k_] := a[n, k] = If[k==1, 1, If[k==2, n, a[n, k-1] + a[n, k-2]]]; Table[a[n-k+1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 15 2017 *)
T[n_, k_]:= n*Fibonacci[k-1] + Fibonacci[k-2]; Table[T[n-k+1, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jul 13 2019 *)

PARI

T(n,k) = n*fibonacci(k-1) + fibonacci(k-2);
for(n=1,15, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Jul 13 2019

Python

from sympy.core.cache import cacheit
@cacheit
def a(n, k):
    return 1 if k==1 else n if k==2 else a(n, k - 1) + a(n, k - 2)
for n in range(1, 21): print([a(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, Aug 19 2017

Sage

f=fibonacci; [[(n-k+1)*f(k-1) + f(k-2) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 13 2019

Formula

The triangle by rows = antidiagonals of an array in which n-th row is generated by a Fibonacci-like operation: (1, n...then a(k+1) = a(k) + a(k-1)).

T(n,k) = n*Fibonacci(k-1) + Fibonacci(k-2). - G. C. Greubel, Jul 13 2019

Extensions

Row sums comment corrected by Philippe Deléham, Nov 18 2013