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

1,5

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

Links

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

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

Example

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; ...

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 *)

Prog

(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
(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
(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
(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

Crossrefs

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

Sequence in context: A177352 A210798 A374435 * A117915 A294453 A097094

Adjacent sequences: A117498 A117499 A117500 * A117502 A117503 A117504

Keyword

nonn,tabl

Author

Gary W. Adamson, Mar 23 2006

Extensions

Row sums comment corrected by Philippe Deléham, Nov 18 2013

Status

approved