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A117501 Triangle generated from an array of generalized Fibonacci-like terms. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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: A327035 A177352 A210798 * 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

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Last modified April 18 22:45 EDT 2021. Contains 343098 sequences. (Running on oeis4.)