A178534 Triangle T(n,k) read by rows. T(n,1) = A000045(n+1), k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) - (Sum_{i=1..k-1} T(n-i,k)).

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 8, 3, 1, 1, 1, 13, 5, 3, 1, 1, 1, 21, 8, 4, 2, 1, 1, 1, 34, 13, 6, 4, 2, 1, 1, 1, 55, 21, 11, 6, 3, 2, 1, 1, 1, 89, 34, 17, 9, 6, 3, 2, 1, 1, 1, 144, 55, 27, 15, 9, 5, 3, 2, 1, 1, 1, 233, 89, 45, 25, 14, 9, 5, 3, 2, 1, 1, 1, 377, 144, 72, 40, 23, 14, 8, 5, 3, 2, 1, 1, 1

Offset: 1

Author

Mats Granvik, May 29 2010

Keywords

• nonn
• tabl

Examples

			Table begins:
   1;
   2,  1;
   3,  1,  1;
   5,  2,  1,  1;
   8,  3,  1,  1,  1;
  13,  5,  3,  1,  1,  1;
  21,  8,  4,  2,  1,  1,  1;
  34, 13,  6,  4,  2,  1,  1,  1;
  55, 21, 11,  6,  3,  2,  1,  1,  1;
  89, 34, 17,  9,  6,  3,  2,  1,  1,  1;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Cf. 1st column=A000045(n+1), 2nd=A000045, 3rd=A093040, 4th=A006498. Matrix inverse of A178535.

Cf. A051731, A129713.

Programs

Maple

A178534 := proc(n, k)
    option remember;
    if k= 1 then
        combinat[fibonacci](n+1) ;
    elif k > n then
        0 ;
    else
        add(procname(n-i, k-1), i=1..k-1)-add(procname(n-i, k), i=1..k-1) ;
    end if;
end proc:
seq(seq(A178534(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Oct 28 2010

Mathematica

T[n_, 1] := Fibonacci[n+1];
T[n_, k_] := T[n, k] = If[k > n, 0, Sum[T[n-i, k-1], {i, 1, k-1}] - Sum[T[n-i, k], {i, 1, k-1}]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 23 2024 *)

PARI

T(n,k)=(n % k==0) + sum(j=1,n\k,fibonacci(n-j*k)) \\ Andrew Howroyd, Feb 23 2024

Python

from sympy.core.cache import cacheit
from sympy import fibonacci
@cacheit
def A(n, k): return fibonacci(n + 1) if k==1 else 0 if k>n else sum([A(n - i, k - 1) for i in range(1, k)]) - sum([A(n - i, k) for i in range(1, k)])
for n in range(1, 13): print([A(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Sep 15 2017

Formula

T(n,1) = A000045(n+1), k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) - Sum_{i=1..k-1} T(n-i,k).

T(n,k) = A129713*A051731. - Mats Granvik, Oct 22 2010

From R. J. Mathar, Sep 16 2017: (Start)

G.f. 3rd column: x^3*(1+x)/((1-x-x^2)*(1+x+x^2)).

G.f. 4th column: x^4/((1-x-x^2)*(1+x^2)) =x^4*(1+x)/((1-x-x^2)*(1+x+x^2+x^3)).

G.f. 5th column: x^5*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4)).

G.f. 6th column: x^6/((1-x-x^2)*(1+x+x^2)*(1-x+x^2)) = x^6*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5)).

G.f. 7th column: x^7*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5+x^6)).

G.f. 8th column: x^8/((1-x-x^2)*(1+x^2)*(1+x^4)) = x^8*(1+x)/((1-x-x^2)*(1+x+x^2+x^3+x^4+x^5+x^6+x^7)).

Conjecture (by extrapolating): G.f. k-th column: x^k*(1-x^2)/((1-x-x^2)*(1-x^k)).

G.f.: (1-x^2)/(1-x-x^2)*Sum_{i>=1} (x*y)^i/(1-x^i) = (1-x^2)/(1-x-x^2)*A051731(x,y). (End)

T(n,k) = A051731(n,k) + Sum_{j=1..floor(n/k)} Fibonacci(n-j*k). - Andrew Howroyd, Feb 23 2024