A178535 -id:A178535 - cp's OEIS frontend

A178536 First column of A178535.

1, -2, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, -1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, -1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 1, -1, -1, 0, -1, 1, 0, 0, 1, -1, -1, 0, 0, 1, -1, 0, 1, 1, 1, 0, -1, 0, 1, 0

Offset: 1

Mats Granvik, May 29 2010

sign

Except for the second term, a(n) seems to be equal to the Mobius function mu(n) = A008683(n) (verified for the first 53 terms).

a(n) = A008683(n) has now been verified for 3 <= n <= 800. - R. J. Mathar, Sep 14 2017

Antti Karttunen, Table of n, a(n) for n = 1..800 (prepared from the b-file of A008683 based on _R. J. Mathar_'s Sep 14 2017 comment)

Cf. A178535, A008683.

Cf. also A181434, A181435.

A178536 := proc(n) A178535(n,1) ; end proc;
seq(A178536(n),n=1..80) ; # R. J. Mathar, Oct 28 2010

Clear[t, n, k, nn, a, A]; nn=92; a = Fibonacci[Range[nn] + 1]; t[n_, 1] = If[n >= 1, a[[n]], 0]; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}] - Sum[t[n - i, k], {i, 1, k - 1}], 0]; MatrixForm[A = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]; Inverse[A][[All, 1]] (* Mats Granvik, Sep 15 2017 *)

a(n) = A178535(n,1).

a(n) = Sum_{k|n} A008683(n/k)*([k = 1] - [2|k]) (conjecture). - Mats Granvik, Jan 24 2021

More terms from R. J. Mathar, Oct 28 2010

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)).

Original entry on oeis.org

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

Mats Granvik, May 29 2010

			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;

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

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

Cf. A051731, A129713.

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

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 *)

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

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

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

cp's OEIS Frontend

A178536 First column of A178535.

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