A168030 Variant of pendular triangle A118340.

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0

Offset

0,1

Comments

Replaced the sums (f(a,b) = a + b) by the operators f(a,b) = a^2 -a*b + b^2 in the construction of triangle in A118340.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

From G. C. Greubel, Jan 12 2023: (Start)

T(n, k) = A118340(n, k) mod 2.

Sum_{k=0..n} T(n, k) = A168148(n). (End)

Example

Triangle begins as:
  1;
  1, 0;
  1, 1, 0;
  1, 0, 1, 0;
  1, 1, 0, 1, 0;
  1, 0, 1, 1, 1, 0;
  1, 1, 1, 0, 0, 1, 0;
  1, 0, 0, 0, 0, 1, 1, 0;
  1, 1, 0, 1, 1, 1, 0, 1, 0;
  1, 0, 1, 0, 0, 1, 1, 1, 1, 0;
  1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0;

Mathematica

t[n_, k_, p_]:= t[n, k, p]= If[k<0 || k>n, 0, If[k==0, 1, If[n<=2*k, t[n, n-k-1, p] +p*t[n-1, k, p], t[n, n-k, p] +t[n-1, k, p]]]]; (* A118340 *)
T[n_, k_, p_]:= Mod[t[n, k, p], 2]; (* A168030 *)
Table[T[n, k, 1], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)

Prog

(Magma)
function t(n, k) // t = A118340
  if k lt 0 or k gt n then return 0;
  elif k eq 0 then return 1;
  elif n gt 2*k then return t(n, n-k) + t(n-1, k);
  else return t(n, n-k-1) + t(n-1, k);
  end if; return t;
end function;
T:= func< n, k | t(n, k) mod 2 >; // A168030
[T(n, k): k in [0..n], n in [0..15]];
(SageMath)
@CachedFunction
def t(n, k): # t = A118340
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (n>2*k): return t(n, n-k) + t(n-1, k)
    else: return t(n, n-k-1) + t(n-1, k)
def A168030(n, k): return t(n, k)%2
flatten([[A168030(n, k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jan 12 2023

Crossrefs

Cf. A118340, A168148 (row sums).

Sequence in context: A190233 A143242 A136442 * A128431 A290452 A215200

Adjacent sequences: A168027 A168028 A168029 * A168031 A168032 A168033

Keyword

nonn,tabl

Author

Philippe Deléham, Nov 17 2009

Status

approved