A186518 Triangle A177517 mod 2. Mahonian numbers modulo 2.

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

Offset

1

Comments

Each column is a palindrome. The left half of the triangle displays chaos, the right half consists of triangles filled with zeros. Pattern is similar to patterns found in elementary cellular automata. The rule is different.

Compare the recurrence for this triangle:

T(1,1)=1, n > 1: T(n,1)=0, k > 1: T(n,k) = (Sum_{i=1..k-1} of T(n-i,k-1)) mod 2

with the recurrence for Dirichlet convolutions:

T(n,1)=1, k > 1: T(n,k) = ((Sum_{i=1..k-1} T(n-i,k-1)) mod 2) - ((Sum_{i=1..k-1} T(n-i,k)) mod 2) which in turn gives triangle A051731.

Links

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

Mats Granvik, Illustration of terms

Formula

T(1,1)=1, n > 1: T(n,1)=0, k > 1: T(n,k) = (Sum_{i=1..k-1} T(n-i,k-1)) mod 2.

Example

Triangle starts:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 0, 0, 0, 1;
  0, 0, 0, 0, 1, 1;
  0, 0, 0, 1, 1, 0, 1;
  0, 0, 0, 0, 0, 1, 1, 1;
  0, 0, 0, 0, 1, 1, 0, 0, 1;
  0, 0, 0, 0, 1, 0, 1, 0, 1, 1;
  0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1;

Maple

T:= proc(n, k) option remember;
      if k=n then 1
    else `mod`( add(T(n-j, k-1), j=1..k-1), 2)
      fi; end:
seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Dec 17 2019

Mathematica

nn = 19; t[n_, 1] = If[n==1, 1, 0]; t[n_, k_]:= t[n, k] = If[n>= k, Mod[Sum[t[n-i, k-1], {i, 1, k-1}], 2], 0]; Flatten[Table[t[n, k], {n, nn}, {k, n}]] (* Mats Granvik, Dec 06 2019 *)

Prog

(Excel cell formula, European) =if(column()=1; if(row()=1; 1; 0); if(row()>=column(); mod(sum(indirect(address(row()-column()+1; column()-1; 4)&":"&address(row()-1; column()-1; 4); 4)); 2); 0))
(Excel cell formula, American) =if(column()=1, if(row()=1, 1, 0), if(row()>=column(), mod(sum(indirect(address(row()-column()+1, column()-1, 4)&":"&address(row()-1, column()-1, 4), 4)), 2), 0))
(PARI) T(n, k) = if(k<1 || k>n, 0, if(k==n, 1, sum(j=1, k-1, T(n-j, k-1))%2 ));
for(n=1, 12, for(k=1, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Dec 17 2019
(Magma)
function T(n, k)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  elif k eq 1 then return 0;
  else return (&+[T(n-j, k-1): j in [1..k-1]]) mod 2;
  end if; return T; end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Dec 17 2019
(Sage)
@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (k==1): return 0
    else: return mod( sum(T(n-j, k-1) for j in (1..k-1)), 2)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Dec 17 2019
(GAP)
T:= function(n, k)
    if k<0 or k>n then return 0;
    elif k=n then return 1;
    elif k=1 then return 0;
    else return Sum([1..k-1], j-> T(n-j, k-1)) mod 2;
    fi;
  end;
Flat(List([1..12], n-> List([1..n], k-> T(n, k) ))); # G. C. Greubel, Dec 17 2019

Crossrefs

Cf. A177517, A008302, A186519 (row sums).

Sequence in context: A173857 A114482 A365799 * A127829 A127831 A164364

Adjacent sequences: A186515 A186516 A186517 * A186519 A186520 A186521

Keyword

nonn,tabl

Author

Mats Granvik, Feb 23 2011

Status

approved