A368311 Symmetric array read by antidiagonals: A(n,k) is the number of sums with carries i + j with abs(i) <= n and abs(j) <= k.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 3, 3, 3, 3, 3, 3, 3, 3, 1, 0, 0, 1, 3, 6, 6, 6, 6, 6, 6, 6, 3, 1, 0, 0, 1, 3, 6, 10, 10, 10, 10, 10, 10, 6, 3, 1, 0

Offset

0,69

Links

Stefano Spezia, First 150 antidiagonals of the array, flattened

David Applegate, Marc LeBrun, and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

Index entries for sequences related to carryless arithmetic

Formula

A(n,0) = A(0,n) = 0.

A(n,k) = A003991(n+1,k+1) - A368310(n,k).

Example

Array begins:
  0, 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, ...
  0, 0, 0, 0,  0,  0,  0,  0,  0,  1,  1,  1, ...
  0, 0, 0, 0,  0,  0,  0,  0,  1,  3,  3,  3, ...
  0, 0, 0, 0,  0,  0,  0,  1,  3,  6,  6,  6, ...
  0, 0, 0, 0,  0,  0,  1,  3,  6, 10, 10, 10, ...
  0, 0, 0, 0,  0,  1,  3,  6, 10, 15, 15, 15, ...
  0, 0, 0, 0,  1,  3,  6, 10, 15, 21, 21, 21, ...
  0, 0, 0, 1,  3,  6, 10, 15, 21, 28, 28, 28, ...
  0, 0, 1, 3,  6, 10, 15, 21, 28, 36, 36, 36, ...
  0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 45, 46, ...
  0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 45, 46, ...
  0, 1, 3, 6, 10, 15, 21, 28, 36, 46, 46, 47, ...
  ...
A(6,5) = 3 since there are three sums with carries having addends almost equal to 6 and 5, respectively: 5 + 5 = 10, 6 + 4 = 10, and 6 + 5 = 11.

Mathematica

len[num_]:=Length[IntegerDigits[num]]; digit[num_, d_] := Part[IntegerDigits[num], d];  B[i_, j_] := Reverse[CoefficientList[Sum[digit[i, c]*x^(len[i]-c), {c, len[i]}]+Sum[digit[j, r]*x^(len[j]-r), {r, len[j]}], x]]; F[n_, k_] := Sum[Sum[Boole[Length[Select[B[i, j], #<10 &]] == IntegerLength[Max[i, j]]], {i, 0, n}], {j, 0, k}]; A[i_, j_]:=(i+1)(j+1)-F[i, j]; Table[A[i - j, j], {i, 0, 13}, {j, 0, i}]//Flatten

Crossrefs

Cf. A003056, A003991, A059692, A169894, A368310 (carryless sums).

Sequence in context: A251552 A324497 A179804 * A102818 A010701 A290858

Adjacent sequences: A368308 A368309 A368310 * A368312 A368313 A368314

Keyword

nonn,base,look,tabl

Author

Stefano Spezia, Dec 21 2023

Status

approved