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
Examples
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.
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
Programs
-
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