A368310 Symmetric array read by antidiagonals: A(n,k) is the number of carryless sums i + j with abs(i) <= n and abs(j) <= k.

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 9, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 16, 15, 12, 7, 8, 14, 18, 20, 20, 18, 14, 8, 9, 16, 21, 24, 25, 24, 21, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 19, 26, 31, 34, 35, 34, 31, 26, 19, 11, 12, 21, 27, 33, 37, 39, 39, 37, 33, 27, 21, 12

Offset

0,2

Comments

A(n,k) differs from A003991(n+1,k+1) starting at the second term of the 11th antidiagonal: A(9,1) = 19 <> A003991(10,2) = 20.

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,k) = A003991(n+1,k+1) for n + k < 10.

A(n,0) = A(0,n) = n + 1.

A(n,k) = A003991(n+1,k+1) - A368311(n,k).

Example

Array begins:
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, ...
   2,  4,  6,  8, 10, 12, 14, 16, 18, 19, 21, ...
   3,  6,  9, 12, 15, 18, 21, 24, 26, 27, 30, ...
   4,  8, 12, 16, 20, 24, 28, 31, 33, 34, 38, ...
   5, 10, 15, 20, 25, 30, 34, 37, 39, 40, 45, ...
   6, 12, 18, 24, 30, 35, 39, 42, 44, 45, 51, ...
   7, 14, 21, 28, 34, 39, 43, 46, 48, 49, 56, ...
   8, 16, 24, 31, 37, 42, 46, 49, 51, 52, 60, ...
   9, 18, 26, 33, 39, 44, 48, 51, 53, 54, 63, ...
  10, 19, 27, 34, 40, 45, 49, 52, 54, 55, 65, ...
  11, 21, 30, 38, 45, 51, 56, 60, 63, 65, 76, ...
  ...
A(6,5) = A003991(7,6) - A368311(6,5) = (6 + 1)*(5 + 1) - 3 = 39 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]]; A[n_, k_] := Sum[Sum[Boole[Length[Select[B[i, j], #<10 &]] == IntegerLength[Max[i, j]]], {i, 0, n}], {j, 0, k}]; Table[A[i - j, j], {i, 0, 11}, {j, 0, i}]//Flatten

Crossrefs

Cf. A003056, A003991, A059692, A169894, A368311 (sums with carries).

Sequence in context: A091257 A216622 A319840 * A003991 A131923 A119457

Adjacent sequences: A368307 A368308 A368309 * A368311 A368312 A368313

Keyword

nonn,base,look,tabl

Author

Stefano Spezia, Dec 21 2023

Status

approved