A087061 Array A(n, k) = lunar sum n + k (n >= 0, k >= 0) read by antidiagonals.
0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 4, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 11, 9, 8, 7, 6, 6, 7, 8, 9, 11, 11, 12, 11, 12, 9, 8, 7, 6, 7, 8, 9, 12, 11, 12, 13, 12, 12, 13, 9, 8
Offset: 0
Examples
Lunar addition table A(n, k) begins:
[0] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 ...
[1] 1 1 2 3 4 5 6 7 8 9 11 11 12 13 ...
[2] 2 2 2 3 4 5 6 7 8 9 12 12 12 13 ...
[3] 3 3 3 3 4 5 6 7 8 9 13 13 13 13 ...
[4] 4 4 4 4 4 5 6 7 8 9 14 14 14 14 ...
[5] 5 5 5 5 5 5 6 7 8 9 15 15 15 15 ...
[6] 6 6 6 6 6 6 6 7 8 9 16 16 16 16 ...
[7] 7 7 7 7 7 7 7 7 8 9 17 17 17 17 ...
[8] 8 8 8 8 8 8 8 8 8 9 18 18 18 18 ...
[9] 9 9 9 9 9 9 9 9 9 9 19 19 19 19 ...
...
Seen as a triangle T(n, k):
[0] 0;
[1] 1, 1;
[2] 2, 1, 2;
[3] 3, 2, 2, 3;
[4] 4, 3, 2, 3, 4;
[5] 5, 4, 3, 3, 4, 5;
[6] 6, 5, 4, 3, 4, 5, 6;
[7] 7, 6, 5, 4, 4, 5, 6, 7;
[8] 8, 7, 6, 5, 4, 5, 6, 7, 8;
[9] 9, 8, 7, 6, 5, 5, 6, 7, 8, 9;
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10010
- D. Applegate, C program for lunar arithmetic and number theory
- D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011.
- Brady Haran and N. J. A. Sloane, Primes on the Moon (Lunar Arithmetic), Numberphile video, Nov 2018.
- Rémy Sigrist, Colored representation of the array for n, k < 1000 (where the color is function of T(n, k))
- Index entries for sequences related to dismal (or lunar) arithmetic
Programs
-
Maple
# Maple programs for lunar arithmetic are in A087062. # Seen as a triangle: T := (n, k) -> if n - k > k then n - k else k fi: for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, May 07 2023
-
Mathematica
ladd[x_, y_] := FromDigits[MapThread[Max, IntegerDigits[#, 10, Max @@ IntegerLength /@ {x, y}] & /@ {x, y}]]; Flatten[Table[ladd[k, n - k], {n, 0, 13}, {k, 0, n}]] (* Davin Park, Sep 29 2016 *) -
PARI
ladd=A087061(m,n)=fromdigits(vector(if(#(m=digits(m))>#n=digits(n),#n=Vec(n,-#m),#m<#n,#m=Vec(m,-#n),#n),k,max(m[k],n[k]))) \\ M. F. Hasler, Nov 12 2017, updated Nov 15 2018
Formula
T(n, k) = n - k if n - k > k, otherwise k, if seen as a triangle. See A004197, which is a kind of dual. In fact T(n, k) + A004197(n, k) = A003056(n, k). - Peter Luschny, May 07 2023
Extensions
Edited by M. F. Hasler, Nov 12 2017
Comments