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A087061
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Array A(n, k) = lunar sum n + k (n >= 0, k >= 0) read by antidiagonals.
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42
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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
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OFFSET
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0,4
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COMMENTS
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There are no carries in lunar arithmetic. For each pair of lunar digits, to Add, take the lArger, but to Multiply, take the sMaller. For example:
169
+ 248
------
269
and
169
x 248
------
168
144
+ 122
--------
12468
Addition and multiplication are associative and commutative and multiplication distributes over addition. E.g., 357 * (169 + 248) = 357 * 269 = 23567 = 13567 + 23457 = (357 * 169) + (357 * 248). Note that 0 + x = x and 9*x = x for all x.
We have changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing. - N. J. A. Sloane, Aug 06 2014
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LINKS
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D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011.
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FORMULA
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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
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EXAMPLE
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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;
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MAPLE
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# 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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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