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A261684 Array T(n,k) = lunar product n*k (n >= 0, k >= 0) read by antidiagonals. 3

%I

%S 0,0,0,0,1,0,0,1,1,0,0,1,2,1,0,0,1,2,2,1,0,0,1,2,3,2,1,0,0,1,2,3,3,2,

%T 1,0,0,1,2,3,4,3,2,1,0,0,1,2,3,4,4,3,2,1,0,0,1,2,3,4,5,4,3,2,1,0,0,10,

%U 2,3,4,5,5,4,3,2,10,0,0,11,10,3,4,5,6,5,4,3,10,11,0

%N Array T(n,k) = lunar product n*k (n >= 0, k >= 0) read by antidiagonals.

%C See A087061 for definition. Note that 0+x = x and 9*x = x for all x.

%H N. J. A. Sloane, <a href="/A261684/b261684.txt">Table of n, a(n) for n = 0..20099</a>

%H D. Applegate, <a href="/A087061/a087061.txt">C program for lunar arithmetic and number theory</a> [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

%H D. Applegate, M. LeBrun and N. J. A. Sloane, <a href="http://arxiv.org/abs/1107.1130">Dismal Arithmetic</a> [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

%H Brady Haran and N. J. A. Sloane, <a href="https://youtu.be/cZkGeR9CWbk">Primes on the Moon (Lunar Arithmetic)</a>, Numberphile video, Nov 2018.

%H <a href="/index/Di#dismal">Index entries for sequences related to dismal (or lunar) arithmetic</a>

%e Lunar multiplication table begins:

%e 0 0 0 0 0 0 ...

%e 0 1 1 1 1 1 ...

%e 0 1 2 2 2 2 ...

%e 0 1 2 3 3 3 ...

%e 0 1 2 3 4 4 ...

%e 0 1 2 3 4 5 ...

%e ....

%p # convert decimal to string:

%p rec := proc(n) local t0,t1,e,l; if n <= 0 then RETURN([[0],1]); fi; t0 := n mod 10; t1 := (n-t0)/10; e := [t0]; l := 1; while t1 <> 0 do t0 := t1 mod 10; t1 := (t1-t0)/10; l := l+1; e := [op(e),t0]; od; RETURN([e,l]); end;

%p # convert string to decimal:

%p cer := proc(ep) local i,e,l,t1; e := ep[1]; l := ep[2]; t1 := 0; if l <= 0 then RETURN(t1); fi; for i from 1 to l do t1 := t1+10^(i-1)*e[i]; od; RETURN(t1); end;

%p # lunar addition:

%p dadd := proc(m,n) local i,r1,r2,e1,e2,l1,l2,l,l3,t0; r1 := rec(m); r2 := rec(n); e1 := r1[1]; e2 := r2[1]; l1 := r1[2]; l2 := r2[2]; l := max(l1,l2); l3 := min(l1,l2); t0 := array(1..l); for i from 1 to l3 do t0[i] := max(e1[i],e2[i]); od; if l>l3 then for i from l3+1 to l do if l1>l2 then t0[i] := e1[i]; else t0[i] := e2[i]; fi; od; fi; cer([t0,l]); end;

%p # lunar multiplication:

%p dmul := proc(m,n) local k,i,j,r1,r2,e1,e2,l1,l2,l,t0; r1 := rec(m); r2 := rec(n); e1 := r1[1]; e2 := r2[1]; l1 := r1[2]; l2 := r2[2]; l := l1+l2-1; t0 := array(1..l); for i from 1 to l do t0[i] := 0; od; for i from 1 to l2 do for j from 1 to l1 do k := min(e2[i],e1[j]); t0[i+j-1] := max(t0[i+j-1],k); od; od; cer([t0,l]); end;

%p # to produce the b-file:

%p M:=199; c:=0; for n from 0 to M do for k from 0 to n do lprint(c,dmul(n-k,k)); c:=c+1; od: od:

%Y Cf. A087061 (addition).

%Y See A087062 for a version that excludes the zero row and column.

%Y Similar to but different from A003983.

%K nonn,tabl,look,hear

%O 0,13

%A _N. J. A. Sloane_, Sep 06 2015

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Last modified July 12 19:26 EDT 2020. Contains 335668 sequences. (Running on oeis4.)