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%I #14 Aug 27 2019 09:50:29
%S 1,2,6,14,29,52,96,160,277,450,712,1086,1657,2448,3636,5280,7635,
%T 10840,15392,21372,29655,40580,55282,74620,100651,134232,178922,
%U 236488,312019,408550,534288,692978,897931,1156256,1485650,1897704,2421635,3071608,3894042
%N Fourth entry in row n of triangle in A169945.
%C Wanted: a recurrence. Are any of A169940-A169954 related to any other entries in the OEIS?
%H <a href="/index/Ca#CARRYLESS">Index entries for sequences related to carryless arithmetic</a>
%F a(n) = A196723(n+1) - A143823(n+1). - _Andrew Howroyd_, Jul 09 2017
%t b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n < 1, 1, b[n - 1, s] + If[m*(m + 1)/2 == Length[Union[Flatten[Table[ sn[[i]] + sn[[j]], {i, 1, m}, {j, i + 1, m + 1}]]]], b[n - 1, sn], 0]]];
%t A196723[n_] := A196723[n] = b[n - 1, {n}] + If[n == 0, 0, A196723[n - 1]];
%t c[n_, s_] := c[n, s] = Module[{sn, m}, If[n < 1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m - 1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m - 1}, {j, i + 1, m}] // Flatten // Union], c[n - 1, sn], 0] + c[n-1, s]]];
%t A143823[n_] := A143823[n] = c[n - 1, {n}] + If[n == 0, 0, A143823[n - 1]];
%t a[n_] := a[n] = A196723[n + 1] - A143823[n + 1];
%t Table[Print[n, " ", a[n]]; a[n], {n, 2, 40}] (* _Jean-François Alcover_, Aug 27 2019, after _Alois P. Heinz_ in A196723 and A143823 *)
%Y Related to thickness: A169940-A169954, A061909.
%Y Cf. A143823, A196723.
%K nonn
%O 2,2
%A _N. J. A. Sloane_, Aug 01 2010
%E a(15)-a(28) from _Nathaniel Johnston_, Nov 12 2010
%E a(29)-a(40) from _Andrew Howroyd_, Jul 09 2017
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