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A254967 Triangle of iterated absolute differences of lucky numbers read by antidiagonals upwards.

%I A254967 #19 Feb 16 2025 08:33:24
%S A254967 1,2,3,2,4,7,0,2,2,9,0,0,2,4,13,0,0,0,2,2,15,2,2,2,2,4,6,21,2,0,2,0,2,
%T A254967 2,4,25,2,0,0,2,2,0,2,6,31,0,2,2,2,0,2,2,4,2,33,0,0,2,0,2,2,0,2,2,4,
%U A254967 37,0,0,0,2,2,0,2,2,0,2,6,43,2,2,2,2,0,2
%N A254967 Triangle of iterated absolute differences of lucky numbers read by antidiagonals upwards.
%C A254967 This sequence is related to the lucky numbers (cf. A000959) in the same way as A036262 is related to the prime numbers;
%H A254967 Reinhard Zumkeller, <a href="/A254967/b254967.txt">Rows n = 0..125 of triangle, flattened</a>
%H A254967 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LuckyNumber.html">Lucky number.</a>
%H A254967 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucky_number">Lucky number</a>
%F A254967 T(n,0) = A054978(n).
%F A254967 T(2*n,n) = A254969(n).
%F A254967 T(n,n-1) = A031883(n) for n > 0.
%F A254967 T(n,n) = A000959(n+1).
%F A254967 T(n,k) = abs(T(n,k+1) - T(n-1,k)) for 0 <= k < n.
%e A254967 .   0:                      1
%e A254967 .   1:                     2 3
%e A254967 .   2:                    2 4 7
%e A254967 .   3:                   0 2 2 9
%e A254967 .   4:                  0 0 2 4 13
%e A254967 .   5:                 0 0 0 2 2 15
%e A254967 .   6:                2 2 2 2 4 6 21
%e A254967 .   7:               2 0 2 0 2 2 4 25
%e A254967 .   8:              2 0 0 2 2 0 2 6 31
%e A254967 .   9:             0 2 2 2 0 2 2 4 2 33
%e A254967 .  10:            0 0 2 0 2 2 0 2 2 4 37
%e A254967 .  11:           0 0 0 2 2 0 2 2 0 2 6 43
%e A254967 .  12:          2 2 2 2 0 2 2 0 2 2 0 6 49
%e A254967 .  13:         0 2 0 2 0 0 2 0 0 2 4 4 2 51 .
%t A254967 nmax = 13; (* max index for triangle rows *)
%t A254967 imax = 25; (* max index for initial lucky array L *)
%t A254967 L = Table[2i + 1, {i, 0, imax}];
%t A254967 For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i -> Nothing, {i, 1, Length[L]/r}]]];
%t A254967 T[n_, n_] := If[n+1 <= Length[L], L[[n+1]], Print["imax should be increased"]; 0];
%t A254967 T[n_, k_] := T[n, k] = Abs[T[n, k+1] - T[n-1, k]];
%t A254967 Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 22 2021 *)
%o A254967 (Haskell)
%o A254967 a254967 n k = a254967_tabl !! n !! k
%o A254967 a254967_row n = a254967_tabl !! n
%o A254967 a254967_tabl = diags [] $
%o A254967    iterate (\lds -> map abs $ zipWith (-) (tail lds) lds) a000959_list
%o A254967    where diags uss (vs:vss) = (map head wss) : diags (map tail wss) vss
%o A254967                               where wss = vs : uss
%Y A254967 Cf. A054978 (left edge), A254969 (central terms), A000959 (right edge), A031883, A036262.
%K A254967 nonn,tabl
%O A254967 0,2
%A A254967 _Reinhard Zumkeller_, Feb 11 2015