A254967 - OEIS

%I #18 Nov 12 2024 22:17:16

%S 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 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 37,0,0,0,2,2,0,2,2,0,2,6,43,2,2,2,2,0,2

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

%C This sequence is related to the lucky numbers (cf. A000959) in the same way as A036262 is related to the prime numbers;

%H Reinhard Zumkeller, <a href="/A254967/b254967.txt">Rows n = 0..125 of triangle, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LuckyNumber.html">Lucky number.</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucky_number">Lucky number</a>

%F T(n,0) = A054978(n).

%F T(2*n,n) = A254969(n).

%F T(n,n-1) = A031883(n) for n > 0.

%F T(n,n) = A000959(n+1).

%F T(n,k) = abs(T(n,k+1) - T(n-1,k)) for 0 <= k < n.

%e .   0:                      1

%e .   1:                     2 3

%e .   2:                    2 4 7

%e .   3:                   0 2 2 9

%e .   4:                  0 0 2 4 13

%e .   5:                 0 0 0 2 2 15

%e .   6:                2 2 2 2 4 6 21

%e .   7:               2 0 2 0 2 2 4 25

%e .   8:              2 0 0 2 2 0 2 6 31

%e .   9:             0 2 2 2 0 2 2 4 2 33

%e .  10:            0 0 2 0 2 2 0 2 2 4 37

%e .  11:           0 0 0 2 2 0 2 2 0 2 6 43

%e .  12:          2 2 2 2 0 2 2 0 2 2 0 6 49

%e .  13:         0 2 0 2 0 0 2 0 0 2 4 4 2 51 .

%t nmax = 13; (* max index for triangle rows *)

%t imax = 25; (* max index for initial lucky array L *)

%t L = Table[2i + 1, {i, 0, imax}];

%t For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i -> Nothing, {i, 1, Length[L]/r}]]];

%t T[n_, n_] := If[n+1 <= Length[L], L[[n+1]], Print["imax should be increased"]; 0];

%t T[n_, k_] := T[n, k] = Abs[T[n, k+1] - T[n-1, k]];

%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 22 2021 *)

%o (Haskell)

%o a254967 n k = a254967_tabl !! n !! k

%o a254967_row n = a254967_tabl !! n

%o a254967_tabl = diags [] $

%o    iterate (\lds -> map abs $ zipWith (-) (tail lds) lds) a000959_list

%o    where diags uss (vs:vss) = (map head wss) : diags (map tail wss) vss

%o                               where wss = vs : uss

%Y Cf. A054978 (left edge), A254969 (central terms), A000959 (right edge), A031883, A036262.

%K nonn,tabl

%O 0,2

%A _Reinhard Zumkeller_, Feb 11 2015