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

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, 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, 37, 0, 0, 0, 2, 2, 0, 2, 2, 0, 2, 6, 43, 2, 2, 2, 2, 0, 2

Offset: 0

Author

Reinhard Zumkeller, Feb 11 2015

Keywords

• nonn
• tabl

Comments

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

Examples

			.   0:                      1
.   1:                     2 3
.   2:                    2 4 7
.   3:                   0 2 2 9
.   4:                  0 0 2 4 13
.   5:                 0 0 0 2 2 15
.   6:                2 2 2 2 4 6 21
.   7:               2 0 2 0 2 2 4 25
.   8:              2 0 0 2 2 0 2 6 31
.   9:             0 2 2 2 0 2 2 4 2 33
.  10:            0 0 2 0 2 2 0 2 2 4 37
.  11:           0 0 0 2 2 0 2 2 0 2 6 43
.  12:          2 2 2 2 0 2 2 0 2 2 0 6 49
.  13:         0 2 0 2 0 0 2 0 0 2 4 4 2 51 .
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• Eric Weisstein's World of Mathematics, Lucky number.
• Wikipedia, Lucky number

Crossrefs

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

Programs

Haskell

a254967 n k = a254967_tabl !! n !! k
a254967_row n = a254967_tabl !! n
a254967_tabl = diags [] $
   iterate (\lds -> map abs $ zipWith (-) (tail lds) lds) a000959_list
   where diags uss (vs:vss) = (map head wss) : diags (map tail wss) vss
                              where wss = vs : uss

Mathematica

nmax = 13; (* max index for triangle rows *)
imax = 25; (* max index for initial lucky array L *)
L = Table[2i + 1, {i, 0, imax}];
For[n = 2, n < Length[L], r = L[[n++]]; L = ReplacePart[L, Table[r*i -> Nothing, {i, 1, Length[L]/r}]]];
T[n_, n_] := If[n+1 <= Length[L], L[[n+1]], Print["imax should be increased"]; 0];
T[n_, k_] := T[n, k] = Abs[T[n, k+1] - T[n-1, k]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 22 2021 *)

Formula

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

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

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

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

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