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

0,2

Comments

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

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

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

T(n,n-1) = A031883(n), n > 0;

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

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

Links

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Eric Weisstein's World of Mathematics, Lucky number.

Wikipedia, Lucky number

Example

.   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 .

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A192298 A341098 A353330 * A229012 A207606 A303845

Adjacent sequences: A254964 A254965 A254966 * A254968 A254969 A254970

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Feb 11 2015

Status

approved