A137171 - OEIS

A137171

Interleaved reading of A000749 and its first to third differences.

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 1, 4, 6, 4, 2, 10, 10, 6, 6, 20, 16, 12, 16, 36, 28, 28, 36, 64, 56, 64, 72, 120, 120, 136, 136, 240, 256, 272, 256, 496, 528, 528, 496, 1024, 1056, 1024, 992, 2080, 2080, 2016, 2016, 4160, 4096, 4032, 4096, 8256, 8128, 8128

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,11

COMMENTS

A000749 is identical to its fourth differences, which implies that the 2nd differences equal the 5th, the 3rd differences the 6th and so on and implies that each of the sequences of these differences obeys the recurrence a(n)=4a(n-1)-6a(n-2)+4a(n-3), n > 3.

The table containing A000749 and its first differences (essentially A038505), 2nd differences (A038504) and 3rd differences (A038503) as the 4 rows is

O, 0, 0, 1, 4, 10, 20, 36, 64, ...

0, 0, 1, 3, 6, 10, 16, 28, 56, ...

0, 1, 2, 3, 4, 6, 12, 28, 64, ...

1, 1, 1, 1, 2, 6, 16, 36, 72, ...

Columns sums are 1, 2, 4, 8, 16, 32 ... = 2^n =A000079. The sequence reads this table column by column.

LINKS

Table of n, a(n) for n=0..62.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 4, 0, 0, 0, -6, 0, 0, 0, 4).

MATHEMATICA

Join[{0, 0, 0, 1}, LinearRecurrence[{0, 0, 0, 4, 0, 0, 0, -6, 0, 0, 0, 4}, {0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 1}, 59]] (* Ray Chandler, Sep 23 2015 *)

CROSSREFS

Cf. A137172, A137173.