A001579, A001550, etc.

[Henry Gould]

Sequence A001579 concerns 3, 14, 70, 368, 2002, 11144, 63010, 360248, .
. . about
which the OEIS gives the formula 3^n + 5^n + g^n. NO reference or other
information is given. Why was this particular sequence chosen to be
listed?

Sequence A001550 which concerns 3, 6, 14, 36, 98, 276, 794, . . .    is
also listed
and a reference is given to Abramowitz and Stegun, Handbook of
Mathematical Tables
and Functions (the valuable old Bureau of Standards book) and the
formula
1^n + 2^n  + 3^n is given. I can understand why  this might be listed,
because
the general series 1^n + 2^n + 3^n + 4^n + . . . + k^n  is  the natural
and useful
sum of the n-th powers of the first k natural numbers.

I also tried to find the sequences with formulas 1^n + 3^n + 6^n,  1^ +
3^n + 5^n,
3^n + 5^n + 7^n,  2^n + 3^n + 7^n,   3^n + 4^n + 5^n, and a number other
examples
of sequences given by the trinomial  a^n + b^n + c^n, and came up quite
empty-
handed.

In view of this I wonder why sequence A001579 is listed?

Seqfans may be slightly amused as to why I am curious about A001579 in
the
first place. While I haven't thought about any combinatorial or
number-theoretic
significance of the sequence, I was getting ready to pay my bill at a
local
restaurant and the amount was $6.53. I have the Hardy-Ramanujanish habit
of
looking for interesting thing about any and all numbers, so I quickly
calculated
the values of many terms of 3, 14, 70, 368, 2002, 11144, 63010, 360248,
. . .
and found not only the present year 2002, but also my house number 368,
and
thereby the sequence became of interest.

With reference to my paper :The Girard-Waring power sum formulas for
symmetric functions and Fibonacci sequences", Fibonacci Quarterly,
37(1999),
No. 2, May, pp. 135-140, it is clear that the sequences of form  a^n +
b^n + c^n

are sums involving the expression (n/n-2k) C(n-2k,k) where C is, of
course.
the usual binomial coefficient, and therefore the sequence a^n + b^n +
c^n
has combinatorial significance.

The question I come down to, then, is this: For which values of a, b, c,
should
we list the sequences values in OEIS? This is one reason I tried a
number of
different values of a, b, and c.

More generally, we should consider  A_n  =  a^n + b^n + c^n + d^n + . .
. +k^n.

Does any one have any keen ideas about these matters?

I wish first of all to know how come 3, 14, 70, 368, 2002, 11144, etc.
was of
sufficient interest to be listed with just a formula, whereas numerous
similar
such sequences are not listed.

Seriatim,

Henry W. Gould