A001579, A001550, etc./Correction

[Henry Gould]

Sequence A001579 concerns 3, 14, 70, 368, 2002, 11144, 63010, 360248, .
. . about which the OEIS gives the formula 3^n + 5^n + 6^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 things 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