Kausler (1805) was one of the first to tabulate pronic numbers, creating a list up to (Dickson 2005, Vol. 1, p. 357;
Vol. 2, p. 233).
Pronic numbers are also known as oblong (Merzbach and Boyer 1991, p. 50) or heteromecic numbers. However, "pronic" seems to be a misspelling of "promic"
(from the Greek promekes, meaning rectangular, oblate, or oblong). However,
no less an authority than Euler himself used the term "pronic," so attempting
to "correct" it at this late date seems inadvisable.
McDaniel (1998ab) proved that the only pronic Fibonacci numbers are and , and the only pronic Lucas number is , rediscovering a result first published by Ming (1995).
The first few
for which
are palindromic are 1, 2, 16, 77, 538, 1621,
... (OEIS A028336), and the first few palindromic
numbers which are pronic are 2, 6, 272, 6006, 289982, ... (OEIS A028337).
De Geest, P. "Palindromic Products of Two Consecutive Integers." http://www.worldofnumbers.com/consec.htm.Dickson,
L. E. History
of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York:
Dover, p. 357, 2005a.Dickson, L. E. History
of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover,
pp. 6, 232-233, 350, and 407, 2005b.Euler, L. Republished in Euler,
L. Opera Omnia, Ser. 1: Opera mathematica, Vol. 15. Basel, Switzerland:
Birkhäuser, 1992.Guy, R. K. "The Second Strong Law of
Small Numbers." Math. Mag63, 3-20, 1990.Kausler,
C. F. Nova Acta Acad. Petrop.14, 268-289, ad annos 1797-8, 1805.McDaniel,
W. L. "Pronic Fibonacci Numbers." Fib. Quart.36, 56-59,
1998a.McDaniel, W. L. "Pronic Lucas Numbers." Fib.
Quart.36, 60-62, 1998b.Merzbach, U. C. and Boyer, C. B.
A
History of Mathematics, 3rd ed. New York: Wiley, p. 50, 1991.Ming,
L. "Nearly Square Numbers in the Fibonacci and Lucas Sequences" [Chinese].
J. Chongqing Teachers College, No. 4, 1-5, 1995.Sloane,
N. J. A. Sequences A002378/M1581,
A028336, and A028337
in "The On-Line Encyclopedia of Integer Sequences."
,
where 
is the 
th
triangular number. The first few are 2, 6, 12,
20, 30, 42, 56, 72, 90, 110, ... (OEIS A002378).
The generating function of the pronic numbers
is
(Dickson 2005, Vol. 1, p. 357;
Vol. 2, p. 233).
and 
, and the only pronic Lucas number is 
, rediscovering a result first published by Ming (1995).
for which 
are palindromic are 1, 2, 16, 77, 538, 1621,
... (OEIS A028336), and the first few palindromic
numbers which are pronic are 2, 6, 272, 6006, 289982, ... (OEIS A028337).
