for . Kausler (1805) was one of the
first to tabulate pronic numbers, creating a list up to (Dickson 2005a, p. 357; 2005b, 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 (1998a, p. 56; 1998b, p. 60) proved that the only pronic Fibonacci numbers are
and , and the only pronic Lucas number
is , rediscovering results previously
published by Luo (1995).
De Geest, P. "Palindromic Products of Two Consecutive Integers." https://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.Luo,
M. "Nearly Square Numbers in the Fibonacci and Lucas Sequences." [Chinese].
J. Chongqing Teachers College12, 1-5, 1995.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.Numberphile.
"Why 1980 Was a Great Year to Be Born... but 2184 Will Be Better." Featuring
M. Parker. June 29, 2015. https://www.youtube.com/watch?v=99stb2mzspI.Sloane,
N. J. A. Sequences A002378/M1581,
A028336, and A028337
in "The On-Line Encyclopedia of Integer Sequences."
,
where 
is the 
th
triangular number and 
is a nonnegative integer.
The first few positive pronic numbers are 2, 6, 12, 20, 30, 42, 56, 72, 90, 110,
... (OEIS A002378, with the initial 0 omitted).
The generating function of the pronic numbers
is
. Kausler (1805) was one of the
first to tabulate pronic numbers, creating a list up to 
(Dickson 2005a, p. 357; 2005b, p. 233).
and 
, and the only pronic Lucas number
is 
, rediscovering results previously
published by Luo (1995).
for which 
are palindromic are
1, 2, 16, 77, 538, 1621, ... (OEIS A028336),
and the first few positive palindromic
numbers which are pronic are 2, 6, 272, 6006, 289982, ... (OEIS A028337).
