A figurate number of the form 
, where 
is the 
th triangular number. The first few are 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, ... (Sloane's A002378). The generating function of the pronic numbers is
, where is the th triangular number. The first few are 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, ... (Sloane's A002378). 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 2005, Vol. 1, p. 357; Vol. 2, p. 233).
(Dickson 2005, Vol. 1, p. 357; Vol. 2, p. 233).
Pronic numbers are also known as oblong 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).
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, ... (Sloane's A028336), and the first few palindromic numbers which are pronic are 2, 6, 272, 6006, 289982, ... (Sloane's A028337).
Ref: mathworld.wolfram.com
for which are palindromic are 1, 2, 16, 77, 538, 1621, ... (Sloane's A028336), and the first few palindromic numbers which are pronic are 2, 6, 272, 6006, 289982, ... (Sloane's A028337).Ref: mathworld.wolfram.com
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