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Pronic Number


Pronic numbers are figurate numbers of the form P_n=2T_n=n(n+1), where T_n is the nth 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

 (2x)/((1-x)^3)=2x+6x^2+12x^3+20x^4+....

Kausler (1805) was one of the first to tabulate pronic numbers, creating a list up to n=1000 (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 F_0=0 and F_3=2, and the only pronic Lucas number is L_0=2, rediscovering a result first published by Ming (1995).

The first few n for which P_n 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).


See also

Figurate Number, Odd Number, Triangular Number

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References

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. Mag 63, 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."

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Pronic Number

Cite this as:

Weisstein, Eric W. "Pronic Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PronicNumber.html

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