A328918 a(n) is the number of ordered pairs of positive integers (x, y) with x + y = 10^n, where x and y each have exactly n-digits but with initial zero digits allowed, and as strings, x and y are permutations of each other.

1, 1, 11, 11, 281, 281, 11181, 11181, 563131, 563131, 32795191, 32795191, 2103687091, 2103687091, 144420919291, 144420919291, 10421915468041, 10421915468041, 781300466839541, 781300466839541, 60358948031151561, 60358948031151561, 4777791013174712961

Offset: 1

Author

Dr. Michael W. Ecker, Oct 30 2019

Keywords

• nonn
• base

Comments

Published with slightly different wording in Mathematics Magazine, Problem 1016, Dec. 1977.

Analyzed for n = 1, 2, 3; computer-verified for n up to 8.

All solutions consist of an even number of digits followed by the digit 5 followed by zero or more 0's. This pattern means that a(2*n-1) = a(2*n). The initial segment consists of pairs of digits that add to 9 (0 with 9, 1 with 8, etc) arranged in arbitrary order and in particular leading 0's are permitted by the definition of the problem. A287317(k) gives the number of such arrangement with k pairs. For example, 339606500 + 660393500 is a solution. - Andrew Howroyd, Nov 03 2019

Examples

			For n = 3, solutions are (095, 905), (185, 815), (275, 725), (365, 635), (455, 545), (500, 500), (545, 455), (635, 365), (725, 275), (815, 185), (905, 095).
		

Links

• Michael W. Ecker, Problem 1016, Mathematics Magazine, Vol. 50, No. 3 (May, 1977), pp. 163-169.

Crossrefs

Cf. A287317.

Programs

PARI

seq(n)={Vec(serlaplace(besseli(0,2*x + O(x*x^n))^5)/(1-x))} \\ Andrew Howroyd, Nov 03 2019

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} A287317(k). - Andrew Howroyd, Nov 03 2019

Extensions

Terms a(9) and beyond from Andrew Howroyd, Nov 03 2019