A328918 - OEIS

%I #21 Nov 03 2019 17:04:10

%S 1,1,11,11,281,281,11181,11181,563131,563131,32795191,32795191,

%T 2103687091,2103687091,144420919291,144420919291,10421915468041,

%U 10421915468041,781300466839541,781300466839541,60358948031151561,60358948031151561,4777791013174712961

%N 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.

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

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

%C 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

%H Michael W. Ecker, <a href="https://www.jstor.org/stable/2689508">Problem 1016</a>, Mathematics Magazine, Vol. 50, No. 3 (May, 1977), pp. 163-169.

%F a(n) = Sum_{k=0..floor((n-1)/2)} A287317(k). - _Andrew Howroyd_, Nov 03 2019

%e 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).

%o (PARI) seq(n)={Vec(serlaplace(besseli(0,2*x + O(x*x^n))^5)/(1-x))} \\ _Andrew Howroyd_, Nov 03 2019

%Y Cf. A287317.

%K nonn,base

%O 1,3

%A _Dr. Michael W. Ecker_, Oct 30 2019

%E Terms a(9) and beyond from _Andrew Howroyd_, Nov 03 2019