A219667 Number of happy numbers without zeros and with digits in nondecreasing order <= 10^n.
1, 2, 10, 30, 111, 296, 711, 1625, 3583, 7088, 13377, 24143, 41376, 68551, 111068, 174887, 269592, 410043, 612350, 900484, 1310108, 1879862, 2664399, 3740002, 5188376, 7121305, 9687674, 13044961, 17402217, 23026128, 30200375, 39290567, 50744867, 65045752, 82799083
Offset: 0
Examples
h(0, 0, 1 <= x <= 9) = 1, h(d, n, x) = 0 for all n < 0, h(d, n, x) = 0 for all n > 81*d.
Links
- Bryan Wolf, Table of n, a(n) for n = 0..10000
- Bryan Wolf, C++ program for generating n, a(n)
Formula
h(d,n,x) = h(d-1,n-x^2,x) + h(d-1,n-(x+1)^2,x+1) + ... + h(n-1,n-9^2,9) where h() is the number of d-digit long happy numbers with digits in nondecreasing order,
n is the sum of the squares of the digits in the happy numbers, and x is the lowest digit in the happy numbers.
a(n) is the sum of all h(d,0 <= x <= 81*d, 1), where x is a happy number.