A055260 Sums of two powers of 9.

2, 10, 18, 82, 90, 162, 730, 738, 810, 1458, 6562, 6570, 6642, 7290, 13122, 59050, 59058, 59130, 59778, 65610, 118098, 531442, 531450, 531522, 532170, 538002, 590490, 1062882, 4782970, 4782978, 4783050, 4783698, 4789530, 4842018, 5314410, 9565938, 43046722

Offset: 0

Author

Henry Bottomley, Jun 22 2000

Keywords

• easy
• nonn
• tabl

Links

• T. D. Noe, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A001019, A052216.

Programs

Mathematica

t = 9^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
Total/@Tuples[9^Range[0,10],2]//Union (* Harvey P. Dale, Jul 03 2019 *)

Python

def valuation(n, b):
  v = 0
  while n > 1: n //= b; v += 1
  return v
def aupto(lim):
  pows = [9**i for i in range(valuation(lim-1, 9) + 1)]
  sum_pows = sorted([a+b for i, a in enumerate(pows) for b in pows[i:]])
  return [s for s in sum_pows if s <= lim]
print(aupto(43046722)) # Michael S. Branicky, Feb 10 2021

Python

from math import isqrt
def A055260(n): return 9**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+9**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 09 2025

Formula

a(n) = 9^(n-trinv(n))+9^trinv(n), where trinv(n) = floor((1+sqrt(1+8*n))/2) = A002262(n) and n-trinv(n) = A003056(n)

Regarded as a triangle T(n, k) = 9^n + 9^k, so as a sequence a(n) = 9^A002262(n) + 9^A003056(n).