A055259 Sums of two powers of 8.

2, 9, 16, 65, 72, 128, 513, 520, 576, 1024, 4097, 4104, 4160, 4608, 8192, 32769, 32776, 32832, 33280, 36864, 65536, 262145, 262152, 262208, 262656, 266240, 294912, 524288, 2097153, 2097160, 2097216, 2097664, 2101248, 2129920, 2359296, 4194304, 16777217

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. A052216.

Programs

Mathematica

Union[Total/@Tuples[8^Range[0,10], {2}]]  (* Harvey P. Dale, Mar 13 2011 *)

Python

def valuation(n, b):
  v = 0
  while n > 1: n //= b; v += 1
  return v
def aupto(lim):
  pows8 = [8**i for i in range(valuation(lim-1, 8) + 1)]
  sum_pows8 = sorted([a+b for i, a in enumerate(pows8) for b in pows8[i:]])
  return [s for s in sum_pows8 if s <= lim]
print(aupto(16777217)) # Michael S. Branicky, Feb 09 2021

Python

from math import isqrt
def A055259(n): return (1<<3*(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1))+(1<<3*(n-1-(a*(a+1)>>1))) # Chai Wah Wu, Apr 08 2025

Formula

a(n) = 8^(n-trinv(n))+8^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) = 8^n + 8^k, so as a sequence a(n) = 8^A002262(n) + 8^A003056(n).