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A055260 Sums of two powers of 9.

%I A055260 #20 Apr 09 2025 19:46:44
%S A055260 2,10,18,82,90,162,730,738,810,1458,6562,6570,6642,7290,13122,59050,
%T A055260 59058,59130,59778,65610,118098,531442,531450,531522,532170,538002,
%U A055260 590490,1062882,4782970,4782978,4783050,4783698,4789530,4842018,5314410,9565938,43046722
%N A055260 Sums of two powers of 9.
%H A055260 T. D. Noe, <a href="/A055260/b055260.txt">Rows n = 0..100 of triangle, flattened</a>
%F A055260 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)
%F A055260 Regarded as a triangle T(n, k) = 9^n + 9^k, so as a sequence a(n) = 9^A002262(n) + 9^A003056(n).
%t A055260 t = 9^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* _T. D. Noe_, Oct 09 2011 *)
%t A055260 Total/@Tuples[9^Range[0,10],2]//Union (* _Harvey P. Dale_, Jul 03 2019 *)
%o A055260 (Python)
%o A055260 def valuation(n, b):
%o A055260   v = 0
%o A055260   while n > 1: n //= b; v += 1
%o A055260   return v
%o A055260 def aupto(lim):
%o A055260   pows = [9**i for i in range(valuation(lim-1, 9) + 1)]
%o A055260   sum_pows = sorted([a+b for i, a in enumerate(pows) for b in pows[i:]])
%o A055260   return [s for s in sum_pows if s <= lim]
%o A055260 print(aupto(43046722)) # _Michael S. Branicky_, Feb 10 2021
%o A055260 (Python)
%o A055260 from math import isqrt
%o A055260 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
%Y A055260 Cf. A001019, A052216.
%K A055260 easy,nonn,tabl
%O A055260 0,1
%A A055260 _Henry Bottomley_, Jun 22 2000