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A029955 Palindromic in base 9.

%I A029955 #33 Jun 14 2024 01:49:30
%S A029955 0,1,2,3,4,5,6,7,8,10,20,30,40,50,60,70,80,82,91,100,109,118,127,136,
%T A029955 145,154,164,173,182,191,200,209,218,227,236,246,255,264,273,282,291,
%U A029955 300,309,318,328,337,346,355,364,373,382,391,400,410,419,428,437
%N A029955 Palindromic in base 9.
%C A029955 Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - _Charles R Greathouse IV_, May 03 2020
%H A029955 T. D. Noe, <a href="/A029955/b029955.txt">Table of n, a(n) for n = 1..10000</a>
%H A029955 Javier Cilleruelo, Florian Luca and Lewis Baxter, <a href="https://doi.org/10.1090/mcom/3221">Every positive integer is a sum of three palindromes</a>, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, <a href="http://arxiv.org/abs/1602.06208">arXiv preprint</a>, arXiv:1602.06208 [math.NT], 2017.
%H A029955 Patrick De Geest, <a href="http://www.worldofnumbers.com/nobase10.htm">Palindromic numbers beyond base 10</a>.
%H A029955 Phakhinkon Phunphayap and Prapanpong Pongsriiam, <a href="https://doi.org/10.13140/RG.2.2.23202.79047">Estimates for the Reciprocal Sum of b-adic Palindromes</a>, 2019.
%H A029955 <a href="/index/Ab#basis_03">Index entries for sequences that are an additive basis</a>, order 3.
%F A029955 Sum_{n>=2} 1/a(n) = 3.29797695... (Phunphayap and Pongsriiam, 2019). - _Amiram Eldar_, Oct 17 2020
%t A029955 f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,9], AppendTo[lst,n]], {n,1000}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jul 08 2009 *)
%o A029955 (Python)
%o A029955 from gmpy2 import digits
%o A029955 def palQgen(l,b): # generator of palindromes in base b of length <= 2*l
%o A029955     if l > 0:
%o A029955         yield 0
%o A029955         for x in range(1,l+1):
%o A029955             for y in range(b**(x-1),b**x):
%o A029955                 s = digits(y,b)
%o A029955                 yield int(s+s[-2::-1],b)
%o A029955             for y in range(b**(x-1),b**x):
%o A029955                 s = digits(y,b)
%o A029955                 yield int(s+s[::-1],b)
%o A029955 A029955_list = list(palQgen(4,9)) # _Chai Wah Wu_, Dec 01 2014
%o A029955 (Python)
%o A029955 from gmpy2 import digits
%o A029955 def A029955(n):
%o A029955     if n == 1: return 0
%o A029955     y = 9*(x:=9**(len(digits(n>>1,9))-1))
%o A029955     return int((c:=n-x)*x+int(digits(c,9)[-2::-1]or'0',9) if n<x+y else (c:=n-y)*y+int(digits(c,9)[-1::-1]or'0',9)) # _Chai Wah Wu_, Jun 14 2024
%o A029955 (PARI) ispal(n,b=9)=my(d=digits(n,b)); d==Vecrev(d) \\ _Charles R Greathouse IV_, May 03 2020
%Y A029955 Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.
%K A029955 nonn,base,easy
%O A029955 1,3
%A A029955 _Patrick De Geest_