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A299539 Numbers n = d_1 d_2 ... d_k (in base 10) such that d_i + d_{k+1-i} = 10 for i = 1..k.

%I A299539 #29 Nov 22 2020 21:25:38
%S A299539 5,19,28,37,46,55,64,73,82,91,159,258,357,456,555,654,753,852,951,
%T A299539 1199,1289,1379,1469,1559,1649,1739,1829,1919,2198,2288,2378,2468,
%U A299539 2558,2648,2738,2828,2918,3197,3287,3377,3467,3557,3647,3737,3827,3917,4196,4286
%N A299539 Numbers n = d_1 d_2 ... d_k (in base 10) such that d_i + d_{k+1-i} = 10 for i = 1..k.
%C A299539 These numbers are also called upside-down numbers.
%C A299539 All terms belong to A052382 (zeroless numbers).
%C A299539 The central digit of the terms with an odd number of digits is always 5.
%C A299539 This sequence can be partitioned into three sets: { 5 }, A083678 and A093472.
%C A299539 This sequence has similarities with A284811: here d_i + d_{k+1-i} = 10, there d_i + d_{k+1-i} = 9.
%C A299539 These numbers have a palindromic Morse code representation (see A060109). To get all numbers with this property one has to include 0 and terms with corresponding "interior" digits 5 replaced by digits 0, e.g., 5 -> 0, 159 -> 109, 555 -> 505, 1559 -> 1009, 15559 -> {10009, 10509, 15059}. - _M. F. Hasler_, Nov 02 2020
%H A299539 Robert E. Kennedy and Curtis N. Cooper, <a href="http://www.jstor.org/stable/2686498">Bach, 5465, and Upside-Down Numbers</a>, The College Mathematics Journal, Vol. 18, No. 2 (Mar., 1987), pp. 111-115.
%H A299539 Giovanni Resta, <a href="http://www.numbersaplenty.com/set/upside-down_number/">Upside-down numbers</a>, Numbers Aplenty.
%e A299539 1 + 9 = 10 and 5 + 5 = 10 and 9 + 1 = 10, hence 159 belongs to this sequence.
%e A299539 4 + 2 = 6, hence 42 does not belong to this sequence.
%p A299539 Res:= NULL;
%p A299539 for d from 1 to 3 do
%p A299539   for x from 0 to 9^(d-1)-1 do
%p A299539     L:= convert(9^(d-1)+x,base,9)[1..d-1];
%p A299539     Res:= Res, 5*10^(d-1)+add((1+L[-i])*10^(2*d-1-i)+(9-L[-i])*10^(i-1),i=1..d-1)
%p A299539   od;
%p A299539   for x from 0 to 9^d-1 do
%p A299539     L:= convert(9^d+x,base,9)[1..d];
%p A299539     Res:= Res, add((1+L[-i])*10^(2*d-i)+(9-L[-i])*10^(i-1),i=1..d)
%p A299539   od
%p A299539 od:
%p A299539 Res; # _Robert Israel_, Mar 06 2018
%t A299539 Select[Range[4300], AllTrue[#1[[1 ;; #2]] + Reverse@ #1[[-#2 ;; -1]], # == 10 &] & @@ {#, Ceiling[Length[#]/2]} &@ IntegerDigits[#] &] (* _Michael De Vlieger_, Nov 04 2020 *)
%o A299539 (PARI) is(n) = my (d=digits(n)); Set(d+Vecrev(d))==Set(10)
%Y A299539 Cf. A052382, A083678, A093472, A284811.
%Y A299539 Cf. also A060109 (Morse code of numbers).
%K A299539 nonn,base,easy
%O A299539 1,1
%A A299539 _Rémy Sigrist_, Mar 05 2018