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A345680 Nonnegative integers whose trajectory under iteration of taking the absolute value of the alternating sum of the squares of the digits (A257588) includes zero.

%I A345680 #38 Feb 24 2022 11:29:42
%S A345680 0,7,11,22,29,33,34,38,43,44,47,49,55,56,59,65,66,70,74,77,83,88,92,
%T A345680 94,95,99,108,110,117,125,126,131,138,142,147,148,149,161,168,171,172,
%U A345680 179,182,184,185,195,196,205,212,220,227,234,237,238,241,258,265,269
%N A345680 Nonnegative integers whose trajectory under iteration of taking the absolute value of the alternating sum of the squares of the digits (A257588) includes zero.
%C A345680 The sequence was initially studied by a group of students at Clifton College, UK.
%C A345680 There are infinitely many terms.
%C A345680 Having checked up to 10^10, there are approximations for the lower and upper density: 0.23 and 0.25 respectively.
%C A345680 Conjecture: there are strings of consecutive terms of arbitrary length.
%C A345680 Any number which is formed by concatenating two-digit multiples of 11 is a term.
%e A345680 For 7, the trajectory under iteration is 7, 49, 65, 11, 0, ..., so 7 is a term.
%e A345680 For 11, the trajectory is 11, 0, ...
%e A345680 For 22, the trajectory is 22, 0, ...
%e A345680 For 29, the trajectory is 29, 77, 0, ...
%e A345680 A non-example is 48. Its trajectory is 48, 48, ...
%t A345680 Select[Range[1000], FixedPoint[ Abs[Sum[(-1)^(n + 1)*Part[IntegerDigits[#]^2, n], {n, 1, Length[IntegerDigits[#]]}]] &, #, 10] == 0 &] (* _Luca Onnis_, Feb 23 2022 *)
%o A345680 (Python)
%o A345680 def happyish_function(number, base: int = 10):  # A257588
%o A345680 # iterates the process
%o A345680   total = 0
%o A345680   times = 0
%o A345680   while number > 0:
%o A345680     total += pow(-1, times) * pow(abs(number) % base, 2)
%o A345680     number = abs(number) // base
%o A345680     times += 1
%o A345680   return abs(total)
%o A345680 def is_happyish(number: int) -> bool:
%o A345680 # determines whether a number is happyish
%o A345680   seen_numbers = set()
%o A345680   while number > 0 and number not in seen_numbers:
%o A345680     seen_numbers.add(number)
%o A345680     number = happyish_function(number)
%o A345680   return number == 0
%o A345680 def happyish_list(number: int):
%o A345680 # creates their list
%o A345680   happyish = []
%o A345680   n = 0
%o A345680   for i in range(number):
%o A345680     if is_happyish(i) == True:
%o A345680       n +=1
%o A345680       happyish.append(i)
%o A345680   return happyish
%o A345680 happyish_list(100) # an example
%Y A345680 Cf. A257588 (iteration step).
%Y A345680 Cf. A007770 (sum of squares not alternating).
%K A345680 nonn,base
%O A345680 1,2
%A A345680 _Stephen Cross_, Jun 23 2021