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A257542 Square-sum pairs: Numbers n such that 0,1, ..., 2n-1 can be partitioned into n pairs, where each pair adds up to a perfect square.

%I A257542 #18 Aug 10 2025 14:42:29
%S A257542 1,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,
%T A257542 29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,
%U A257542 52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75
%N A257542 Square-sum pairs: Numbers n such that 0,1, ..., 2n-1 can be partitioned into n pairs, where each pair adds up to a perfect square.
%C A257542 Kilkelly uses induction to prove that all integers greater than 20 are in the sequence after using various methods on smaller cases.
%C A257542 The positive integers except 2, 3, and 6.
%C A257542 The positive integers except the strong divisors of 6. - _Omar E. Pol_, Apr 30 2015
%D A257542 T. Kilkelly, The ARML Power Contest, American Mathematical Society, 2015, chapter 11.
%H A257542 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A257542 From _Chai Wah Wu_, Aug 13 2020: (Start)
%F A257542 a(n) = 2*a(n-1) - a(n-2) for n > 5.
%F A257542 G.f.: x*(-x^4 + x^3 - 2*x^2 + 2*x + 1)/(x - 1)^2. (End)
%e A257542 For n = 4: (0, 1), (2, 7), (3, 6), (4, 5)
%e A257542 For n = 7: (0, 9), (1, 8), (2, 7), (3, 13), (4, 12), (5, 11), (6, 10)
%o A257542 (PARI) is(n)=n>6 || n==1 || n==4 || n==5 \\ _Charles R Greathouse IV_, Apr 30 2015
%Y A257542 Cf. A253472, A252897.
%Y A257542 Essentially the same as A055495.
%K A257542 nonn,easy
%O A257542 1,2
%A A257542 _Brian Hopkins_, Apr 28 2015