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A325173 Perfect squares of the form a + b^2 + c^3, where a,b,c are consecutive numbers.

%I A325173 #48 Sep 11 2019 08:49:41
%S A325173 9,144,1089,5184,18225,51984,127449,278784,558009,1040400,1830609,
%T A325173 3069504,4941729,7683984,11594025,17040384,24472809,34433424,47568609,
%U A325173 64641600,86545809,114318864,149157369,192432384,245705625,310746384,389549169,484352064,597655809,732243600
%N A325173 Perfect squares of the form a + b^2 + c^3, where a,b,c are consecutive numbers.
%H A325173 Colin Barker, <a href="/A325173/b325173.txt">Table of n, a(n) for n = 1..1000</a>
%H A325173 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A325173 a(n) = A000290(A054602(n)). - _Michel Marcus_, Sep 05 2019
%F A325173 From _Colin Barker_, Sep 05 2019: (Start)
%F A325173 G.f.: 9*x*(1 + x)*(1 + 8*x + 22*x^2 + 8*x^3 + x^4) / (1 - x)^7.
%F A325173 a(n) = n^2*(2 + n^2)^2.
%F A325173 a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
%F A325173 (End)
%F A325173 E.g.f.: exp(x)*x*(9 + 63*x + 114*x^2 + 69*x^3 + 15*x^4 + x^5). - Conjectured by _Stefano Spezia_, Sep 05 2019 after _Colin Barker_
%F A325173 From _Chai Wah Wu_, Sep 10 2019: (Start)
%F A325173 Above conjectures are true. Proof: k + (k+1)^2 + (k+2)^3 = (k+1)*(k+3)^2 and thus is a perfect square if and only if k+1 = n^2 is a perfect square. This implies that (k+1)*(k+3)^2 = n^2*(n^2+2)^2.
%F A325173 (End)
%e A325173 9 = 0 + 1^2 + 2^3. 0,1,2 are consecutive numbers and 9 is a perfect square. Hence, 9 is a member of the sequence.
%e A325173 18225 = 24 + 25^2 + 26^3. 24,25,26 are consecutive numbers and 18225 is a perfect square. Hence 18225 is a member of the sequence.
%o A325173 (PARI) a(n) = n^2*(2 + n^2)^2 \\ _David A. Corneth_, Sep 11 2019
%o A325173 (PARI) Vec(9*x*(1 + x)*(1 + 8*x + 22*x^2 + 8*x^3 + x^4) / (1 - x)^7 + O(x^40)) \\ _Colin Barker_, Sep 11 2019
%Y A325173 Intersection of A000290 and A027620.
%Y A325173 Cf. A005563 (the indices n that give these squares), A054602.
%K A325173 nonn,easy
%O A325173 1,1
%A A325173 _Philip Mizzi_, Sep 05 2019