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A272775 Squares of the form P(n, 5) + n, where P(x,k) is the Pochhammer function and n = square (A000290).

%I A272775 #16 May 19 2025 23:45:19
%S A272775 121,6724,154449,1860496,14250625,78960996,344362249,1250895424,
%T A272775 3936182121,11035502500,28143753121,66322731024,146186169649,
%U A272775 304278004996,602680505625,1143051786496,2086600473049,3681862517124,6302555019121,10498248010000,17061121121121
%N A272775 Squares of the form P(n, 5) + n, where P(x,k) is the Pochhammer function and n = square (A000290).
%C A272775 Theorem: Only for a square n is the number M(n) = P(n, 5) + n also square, where P(x,k) = x*(x+1)*...*(x+k-1) is the Pochhammer function (rising factorial).
%C A272775 This sequence contains squares M(n) for the squares n from A000290.
%H A272775 Colin Barker, <a href="/A272775/b272775.txt">Table of n, a(n) for n = 1..1000</a>
%H A272775 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pochhammer Symbol.html">Pochhammer Symbol</a>.
%H A272775 Wikipedia, <a href="https://en.wikipedia.org/wiki/Metallic_mean">Metallic mean</a>.
%H A272775 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F A272775 a(n) = (A261391(n))^2 = ((n-th metallic mean)^5 - 1/(n-th metallic mean)^5)^2.
%F A272775 a(n) = n^10 + 10*n^8 + 35*n^6 + 50*n^4 + 25*n^2 = (n^5 + 5*n^3 + 5*n)^2.
%F A272775 G.f.: x*(1 +x)*(121 +5272*x +81868*x^2 +429544*x^3 +780790*x^4 +429544*x^5 +81868*x^6 +5272*x^7 +121*x^8) / (1-x)^11. - _Colin Barker_, May 06 2016
%o A272775 (Magma) [n*(n+1)*(n+2)*(n+3)*(n+4) + n: n in [1..7000] | IsSquare(n*(n+1)*(n+2)*(n+3)*(n+4) + n)];
%o A272775 (PARI) Vec(x*(1 +x)*(121 +5272*x +81868*x^2 +429544*x^3 +780790*x^4 +429544*x^5 +81868*x^6 +5272*x^7 +121*x^8)/(1-x)^11 + O(x^50)) \\ _Colin Barker_, May 06 2016
%Y A272775 Cf. A000290, A261391.
%K A272775 nonn,easy
%O A272775 1,1
%A A272775 _Jaroslav Krizek_, May 06 2016