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

121, 6724, 154449, 1860496, 14250625, 78960996, 344362249, 1250895424, 3936182121, 11035502500, 28143753121, 66322731024, 146186169649, 304278004996, 602680505625, 1143051786496, 2086600473049, 3681862517124, 6302555019121, 10498248010000, 17061121121121

Offset: 1

Author

Jaroslav Krizek, May 06 2016

Keywords

• nonn
• easy

Comments

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).

This sequence contains squares M(n) for the squares n from A000290.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Pochhammer Symbol.
• Wikipedia, Metallic mean.
• Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Crossrefs

Cf. A000290, A261391.

Programs

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)];

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

Formula

a(n) = (A261391(n))^2 = ((n-th metallic mean)^5 - 1/(n-th metallic mean)^5)^2.

a(n) = n^10 + 10*n^8 + 35*n^6 + 50*n^4 + 25*n^2 = (n^5 + 5*n^3 + 5*n)^2.

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