A306380 Squares of the form 5*k^2 + 5.

25, 7225, 2325625, 748843225, 241125192025, 77641562988025, 25000342156951225, 8050032532975305625, 2592085475275891459225, 834643473006304074564025, 268752606222554636118156025, 86537504560189586525971675225

Offset: 1

Author

Stefano Spezia, Feb 13 2019

Keywords

• nonn
• easy

Comments

All terms of this sequence are odd, hence they end with 5.

Links

• Stefano Spezia, Table of n, a(n) for n = 1..390
• Eric Weisstein's World of Mathematics, Pell Equation.
• Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Crossrefs

Cf. A000290, A075796 (associated k).

Programs

GAP

a:=[25,7225,2325625];; for n in [4..20] do a[n]:=323*a[n-1]-323*a[n-2]+a[n-3]; od; a;

Magma

I:=[25, 7225, 2325625]; [n le 3 select I[n] else 323*Self(n-1)-323*Self(n-2)+Self(n-3): n in [1..20]];

Maple

a := n ->(5/4)*(2+(9-4*sqrt(5))^(2*n-2)*(9+sqrt(5))+(9+4*sqrt(5))^(2*n-2)*(9-sqrt(5))): op(map(simplify, [seq(a(n), n = 1 .. 20)]))

Mathematica

LinearRecurrence[{323, -323, 1}, {25, 7225, 2325625}, 30]

Maxima

a[1]:25$ a[2]:7225$ a[3]:2325625$ a[n]:=323*a[n-1]-323*a[n-2]+a[n-3]$ create_list(a[n], n, 1, 20);

PARI

Vec(25*x*(1-34*x+x^2)/((1-x)*(1-322*x+x^2)) + O(x^20))

Formula

O.g.f.: 25*x*(1 - 34*x + x^2)/((1 - x)*(1 - 322*x + x^2)).

E.g.f.: (5/4)*x*(2*exp(x) + (9 - 4*sqrt(5))*exp((9 - 4*sqrt(5))^2*x) + (9 + 4*sqrt(5))*exp((9 + 4*sqrt(5))^2*x)).

a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3) for n > 3.

a(n) = (5/4)*(2 + (9 - 4*sqrt(5))^(2*n)*(9 + 4*sqrt(5)) + (9 - 4*sqrt(5))*(9 + 4*sqrt(5))^(2*n)).

a(n) = 5*A000290(A075796(n)) + 5.