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A306380 Squares of the form 5*k^2 + 5. 2
25, 7225, 2325625, 748843225, 241125192025, 77641562988025, 25000342156951225, 8050032532975305625, 2592085475275891459225, 834643473006304074564025, 268752606222554636118156025, 86537504560189586525971675225 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All terms of this sequence are odd, hence they end with 5.
LINKS
Eric Weisstein's World of Mathematics, Pell Equation.
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.
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]
PROG
(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]];
(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))
CROSSREFS
Cf. A000290, A075796 (associated k).
Sequence in context: A012748 A337725 A036512 * A053860 A369582 A091744
KEYWORD
nonn,easy
AUTHOR
Stefano Spezia, Feb 13 2019
STATUS
approved

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Last modified July 21 20:15 EDT 2024. Contains 374475 sequences. (Running on oeis4.)