A220082 Numbers k such that 10*k-1 is a square.

1, 5, 17, 29, 53, 73, 109, 137, 185, 221, 281, 325, 397, 449, 533, 593, 689, 757, 865, 941, 1061, 1145, 1277, 1369, 1513, 1613, 1769, 1877, 2045, 2161, 2341, 2465, 2657, 2789, 2993, 3133, 3349, 3497, 3725, 3881, 4121, 4285, 4537, 4709, 4973, 5153, 5429, 5617, 5905

Offset: 1

Author

Bruno Berselli, Dec 05 2012

Keywords

• nonn
• easy

Comments

Equivalently, numbers of the form m*(10*m+6)+1, where m=0,-1,1,-2,2,-3,3,...

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A085787, A132356 (numbers n such that 10*n+1 is a square).

Cf. numbers n such that k*n-1 is a square: A002522 (k=1), A001844 (k=2), A062317 (k=5).

Programs

Magma

[n: n in [1..6000] | IsSquare(10*n-1)]; /* or (see the first comment): */ [1] cat [m*(10*m+6)+1: m in [-n,n], n in [1..24]];

Magma

I:=[1,5,17,29,53]; [n le 5 select I[n] else Self(n-1) +2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Aug 18 2013

Maple

A220082:=proc(q)
local n;
for n from 1 to q do if type(sqrt(10*n-1), integer) then print(n);
fi; od; end:
A220082(1000); # Paolo P. Lava, Feb 19 2013

Mathematica

Select[Range[0, 6000], IntegerQ[Sqrt[10 # - 1]] &]
CoefficientList[Series[(1 + 4 x + 10 x^2 + 4 x^3 + x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,5,17,29,53},50] (* Harvey P. Dale, Nov 19 2023 *)

Formula

G.f.: x*(1+4*x+10*x^2+4*x^3+x^4)/((1+x)^2*(1-x)^3).

a(n) = a(-n+1) = (10*n*(n-1)-(2*n-1)*(-1)^n+3)/4.

For the definition: 10*a(n)-1 = ((10*n-(-1)^n-5)/2)^2.

a(n) = A212570(n)-A212570(n-1) = 4*A085787(n-1)+1 = A132356(n-1)-(2*n-1)*(-1)^n.