A222964 Numbers k such that 25*k+36 is a square.

0, 13, 37, 76, 124, 189, 261, 352, 448, 565, 685, 828, 972, 1141, 1309, 1504, 1696, 1917, 2133, 2380, 2620, 2893, 3157, 3456, 3744, 4069, 4381, 4732, 5068, 5445, 5805, 6208, 6592, 7021, 7429, 7884, 8316, 8797, 9253, 9760, 10240, 10773, 11277, 11836, 12364, 12949, 13501, 14112, 14688

Offset: 1

Author

Vincenzo Librandi, Apr 07 2013

Keywords

• nonn
• easy

Comments

Also, numbers of the form 25m^2+12*m, where m = 0,-1,1,-2,2,-3,3,... - Bruno Berselli, Apr 07 2013

Links

• Vincenzo Librandi, 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. numbers n such that k^2*n+(k+1)^2 is a square: A028552 (k=2), A218864 (k=3), A165717 (k=4).

Cf. numbers of the form k^2*m^2+floor(k^2/2)*m, where m=0,-1,1,-2,2,-3,3,...: A002378 (k=2), A185039 (k=3), A033996 (k=4), this sequence (k=5), A163758 (k=6).

Programs

Magma

[n: n in [0..15000] | IsSquare(25*n+36)];

Magma

I:=[0, 13, 37, 76, 124]; [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..50]];

Magma

[0] cat [25*m^2+12*m where m is n*t: t in [-1, 1], n in [1..20]]; // Bruno Berselli, Apr 07 2013

Mathematica

Select[Range[0, 10000], IntegerQ[Sqrt[25 # + 36]]&] (* or *) CoefficientList[Series[x (13 + 24 x + 13 x^2)/((1+x)^2(1-x)^3), {x, 0, 40}], x]
LinearRecurrence[{1,2,-2,-1,1},{0,13,37,76,124},50] (* Harvey P. Dale, Jan 23 2025 *)

Formula

G.f.: x^2*(13+24*x+13*x^2)/((1+x)^2*(1-x)^3).

a(n) = (50*n*(n-1)+(2*n-1)*(-1)^n+1)/8.

a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).

Sum_{n>=2} 1/a(n) = 25/144 - tan(Pi/50)*Pi/12. - Amiram Eldar, Feb 16 2023