A247375 Numbers m such that floor(m/2) is a square.

0, 1, 2, 3, 8, 9, 18, 19, 32, 33, 50, 51, 72, 73, 98, 99, 128, 129, 162, 163, 200, 201, 242, 243, 288, 289, 338, 339, 392, 393, 450, 451, 512, 513, 578, 579, 648, 649, 722, 723, 800, 801, 882, 883, 968, 969, 1058, 1059, 1152, 1153, 1250, 1251, 1352, 1353

Offset: 0

Author

Bruno Berselli, Sep 15 2014

Keywords

• nonn
• easy

Comments

Union of A001105 and A058331.

Squares of the sequence are listed in A055792.

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A001105, A055792, A058331, A213037.

Cf. A130404 (numbers m such that floor(m/2) is a triangular number).

Programs

Magma

[n: n in [0..1400] | IsSquare(Floor(n div 2))];

Mathematica

Select[Range[0, 1400], IntegerQ[Sqrt[Floor[#/2]]] &]
LinearRecurrence[{1,2,-2,-1,1},{0,1,2,3,8},70] (* Harvey P. Dale, Oct 21 2021 *)

Sage

[n for n in [0..1400] if is_square(floor(n/2))]

Formula

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

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

a(n+1) = 1 + A213037(n).

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n >= 5. - Wesley Ivan Hurt, Dec 18 2020

Sum_{n>=1} 1/a(n) = Pi^2/12 + coth(Pi/sqrt(2))*Pi/(2*sqrt(2)) + 1/2. - Amiram Eldar, Sep 24 2022