A358682 Numbers k such that 8*k^2 + 8*k - 7 is a square.

1, 7, 43, 253, 1477, 8611, 50191, 292537, 1705033, 9937663, 57920947, 337588021, 1967607181, 11468055067, 66840723223, 389576284273, 2270616982417, 13234125610231, 77134136678971, 449570694463597, 2620290030102613, 15272169486152083, 89012726886809887, 518804191834707241

Offset: 1

Author

Stefano Spezia, Nov 26 2022

Keywords

• nonn
• easy

Comments

a(n) is the n-th almost cobalancing number of second type (see Tekcan and Erdem).

Examples

			a(2) = 7 is a term since 8*7^2 + 8*7 - 7 = 441 = 21^2.
		

Links

• Ahmet Tekcan and Alper Erdem, General Terms of All Almost Balancing Numbers of First and Second Type, arXiv:2211.08907 [math.NT], 2022.
• Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Crossrefs

Cf. A002315, A006452, A077443, A077446, A106328, A106329, A216134.

Programs

Mathematica

LinearRecurrence[{7,-7,1},{1,7,43},24]

Formula

a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 3.

a(n) = (3*(3 - 2*sqrt(2))^n*(2 + sqrt(2)) + 3*(2 - sqrt(2))*(3 + 2*sqrt(2))^n - 4)/8.

O.g.f.: x*(1 + x^2)/((1 - x)*(1 - 6*x + x^2)).

E.g.f.: (3*(2 + sqrt(2))*(cosh(3*x - 2*sqrt(2)*x) + sinh(3*x - 2*sqrt(2)*x)) + 3*(2 - sqrt(2))*(cosh(3*x + 2*sqrt(2)*x) + sinh(3*x + 2*sqrt(2)*x)) - 4*(cosh(x) + sinh(x)) - 8)/8.

a(n) = 3*A011900(n) - 2 = 6*A053142(n) + 1. - Hugo Pfoertner, Nov 26 2022