A273182 a(n) is the second number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of a magic square of squares.

14, 84, 490, 2856, 16646, 97020, 565474, 3295824, 19209470, 111960996, 652556506, 3803378040, 22167711734, 129202892364, 753049642450, 4389094962336, 25581520131566, 149100025827060, 869018634830794, 5065011783157704, 29521052064115430, 172061300601534876

Offset: 0

Author

Eddie Gutierrez, May 17 2016

Keywords

• nonn
• easy

Comments

The multiplying factor 6 appears to come from the ratio of a(1)/a(0) of the sequence. Each of the lines of tables (V vs VII) or (VI vs VIII) in oddwheel.com/ImaginaryB.html generates this factor.

Examples

			a(2) = 84*6 -14 = 490; a(3) = 490*6 - 84 = 2856; a(4) = 2856*6 - 490 = 16646.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
• E. Gutierrez, Recursion Methods to Generate New Integer Sequences (Part VIF)
• E. Gutierrez, Table of Tuples and Use of Magic Ratio for Tuple Conversion (Part IB)
• E. Gutierrez, Table of Tuples for Square of Squares (Part IC)
• Index entries for linear recurrences with constant coefficients, signature (6,-1).

Crossrefs

Cf. A178218, A273187, A273189.

Programs

Mathematica

CoefficientList[Series[14/(1 - 6 x + x^2), {x, 0, 21}], x] (* Michael De Vlieger, May 18 2016 *)

PARI

Vec(14/(1-6*x+x^2) + O(x^50)) \\ Colin Barker, May 18 2016

Formula

a(0)=14, a(1)= 84, a(n+1)= a(n)*6 - a(n-1).

G.f.: 14 / (1-6*x+x^2). - Colin Barker, May 18 2016

E.g.f.: 7*(3*sqrt(2)*sinh(2*sqrt(2)*x) + 4*cosh(2*sqrt(2)*x))*exp(3*x)/2. - Ilya Gutkovskiy, May 18 2016