A273189 -id:A273189 - cp's OEIS frontend

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

Eddie Gutierrez, May 17 2016

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.

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

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).

Cf. A178218, A273187, A273189.

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

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

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

A273187 a(n) is the third number in a triple consisting of 3 numbers, which when squared are part of a right diagonal of magic square of squares.

Original entry on oeis.org

99, 449, 2499, 14449, 84099, 490049, 2856099, 16646449, 97022499, 565488449, 3295908099, 19209960049, 111963852099, 652573152449, 3803475062499, 22168277222449, 129206188272099, 753068852410049, 4389206926188099, 25582172704718449, 149103829302122499

Offset: 0

Eddie Gutierrez, May 17 2016

			a(2)= 449*6 - (99 + 96)= 2499;
a(3)= 2499*6 - (449 + 96)= 14449.

Colin Barker, Table of n, a(n) for n = 0..1000
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 (7,-7,1).

Cf. A273182, A273189.

CoefficientList[Series[(99 - 244 x + 49 x^2)/((1 - x) (1 - 6 x + x^2)), {x, 0, 20}], x] (* Michael De Vlieger, May 20 2016 *)

Vec((99-244*x+49*x^2)/((1-x)*(1-6*x+x^2)) + O(x^50)) \\ Colin Barker, May 18 2016

a(0)= 99, a(1)= 449, a(n+1)= a(n)*6 - a(n-1) - k where k=96.
From Colin Barker, May 18 2016: (Start)
a(n) = (24+25/2*(3-2*sqrt(2))^(1+n)+25/2*(3+2*sqrt(2))^(1+n)).
a(n) = 7*a(n-1)-7*a(n-2)+a(n-3) for n>2.
G.f.: (99-244*x+49*x^2) / ((1-x)*(1-6*x+x^2)).
(End)
a(n) = 24 + 25*A001541(n+1). - R. J. Mathar, Jun 07 2016

cp's OEIS Frontend

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.

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