A094347 a(n) = 14*a(n-1) - a(n-2); a(0) = a(1) = 2.

2, 2, 26, 362, 5042, 70226, 978122, 13623482, 189750626, 2642885282, 36810643322, 512706121226, 7141075053842, 99462344632562, 1385331749802026, 19295182152595802, 268747218386539202, 3743165875258953026

Offset: 0

Author

Lekraj Beedassy, Jun 03 2004

Keywords

• nonn
• easy

Comments

Even x satisfying the Pellian x^2 - 3*y^2 = 1. For corresponding y see A028230.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..874
• Christian Aebi and Grant Cairns, Less than Equable Triangles on the Eisenstein lattice, arXiv:2312.10866 [math.CO], 2023.
• R. K. Guy, Letter to N. J. A. Sloane concerning A001075, A011943, A094347 [Scanned and annotated letter, included with permission]
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (14,-1).

Crossrefs

a(n) = 2*A001570(n).

Bisection of A001075.

Cf. A028230.

Programs

Mathematica

LinearRecurrence[{14,-1},{2,2},40] (* or *) CoefficientList[ Series[2(1-13x)/(1-14x+x^2),{x,0,39}],x] (* Harvey P. Dale, Apr 23 2011 *)

Maxima

(a[0]:2, a[1]:2, a[n] := 14*a[n - 1] - a[n-2], makelist(a[n], n, 0, 50)); /* Franck Maminirina Ramaharo, Nov 12 2018 */

Formula

G.f.: 2*(1 - 13*x)/(1 - 14*x + x^2). [Philippe Deléham, Nov 17 2008]

a(n) = ((2 + sqrt(3))^(2*n - 1) + (2 - sqrt(3))^(2*n - 1))/2. - Gerry Martens, Jun 03 2015

a(n) = (1/2)*sqrt(4 + (-2*sqrt(-2 + (7 - 4*sqrt(3))^(2*n) + (7 + 4*sqrt(3))^(2*n)) + sqrt(3)*sqrt(2 + (7 - 4*sqrt(3))^(2*n) + (7 + 4*sqrt(3))^(2*n)))^2). - Gerry Martens, Jun 03 2015

E.g.f.: exp(7*x)*(2*cosh(4*sqrt(3)*x) - sqrt(3)*sinh(4*sqrt(3)*x)). - Franck Maminirina Ramaharo, Nov 12 2018

Extensions

Corrected by Lekraj Beedassy, Jun 11 2004