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A094347 a(n) = 14*a(n-1) - a(n-2); a(0) = a(1) = 2. 6

%I #40 Dec 21 2023 11:57:41

%S 2,2,26,362,5042,70226,978122,13623482,189750626,2642885282,

%T 36810643322,512706121226,7141075053842,99462344632562,

%U 1385331749802026,19295182152595802,268747218386539202,3743165875258953026

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

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

%H Michael De Vlieger, <a href="/A094347/b094347.txt">Table of n, a(n) for n = 0..874</a>

%H Christian Aebi and Grant Cairns, <a href="https://arxiv.org/abs/2312.10866">Less than Equable Triangles on the Eisenstein lattice</a>, arXiv:2312.10866 [math.CO], 2023.

%H R. K. Guy, <a href="/A001075/a001075.pdf">Letter to N. J. A. Sloane concerning A001075, A011943, A094347</a> [Scanned and annotated letter, included with permission]

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-1).

%F G.f.: 2*(1 - 13*x)/(1 - 14*x + x^2). [_Philippe Deléham_, Nov 17 2008]

%F a(n) = ((2 + sqrt(3))^(2*n - 1) + (2 - sqrt(3))^(2*n - 1))/2. - _Gerry Martens_, Jun 03 2015

%F 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

%F 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

%t 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 *)

%o (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 */

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

%Y Bisection of A001075.

%Y Cf. A028230.

%K nonn,easy

%O 0,1

%A _Lekraj Beedassy_, Jun 03 2004

%E Corrected by _Lekraj Beedassy_, Jun 11 2004

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Last modified March 29 07:27 EDT 2024. Contains 371265 sequences. (Running on oeis4.)