This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A105309 #66 Feb 16 2025 08:32:57
%S A105309 1,1,2,5,9,20,41,85,178,369,769,1600,3329,6929,14418,30005,62441,
%T A105309 129940,270409,562725,1171042,2436961,5071361,10553600,21962241,
%U A105309 45703841,95110562,197926885,411889609,857150100,1783745641,3712008565
%N A105309 a(n) = |b(n)|^2 = x^2 + 3*y^2 where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)*(0,c,c,c) where c = 1/sqrt(3).
%C A105309 Prepending 0 and keeping the offset at 0, turns this into a divisibility sequence with g.f. x(1-x^2)/(1-x-2x^2-x^3+x^4). - _T. D. Noe_, Dec 22 2008
%C A105309 Equals INVERT transform of (1, 1, 2, 0, 2, 0, 2, ...). - _Gary W. Adamson_, Apr 28 2009
%C A105309 Sequence gives the norm of the coefficients in 1/(1 - I*x - I*x^2), where I^2=-1. - _Paul D. Hanna_, Dec 06 2011
%C A105309 This is the case P1 = 1, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - _Peter Bala_, Mar 27 2014
%H A105309 Peter Bala, <a href="/A100047/a100047.pdf">Linear divisibility sequences and Chebyshev polynomials</a>
%H A105309 Ricky X. F. Chen and Louis W. Shapiro, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL10/Chen/chen509.html">On Sequences G(n) satisfying G(n) = (d+2)G(n-1)-G(n-2)</a>, J. Int. Seq. 10 (2007) 07.8.1, Theorem 16.
%H A105309 Richard J. Mathar, <a href="https://arxiv.org/abs/2404.18806">Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards</a>, arXiv:2404.18806 [math.CO], 2024. See p. 8.
%H A105309 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Quaternion.html">Quaternion</a>
%H A105309 H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H A105309 H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume
%H A105309 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,1,-1).
%F A105309 a(n) = A092886(n+1) - A092886(n-1), n > 0.
%F A105309 a(n) = A201837(n)^2 + A201838(n)^2. - _Paul D. Hanna_, Dec 06 2011
%F A105309 From _Peter Bala_, Mar 27 2014: (Start)
%F A105309 a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(17))/4 and beta = (1 - sqrt(17))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
%F A105309 a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 1/2].
%F A105309 a(n) = U(n-1,(1 + i)/sqrt(8))*U(n-1,(1 - i)/sqrt(8)), where U(n,x) denotes the Chebyshev polynomial of the second kind.
%F A105309 The o.g.f. is the Chebyshev transform of the rational function x/(1 - x + 4*x^2) = x + x^2 + 5*x^2 + 9*x^4 + 29*x^5 + ... (see A006131), where the Chebyshev transform takes the function A(x) to the function (1 - x^2)/(1 + x^2)*A(x/(1 + x^2)).
%F A105309 See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
%F A105309 a(n) = abs(((sqrt(4*i - 1) + i)^(n+1) - (i - sqrt(4*i - 1))^(n+1)) / 2^(n+1) / sqrt(4*i - 1))^2. - _Daniel Suteu_, Dec 20 2016
%F A105309 a(n) = a(-2-n) for all n in Z. - _Michael Somos_, Dec 20 2016
%F A105309 G.f.: (1+x)*(1-x)/(1-x-2*x^2-x^3+x^4). - _R. J. Mathar_, Apr 26 2024
%e A105309 G.f. = 1 + x + 2*x^2 + 5*x^3 + 9*x^4 + 20*x^5 + 41*x^6 + 85*x^7 + 178*x^8 + ...
%t A105309 a[ n_] := (ChebyshevT[n + 1, (1 + Sqrt[17])/4] - ChebyshevT[n + 1, (1 - Sqrt[17])/4]) 2 / Sqrt[17] // Simplify; (* _Michael Somos_, Dec 20 2016 *)
%o A105309 (PARI) {a(n) = my(A); n = abs(n+1)-1; if( n<2, n>=0, n++; A = vector(n, i, 1); for(i=3, n, A[i] = A[i-1] + A[i-2]*I); norm(A[n]))}; /* _Michael Somos_, Apr 28 2005 */
%o A105309 (PARI) {a(n)=norm(polcoeff(1/(1-I*x-I*x^2+x*O(x^n)), n))} /* _Paul D. Hanna_ */
%o A105309 (PARI) {a(n)=polcoeff((1-x^2)/(1-x-2*x^2-x^3+x^4)+x*O(x^n),n)}
%Y A105309 Cf. A092886, A201837, A201838.
%Y A105309 Cf. A006131, A100047, A240513
%K A105309 nonn,easy
%O A105309 0,3
%A A105309 _Gerald McGarvey_, Apr 25 2005