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 A105968 #27 Apr 03 2024 11:51:23
%S A105968 1,4,13,50,185,692,2581,9634,35953,134180,500765,1868882,6974761,
%T A105968 26030164,97145893,362553410,1353067745,5049717572,18845802541,
%U A105968 70333492594,262488167833,979619178740,3655988547125,13644335009762,50921351491921,190041070957924
%N A105968 a(n) = 4*a(n-1) - a(n-2) - 2*(-1)^n, a(0) = 1, a(1) = 4.
%C A105968 This sequence is the (type 1A) "jbasejfor" transformation of the sequence (-1, -1, -1, -1, ..) with respect to the floretion given in the program code. Under the same conditions, the jbasejfor transformation of the sequence (1, 1, 1, 1, ...) is A006253 [Number of perfect matchings (or domino tilings) in C_4 X P_n]; the jbasejfor transformation of the sequence (1, -1, 1, -1, ...) is A001075 [Chebyshev's T(n,x) polynomials evaluated at x=2]; the jbasejfor transformation of the sequence (-1, 1, -1, 1, ...) is A001353 [3*a(n)^2 + 1 is a perfect square]. In this sense, the sequences (a(n)), A006253, A001075 and A001353 form a "quartett".
%C A105968 Floretion Algebra Multiplication Program, FAMP Code: 4jbasejforseq[ + .25'i + .25'j + .25'k + .25i' + .25j' + .25k' + .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' + .25'ki' + .25'kj' + .25e]. ForType: 1A. 1vesforseq = (-1, -1, -1, -1, ..).
%H A105968 Colin Barker, <a href="/A105968/b105968.txt">Table of n, a(n) for n = 0..1000</a>
%H A105968 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-1).
%F A105968 G.f.: (1-x)*(1+2*x)/((1+x)*(1-4*x+x^2)).
%F A105968 a(n) + a(n+1) = A054491(n+1) - A054491(n).
%F A105968 a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3). - _Colin Barker_, May 25 2015
%F A105968 a(n) = ( 4*ChebyshevU(n,2) - 5*ChebyshevU(n-1,2) - (-1)^n )/3. - _G. C. Greubel_, Jan 15 2020
%F A105968 E.g.f.: (exp(2*x)*(4*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - cosh(x) + sinh(x))/3. - _Stefano Spezia_, Sep 19 2023
%p A105968 seq( simplify((4*ChebyshevU(n,2) -5*ChebyshevU(n-1,2) -(-1)^n)/3), n = 0..30); # _G. C. Greubel_, Jan 15 2020
%t A105968 Table[(4*ChebyshevU[n, 2] -5*ChebyshevU[n-1, 2] -(-1)^n)/3, {n,0,30}] (* _G. C. Greubel_, Jan 15 2020 *)
%t A105968 nxt[{n_,a_,b_}]:={n+1,b,4b-a-2(-1)^(n+1)}; NestList[nxt,{1,1,4},30][[;;,2]] (* or *) LinearRecurrence[ {3,3,-1},{1,4,13},30] (* _Harvey P. Dale_, Apr 03 2024 *)
%o A105968 (PARI) Vec((1-x)*(1+2*x)/((1+x)*(1-4*x+x^2)) + O(x^30)) \\ _Colin Barker_, May 25 2015
%o A105968 (Magma) I:=[1,4,13]; [n le 3 select I[n] else 3*Self(n-1) +3*Self(n-2) -Self(n-3): n in [1..30]]; // _G. C. Greubel_, Jan 15 2020
%o A105968 (SageMath) [(4*chebyshev_U(n,2) -5*chebyshev_U(n-1,2) -(-1)^n)/3 for n in (0..30)] # _G. C. Greubel_, Jan 15 2020
%o A105968 (GAP) a:=[1,4,13];; for n in [4..30] do a[n]:=3*a[n-1]+3*a[n-2]-a[n-3]; od; a; # _G. C. Greubel_, Jan 15 2020
%Y A105968 Cf. A001075, A001353, A006253, A054491.
%K A105968 easy,nonn
%O A105968 0,2
%A A105968 _Creighton Dement_, Apr 28 2005