A105968 a(n) = 4*a(n-1) - a(n-2) - 2*(-1)^n, a(0) = 1, a(1) = 4.

1, 4, 13, 50, 185, 692, 2581, 9634, 35953, 134180, 500765, 1868882, 6974761, 26030164, 97145893, 362553410, 1353067745, 5049717572, 18845802541, 70333492594, 262488167833, 979619178740, 3655988547125, 13644335009762, 50921351491921, 190041070957924

Offset: 0

Author

Creighton Dement, Apr 28 2005

Keywords

• easy
• nonn

Comments

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".

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, ..).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Crossrefs

Cf. A001075, A001353, A006253, A054491.

Programs

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

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

Maple

seq( simplify((4*ChebyshevU(n,2) -5*ChebyshevU(n-1,2) -(-1)^n)/3), n = 0..30); # G. C. Greubel, Jan 15 2020

Mathematica

Table[(4*ChebyshevU[n, 2] -5*ChebyshevU[n-1, 2] -(-1)^n)/3, {n,0,30}] (* G. C. Greubel, Jan 15 2020 *)
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 *)

PARI

Vec((1-x)*(1+2*x)/((1+x)*(1-4*x+x^2)) + O(x^30)) \\ Colin Barker, May 25 2015

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

Formula

G.f.: (1-x)*(1+2*x)/((1+x)*(1-4*x+x^2)).

a(n) + a(n+1) = A054491(n+1) - A054491(n).

a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3). - Colin Barker, May 25 2015

a(n) = ( 4*ChebyshevU(n,2) - 5*ChebyshevU(n-1,2) - (-1)^n )/3. - G. C. Greubel, Jan 15 2020

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