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A105968 a(n) = 4*a(n-1) - a(n-2) - 2*(-1)^n, a(0) = 1, a(1) = 4. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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".

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

FORMULA

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

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

a(n) = 1/6*sqrt(3)*((2+sqrt(3))^n-(2-sqrt(3))^n)+2/3*((2-sqrt(3))^n+(2+sqrt(3))^n) -1/3*(-1)^n. - Paolo P. Lava, Jul 27 2011

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

PROG

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, ..).

(PARI) Vec(-(x-1)*(2*x+1) / ((x+1)*(x^2-4*x+1)) + O(x^100)) \\ Colin Barker, May 25 2015

CROSSREFS

Cf. A006253, A001075, A001353, A054491.

Sequence in context: A230081 A056275 A149455 * A149456 A149457 A149458

Adjacent sequences:  A105965 A105966 A105967 * A105969 A105970 A105971

KEYWORD

easy,nonn

AUTHOR

Creighton Dement, Apr 28 2005

STATUS

approved

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Last modified September 26 15:27 EDT 2017. Contains 292531 sequences.