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A109437
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a(-1) = a(0) = 0, a(1) = 1; a(n) = 5a(n-1) - 5a(n-2) + a(n-3) + 2*(-1)^(n+1), alternatively a(n) = 3a(n-1) + 3a(n-2) - a(n-3).
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6
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0, 1, 3, 12, 44, 165, 615, 2296, 8568, 31977, 119339, 445380, 1662180, 6203341, 23151183, 86401392, 322454384, 1203416145, 4491210195, 16761424636, 62554488348, 233456528757, 871271626679, 3251629977960, 12135248285160
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OFFSET
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0,3
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COMMENTS
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See A105968 for a similar sequence. Observe the four periodic sequences (1,1,1,1,); (-1,-1,-1,-1); (1,-1,1,-1,); (-1,1,-1,1,); (a(n)) is the (Type 1A) jbasejfor-transform of the periodic sequence (1,1,1,1) with respect to the floretion given in the program code. A109438 is the (Type 1A) jbasejfor-transform of the periodic sequence (-1,-1,-1,-1) with respect to the floretion given in the program code. A001834 is the (Type 1A) jbasejfor-transform of the periodic sequence (1,-1,1,-1) with respect to the floretion given in the program code. A102871 is the (Type 1A) jbasejfor-transform of the periodic sequence (-1,1,-1,1) with respect to the floretion given in the program code.
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REFERENCES
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R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., 58:2 (2020), 140-142.
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LINKS
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Table of n, a(n) for n=0..24.
P. Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6
Index entries for linear recurrences with constant coefficients, signature (3,3,-1).
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FORMULA
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G.f. x/((x+1)(x^2-4x+1))
a(n) = A002530(n)*A002530(n+1). - Zerinvary Lajos, Feb 08 2007
a(-1 - n) = -a(n). a(2*n) = A011916(n). a(2*n + 1) = -A011916(-1 -n). - Michael Somos, Jul 27 2012
6*a(n) = A001353(n)+A001353(n+1)-(-1)^n. - R. J. Mathar, Sep 07 2016
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EXAMPLE
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x + 3*x^2 + 12*x^3 + 44*x^4 + 165*x^5 + 615*x^6 + 2296*x^7 + 8568*x^8 + ...
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MAPLE
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with(numtheory):a := cfrac (tan(Pi/3), 60): > b := cfrac (tan(Pi/6), 60): > seq(nthnumer (b, i)*nthdenom (a, i), i=0..24 ); # Zerinvary Lajos, Feb 08 2007
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MATHEMATICA
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LinearRecurrence[{3, 3, -1}, {0, 1, 3}, 40] (* Harvey P. Dale, Apr 21 2018 *)
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PROG
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Floretion Algebra Multiplication Program, FAMP Code: (-1)^(n+1)jbasejfor[ + .5'ii' + .5'kk' + .5'ij' + .5'ji' + .5'jk' + .5'kj'] 1vesfor = (1, 1, 1, 1, )
(PARI) {a(n) = local(s=1); if( n<0, n = -1 - n; s=-1); s * polcoeff( x / ((x + 1) * (x^2 -4*x + 1)) + x * O(x^n), n)} /* Michael Somos, Jul 27 2012 */
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CROSSREFS
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Cf. A001834, A002530, A011916, A102871, A109438.
Sequence in context: A190051 A220633 A296225 * A331473 A005656 A339066
Adjacent sequences: A109434 A109435 A109436 * A109438 A109439 A109440
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KEYWORD
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nonn,easy
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AUTHOR
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Creighton Dement, Jun 28 2005
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STATUS
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approved
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