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A073817
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Tetranacci numbers with different initial conditions: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) starting with a(0)=4, a(1)=1, a(2)=3, a(3)=7.
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42
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4, 1, 3, 7, 15, 26, 51, 99, 191, 367, 708, 1365, 2631, 5071, 9775, 18842, 36319, 70007, 134943, 260111, 501380, 966441, 1862875, 3590807, 6921503, 13341626, 25716811, 49570747, 95550687, 184179871, 355018116, 684319421, 1319068095, 2542585503
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OFFSET
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0,1
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COMMENTS
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These tetranacci numbers follow the same pattern as Lucas and generalized tribonacci(A001644) numbers: Binet's formula is a(n) = r1^n + r^2^n + r3^n + r4^n, with r1, r2, r3, r4 roots of the characteristic polynomial.
For n >= 4, a(n) is the number of cyclic sequences consisting of n zeros and ones that do not contain four consecutive ones provided the positions of the zeros and ones are fixed on a circle. This is proved in Charalambides (1991) and Zhang and Hadjicostas (2015). For example, a(4)=15 because only the sequences 1110, 1101, 1011, 0111, 0011, 0101, 1001, 1010, 0110, 1100, 0001, 0010, 0100, 1000, 0000 avoid four consecutive ones on a circle. (For n=1,2,3 the statement is still true provided we allow the sequence to wrap around itself on a circle. For example, a(2)=3 because only the sequences 00, 01, 10 avoid four consecutive ones when wrapped around on a circle.) - Petros Hadjicostas, Dec 18 2016
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.
Spiros D. Dafnis, Andreas N. Philippou, and Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
P. Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), #16.8.2.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
J. L. Ramírez and V. F. Sirvent, A Generalization of the k-Bonacci Sequence from Riordan Arrays, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38.
Yüksel Soykan, Gaussian Generalized Tetranacci Numbers, arXiv:1902.03936 [math.NT], 2019.
Yüksel Soykan, Tetranacci and Tetranacci-Lucas Quaternions, arXiv:1902.05868 [math.RA], 2019.
Yüksel Soykan, Summation Formulas for Generalized Tetranacci Numbers, Asian Journal of Advanced Research and Reports (2019) Vol. 7, No. 2, Article No. AJARR.52434, 1-12.
Yüksel Soykan, Properties of Generalized (r, s, t, u)-Numbers, Earthline J. of Math. Sci. (2021) Vol. 5, No. 2, 297-327.
Kai Wang, Identities, generating functions and Binet formula for generalized k-nacci sequences, 2020.
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
E. Weisstein, Fibonacci n-Step
L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).
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FORMULA
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G.f.: (4 - 3*x - 2*x^2 - x^3)/(1 - x - x^2 - x^3 - x^4).
a(n) = 2*a(n-1) - a(n-5), with a(0)=4, a(1)=1, a(2)=3, a(3)=7, a(4)=15. - Vincenzo Librandi, Dec 20 2010
a(n) = A000078(n+2) + 2*A000078(n+1) + 3*A000078(n) + 4*A000078(n-1). - Advika Srivastava, Aug 22 2019
a(n) = 8*a(n-3) - a(n-5) - 2*a(n-6) - 4*a(n-7). - Advika Srivastava, Aug 25 2019
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MATHEMATICA
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a[0]=4; a[1]=1; a[2]=3; a[3]=7; a[4]=15; a[n_]:= 2*a[n-1] -a[n-5]; Array[a, 34, 0]
CoefficientList[Series[(4-3x-2x^2-x^3)/(1-x-x^2-x^3-x^4), {x, 0, 40}], x]
LinearRecurrence[{1, 1, 1, 1}, {4, 1, 3, 7}, 40] (* Harvey P. Dale, Jun 01 2015 *)
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PROG
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(PARI) Vec((4-3*x-2*x^2-x^3)/(1-x-x^2-x^3-x^4) + O(x^40)) \\ Michel Marcus, Jan 29 2016
(MAGMA) I:=[4, 1, 3, 7]; [n le 4 select I[n] else Self(n-1) +Self(n-2) +Self(n-3) +Self(n-4): n in [1..40]]; // G. C. Greubel, Feb 19 2019
(Sage) ((4-3*x-2*x^2-x^3)/(1-x-x^2-x^3-x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 19 2019
(GAP) a:=[4, 1, 3, 7];; for n in [5..40] do a[n]:=a[n-1]+a[n-2]+a[n-3] +a[n-4]; od; a; # G. C. Greubel, Feb 19 2019
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CROSSREFS
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Cf. A000078, A001630, A001644, A000032, A106295 (Pisano periods). Two other versions: A001648, A074081.
Sequence in context: A200171 A109531 A200132 * A074081 A132703 A176217
Adjacent sequences: A073814 A073815 A073816 * A073818 A073819 A073820
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KEYWORD
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nonn,easy
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AUTHOR
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Mario Catalani (mario.catalani(AT)unito.it), Aug 12 2002
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EXTENSIONS
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Typo in definition corrected by Vincenzo Librandi, Dec 20 2010
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STATUS
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approved
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