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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 + r2^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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A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.
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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) = Trace(M^n), where M is the 4 X 4 matrix [0, 0, 0, 1; 1, 0, 0, 1; 0, 1, 0, 1; 0, 0, 1, 1], the companion matrix to the monic polynomial x^4 - x^3 - x^2 - x - 1. It follows that the sequence satisfies the Gauss congruences: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. - Peter Bala, Dec 31 2022
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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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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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STATUS
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approved
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