A094864 a(0)=1, a(1)=2, a(2)=6, a(3)=18; for n >= 4, a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
1, 2, 6, 18, 53, 154, 443, 1264, 3582, 10092, 28291, 78962, 219541, 608318, 1680438, 4629414, 12722033, 34882954, 95451407, 260698732, 710802606, 1934955072, 5259642751, 14277467618, 38707663273, 104816737274, 283521290598, 766112145594, 2068131437357
Offset: 0
Links
- Jinyuan Wang, Table of n, a(n) for n = 0..1000
- Bayo Musa Ahmed, Nadia Aldhamri, Fatma Al-Kharousi, Georg Klein, and Abdullahi Umar, Combinatorial results for order-preserving partial injective contraction mappings, arXiv:2203.06417 [math.CO], 2022. See also Contrib. Disc. Math. (2024) Vol. 19, No. 2, 1-19. See pp. 2, 11, 12.
- Elena Barcucci, Renzo Pinzani, and Renzo Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.
- Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 23.
- S. Rinaldi and D. G. Rogers, How the odd terms in the Fibonacci sequence stack up, 8th Nordic Comb. Conf, Aalborg, Denmark, Oct 20 2004. [Broken link]
- S. Rinaldi and D. G. Rogers, How the odd terms in the Fibonacci sequence stack up, Math. Gaz. vol 90, no 519 (2006) pp. 431-442.
- Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).
Programs
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Magma
I:=[1,2,6,18]; [n le 4 select I[n] else 6*Self(n-1)-11*Self(n-2)+6*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jul 30 2019
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Mathematica
Table[((n - 1)*LucasL[2n-1] + 2*Fibonacci[2n+3])/5, {n, 0, 28}] (* Rigoberto Florez, Jul 29 2019 *) LinearRecurrence[{6, -11, 6, -1}, {1, 2, 6, 18}, 40] (* Vincenzo Librandi, Jul 30 2019 *) -
PARI
Vec(-(2*x-1)*(x-1)^2/(x^2-3*x+1)^2 + O(x^40)) \\ Michel Marcus, Feb 14 2016
Formula
O.g.f: -(2*x-1)*(x-1)^2/(x^2-3*x+1)^2 = (-1-2*x)/(x^2-3*x+1)+(2-5*x)/(x^2-3*x+1)^2. - R. J. Mathar, Dec 02 2007
a(n) = (2*F(2*n+3)+(n-1)*L(2*n-1))/5, where F(n) is the n-th Fibonacci number and L(n) is the n-th Lucas number. - Rigoberto Florez, Jul 29 2019