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A002203
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Companion Pell numbers: a(n) = 2*a(n-1) + a(n-2), a(0) = a(1) = 2.
(Formerly M0360 N0136)
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158
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2, 2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, 39202, 94642, 228486, 551614, 1331714, 3215042, 7761798, 18738638, 45239074, 109216786, 263672646, 636562078, 1536796802, 3710155682, 8957108166, 21624372014, 52205852194, 126036076402, 304278004998
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OFFSET
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0,1
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COMMENTS
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Also the number of matchings (independent edge sets) of the n-sunlet graph. - Eric W. Weisstein, Mar 09 2016
The signed sequence 2, -2, 6, -14, 34, -82, 198, -478, 1154, -2786, ... is the Lucas V(-2,-1) sequence. - R. J. Mathar, Jan 08 2013
Also named "Pell-Lucas numbers", apparently by Hoggatt and Alexanderson (1976), after the English mathematician John Pell (1611-1685) and the French mathematician Édouard Lucas (1842-1891). - Amiram Eldar, Oct 02 2023
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REFERENCES
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Paul Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 76.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 43.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Gerald L. Alexanderson, Problem B-102, Fib. Quart., 4 (1966), 373.
Ezgi Kantarcı Oguz, Cem Yalım Özel, and Mohan Ravichandran, Chainlink Polytopes, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #67.
Hideyuki Ohtsuka, Problem 12090, The American Mathematical Monthly, Vol. 126, No. 2 (2019), p. 180; A Pell-Lucas Computation of Pi, Solution to Problem 12090 by M. Vowe, ibid., Vol. 127, No. 7 (2020), pp. 666-667.
Eric Weisstein's World of Mathematics, Matching.
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FORMULA
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O.g.f.: (2 - 2*x)/(1 - 2*x - x^2). - Simon Plouffe in his 1992 dissertation
a(n) = (1 + sqrt(2))^n + (1 - sqrt(2))^n. - Mario Catalani (mario.catalani(AT)unito.it), Mar 17 2003
Given F(n) = A000129(n), the Pell numbers, and L(n) = a(n), then:
L(n+m) + (-1)^m*L(n-m) = L(n)*L(m).
L(n+m) - (-1)^m*L(n-m) = 8*F(n)*F(m).
L(n+m+k) + (-1)^k*L(n+m-k) + (-1)^m*(L(n-m+k) + (-1)^k*L(n-m-k)) = L(n)*L(m)*L(k).
L(n+m+k) - (-1)^k*L(n+m-k) + (-1)^m*(L(n-m+k) - (-1)^k*L(n-m-k)) = 8*F(n)*L(m)*F(k).
L(n+m+k) + (-1)^k*L(n+m-k) - (-1)^m*(L(n-m+k) + (-1)^k*L(n-m-k)) = 8*F(n)*F(m)*L(k).
L(n+m+k) - (-1)^k*L(n+m-k) - (-1)^m*(L(n-m+k) - (-1)^k*L(n-m-k)) = 8*L(n)*F(m)*F(k).
(End)
G.f.: G(0), where G(k) = 1 + 1/(1 - x*(2*k - 1)/(x*(2*k + 1) - 1/G(k + 1))); (continued fraction). - Sergei N. Gladkovskii, Jun 19 2013
a(n) = [x^n] ( (1 + 2*x + sqrt(1 + 4*x + 8*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015
a(n)^2 - a(n + 1) * a(n - 1) = (-1)^(n) * 8.
a(n)^2 - a(n + r) * a(n - r) = (-1)^(n - r - 1) * 8 * A000129(r)^2.
a(m) * a(n + 1) - a(m + 1) * a(n) = (-1)^(n - 1) * 8 * A000129(m - n).
(End)
a(n) = (-1)^n * (a(n)^3 - a(3*n))/3. - Greg Dresden, Jun 16 2021
a(n) = (a(n+2) + a(n-2))/6 for n >= 2. - Greg Dresden, Jun 23 2021
a(3n+2)/a(3n-1) = [14, ..., 14, -3] with (n+1) 14's.
a(3n+3)/a( 3n ) = [14, ..., 14, 7] with n 14's.
a(3n+4)/a(3n+1) = [14, ..., 14, 17] with n 14's. (End)
a(n) = trace([2, 1; 1, 0]^n) for n >= 1.
The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k) for all positive integers n and k and all primes p.
a(3^n) == A271222(n) (mod 3^n). (End)
Sum_{n>=1} arctan(2/a(n))*arctan(2/a(n+1)) = Pi^2/32 (A244854) (Ohtsuka, 2019). - Amiram Eldar, Feb 11 2024
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MAPLE
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option remember;
if n <= 1 then
2;
else
2*procname(n-1)+procname(n-2) ;
end if;
# second Maple program:
a:= n-> (<<0|1>, <1|2>>^n. <<2, 2>>)[1, 1]:
a := n -> 2*I^n*ChebyshevT(n, -I):
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MATHEMATICA
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Table[(1 - Sqrt[2])^n + (1 + Sqrt[2])^n, {n, 0, 20}] // Expand (* Eric W. Weisstein, Oct 03 2017 *)
CoefficientList[Series[(2 (1 - x))/(1 - 2 x - x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Oct 03 2017 *)
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PROG
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(Sage) [lucas_number2(n, 2, -1) for n in range(0, 29)] # Zerinvary Lajos, Apr 30 2009
(Haskell)
a002203 n = a002203_list !! n
a002203_list =
2 : 2 : zipWith (+) (map (* 2) $ tail a002203_list) a002203_list
(Magma) I:=[2, 2]; [n le 2 select I[n] else 2*Self(n-1)+Self(n-2): n in [1..35]] // Vincenzo Librandi, Aug 15 2015
(PARI) first(m)=my(v=vector(m)); v[1]=2; v[2]=2; for(i=3, m, v[i]=2*v[i-1]+v[i-2]); v; \\ Anders Hellström, Aug 15 2015
(PARI) a(n) = my(w=quadgen(8)); (1+w)^n + (1-w)^n; \\ Michel Marcus, Jun 17 2021
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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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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Dec 03 2001
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STATUS
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approved
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