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 A000285 a(0) = 1, a(1) = 4, and a(n) = a(n-1) + a(n-2) for n >= 2. (Formerly M3246 N1309) 46
 1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, 411, 665, 1076, 1741, 2817, 4558, 7375, 11933, 19308, 31241, 50549, 81790, 132339, 214129, 346468, 560597, 907065, 1467662, 2374727, 3842389, 6217116, 10059505, 16276621, 26336126, 42612747, 68948873, 111561620, 180510493, 292072113, 472582606 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(4;n-1-k,k), n >= 1, with a(-1)=3. These are the sums over the SW-NE diagonals in P(4;n,k), the (4,1) Pascal triangle A093561. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also SW-NE diagonal sums in the Pascal (1,3) triangle A095660. In general, for a Fibonacci sequence beginning with 1,b we have a(n) = (2^(-1-n)*((1-sqrt(5))^n*(1+sqrt(5)-2b) + (1+sqrt(5))^n*(-1+sqrt(5)+2b)))/sqrt(5). In this case we have b=4. - Herbert Kociemba, Dec 18 2011 Pisano period lengths: 1, 3, 8, 6, 20, 24, 16, 12, 24, 60, 5, 24, 28, 48, 40, 24, 36, 24, 18, 60, ... - R. J. Mathar, Aug 10 2012 a(n) = number of independent vertex subsets (i.e., the Merrifield-Simmons index) of the tree obtained from the path tree P_{n-1} by attaching two pendant edges to one of its endpoints (n >= 2). Example: if n=3, then we have the star tree with edges ab, ac, ad; it has 9 independent vertex susbsets: empty, a, b, c, d, bc, cd, bd, bcd. For n >= 2, the number a(n-1) is the dimension of a commutative Hecke algebra of type D_n with independent parameters. See Theorem 1.4 and Corollary 1.5 in the link "Hecke algebras with independent parameters". - Jia Huang, Jan 20 2019 REFERENCES R. E. Merrifield, H. E. Simmons, Topological Methods in Chemistry, Wiley, New York, 1989. pp. 131. J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 224. 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). LINKS T. D. Noe, Table of n, a(n) for n = 0..500 B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.8.5. A. Brousseau, Seeking the lost gold mine or exploring Fibonacci factorizations, Fib. Quart., 3 (1965), 129-130. Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 53. Jia Huang, Hecke algebras with independent parameters, arXiv preprint arXiv:1405.1636 [math.RT], 2014; Journal of Algebraic Combinatorics 43 (2016) 521-551. Tanya Khovanova, Recursive Sequences Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012. Index entries for linear recurrences with constant coefficients, signature (1,1). FORMULA G.f.: (1+3*x)/(1-x-x^2). - Simon Plouffe in his 1992 dissertation Row sums of A131775 starting (1, 4, 5, 9, 14, 23, ...). - Gary W. Adamson, Jul 14 2007 a(n) = 2*Fibonacci(n) + Fibonacci(n+2). - Zerinvary Lajos, Oct 05 2007 a(n) = ((1+sqrt(5))^n - (1-sqrt(5))^n)/(2^n*sqrt(5)) + (3/2)* ((1+sqrt(5))^(n-1) - (1-sqrt(5))^(n-1))/(2^(n-2)*sqrt(5)). Offset 1. a(3)=5. - Al Hakanson (hawkuu(AT)gmail.com), Jan 14 2009 a(n) = 3*Fibonacci(n+2) - 2*Fibonacci(n+1). - Gary Detlefs, Dec 21 2010 a(n) = A104449(n+1). - Michael Somos, Apr 07 2012 From Michael Somos, May 28 2014: (Start) a(n) = A101220(3, 0, n+1). a(n) = A109754(3, n+1). a(k) = A090888(2, k-1), for k > 0. a(-1 - n) = (-1)^n * A013655(n). a(n) = Fibonacci(n) + Lucas(n+1), see Mathematica field. (End) 11*Fibonacci(n+1) = a(n+3) - a(n-2) = 3*a(n-1) + 2*a(n). - Manfred Arens and Michel Marcus, Jul 14 2014 a(n) = 4*A000045(n) + A000045(n-1). - Paolo P. Lava, May 18 2015 a(n) = (9*F(n) + F(n-3))/2. - J. M. Bergot, Jul 15 2017 EXAMPLE G.f. = 1 + 4*x + 5*x^2 + 9*x^3 + 14*x^4 + 23*x^5 + 37*x^6 + 60*x^7 + ... MAPLE with(combinat):a:=n->2*fibonacci(n)+fibonacci(n+2): seq(a(n), n=0..34); MATHEMATICA LinearRecurrence[{1, 1}, {1, 4}, 40] (* or *) Table[(3*LucasL[n]- Fibonacci[n])/2, {n, 40}] (* Harvey P. Dale, Jul 18 2011 *) a[ n_]:= Fibonacci[n] + LucasL[n+1]; (* Michael Somos, May 28 2014 *) PROG (Haskell) a000285 n = a000285_list !! n a000285_list = 1 : 4 : zipWith (+) a000285_list (tail a000285_list) -- Reinhard Zumkeller, Apr 28 2011 (Maxima) a:1\$ a:4\$ a[n]:=a[n-1]+a[n-2]\$ makelist(a[n], n, 0, 30); /* Martin Ettl, Oct 25 2012 */ (PARI) Vec((1+3*x)/(1-x-x^2)+O(x^40)) \\ Charles R Greathouse IV, Nov 20 2012 (Magma) a0:=1; a1:=4; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // Bruno Berselli, Feb 12 2013 (Sage) f=fibonacci; [f(n+2) +2*f(n) for n in (0..40)] # G. C. Greubel, Nov 08 2019 (GAP) F:=Fibonacci;; List([0..40], n-> F(n+2) +2*F(n) ); // G. C. Greubel, Nov 08 2019 CROSSREFS Essentially the same as A104449, which only has A104449(0)=3 prefixed. Cf. A090888, A101220, A109754, A091157 (subsequence of primes). Cf. A013655, A131775. Sequence in context: A120740 A274282 A347553 * A042031 A041493 A352321 Adjacent sequences:  A000282 A000283 A000284 * A000286 A000287 A000288 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified October 2 05:50 EDT 2022. Contains 357191 sequences. (Running on oeis4.)