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 A046717 a(n) = 2*a(n-1) + 3*a(n-2), a(0) = a(1) = 1. 53
 1, 1, 5, 13, 41, 121, 365, 1093, 3281, 9841, 29525, 88573, 265721, 797161, 2391485, 7174453, 21523361, 64570081, 193710245, 581130733, 1743392201, 5230176601, 15690529805, 47071589413, 141214768241, 423644304721, 1270932914165, 3812798742493, 11438396227481 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Form the digraph with matrix A = [0,1,1,1;1,0,1,1;1,1,0,1;1,0,1,1]. Then the sequence 0,1,1,5,... or (3^(n-1)-(-1)^n)/2+0^n/3 with g.f. x(1-x)/(1-2x-3x^2) corresponds to the (1,2) term of A^n. - Paul Barry, Oct 02 2004 3*a(n+1) + a(n) = 4*A060925(n); a(n+1) = A015518(n) + A060925(n); a(n+1) - 6*A015518(n) = (-1)^n. - Creighton Dement, Nov 15 2004 The sequence corresponds to the (1,1) term of the matrix [1,2;2,1]^n. - Simone Severini, Dec 04 2004 The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is 2. - Cino Hilliard, Sep 25 2005 a(n)^2 + (2*A015518(n))^2 = a(2n). E.g., a(3) = 13, 2*A015518(3) = 14, A046717(6) = 365. 13^2 + 14^2 = 365. - Gary W. Adamson, Jun 17 2006 Equals INVERTi transform of A104934: (1, 2, 8, 28, 100, 356, 1268, ...). - Gary W. Adamson, Jul 21 2010 a(n) is the number of compositions of n when there are 1 type of 1 and 4 types of other natural numbers. - Milan Janjic, Aug 13 2010 An elephant sequence, see A175655. For the central square just one A vector, with decimal value 341, leads to this sequence (without the first leading 1). For the corner squares this vector leads to the companion sequence A015518 (without the leading 0). - Johannes W. Meijer, Aug 15 2010 Pisano period lengths: 1, 1, 2, 1, 4, 2, 6, 4, 2, 4, 10, 2, 6, 6, 4, 8, 16, 2, 18, 4, ... - R. J. Mathar, Aug 10 2012 REFERENCES John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002. P. D. Jarvis, J. G. Sumner, Matrix group structure and Markov invariants in the strand symmetric phylogenetic substitution model, arXiv preprint arXiv:1307.5574 [q-bio.PE], 2013. Index entries for linear recurrences with constant coefficients, signature (2,3) FORMULA G.f.: (1-x)/((1+x)*(1-3*x)). a(n) = (3^n + (-1)^n)/2. a(n) = Sum_{k=0..n} binomial(n, 2k)2^(2k). - Paul Barry, Feb 26 2003 Binomial transform of A000302 (powers of 4) with interpolated zeros. Inverse binomial transform of A081294. - Paul Barry, Mar 17 2003 E.g.f.: exp(x)cosh(2x). - Paul Barry, Mar 17 2003 a(n) = ceiling(3^n/4) + floor(3^n/4) = ceiling(3^n/4)^2 - floor(3^n/4)^2. - Paul Barry, Jan 17 2005 a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n,j)C(n-j,k)*(1+(-1)^(j-k))/2. - Paul Barry, May 21 2006 a(n) = Sum_{k=0..n} A098158(n,k)*4^(n-k). - Philippe Deléham, Dec 26 2007 a(n) = (3^n + (-1)^n)/2. - M. F. Hasler, Mar 20 2008 a(n) = 2 A015518(n) + (-1)^n; for n > 0, a(n) = A080925(n). - M. F. Hasler, Mar 20 2008 ((1 + sqrt4)^n + (1 - sqrt4)^n)/2. The offset is 0. a(3)=13. - Al Hakanson (hawkuu(AT)gmail.com), Nov 22 2008 If p=1 and p[i]=4 (i > 1), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j] = -1, (i = j+1), and A[i,j] = 0 otherwise, then, for n >= 1, a(n) = det A. - Milan Janjic, Apr 29 2010 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(4*k-1)/(x*(4*k+3) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013 G.f.: G(0)/2, where G(k) = 1 + (-1)^k/(3^k - 3*9^k*x/(3*3^k*x + (-1)^k/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013 MAPLE a:=1:a:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2] od: seq(a[n], n=0..33); # Zerinvary Lajos, Dec 14 2008 seq(denom(((-2)^(2*n)+6^(2*n))/((-2)^n+6^n)), n=0..26) MATHEMATICA Table[(3^n + (-1)^n)/2, {n, 0, 30}] (* Artur Jasinski, Dec 10 2006 *) CoefficientList[ Series[(1 - x)/(1 - 2x - 3x^2), {x, 0, 30}], x]  (* Robert G. Wilson v, Apr 04 2011 *) Table[ MatrixPower[{{1, 2}, {1, 1}}, n][[1, 1]], {n, 0, 30}] (* Robert G. Wilson v, Apr 04 2011 *) PROG (PARI) {a(n) = (3^n+(-1)^n)/2}; for(n=0, 30, print1(a(n), ", ")) /* modified by G. C. Greubel, Jan 07 2018 */ (Sage) [lucas_number2(n, 2, -3)/2 for n in range(0, 27)] # Zerinvary Lajos, Apr 30 2009 (MAGMA) [n le 2 select 1 else 2*Self(n-1)+3*Self(n-2): n in [1..35]]; // Vincenzo Librandi, Jul 21 2013 (PARI) x='x+O('x^30); Vec((1-x)/((1+x)*(1-3*x))) \\ G. C. Greubel, Jan 07 2018 (MAGMA) [(3^n + (-1)^n)/2: n in [0..30]]; // G. C. Greubel, Jan 07 2018 CROSSREFS The first difference sequence of A015518. Row sums of triangle A080928. The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519. Cf. A015518. Cf. A104934. - Gary W. Adamson, Jul 21 2010 Sequence in context: A182300 A080925 A164907 * A085601 A147718 A111009 Adjacent sequences:  A046714 A046715 A046716 * A046718 A046719 A046720 KEYWORD nonn,easy AUTHOR Gervais Deroo and M. Deroo EXTENSIONS Description corrected by and more terms from Michael Somos STATUS approved

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Last modified October 21 19:16 EDT 2021. Contains 348155 sequences. (Running on oeis4.)