A147703 -id:A147703 - cp's OEIS frontend

A001519 a(n) = 3*a(n-1) - a(n-2) for n >= 2, with a(0) = a(1) = 1.

1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445, 365435296162, 956722026041

Offset: 0

N. J. A. Sloane

This is a bisection of the Fibonacci sequence A000045. a(n) = F(2*n-1), with F(n) = A000045(n) and F(-1) = 1.
Number of ordered trees with n+1 edges and height at most 3 (height=number of edges on a maximal path starting at the root). Number of directed column-convex polyominoes of area n+1. Number of nondecreasing Dyck paths of length 2n+2. - Emeric Deutsch, Jul 11 2001
Terms are the solutions x to: 5x^2-4 is a square, with 5x^2-4 in A081071 and sqrt(5x^2-4) in A002878. - Benoit Cloitre, Apr 07 2002
a(0) = a(1) = 1, a(n+1) is the smallest Fibonacci number greater than the n-th partial sum. - Amarnath Murthy, Oct 21 2002
The fractional part of tau*a(n) decreases monotonically to zero. - Benoit Cloitre, Feb 01 2003
Numbers k such that floor(phi^2*k^2) - floor(phi*k)^2 = 1 where phi=(1+sqrt(5))/2. - Benoit Cloitre, Mar 16 2003
Number of leftist horizontally convex polyominoes with area n+1.
Number of 31-avoiding words of length n on alphabet {1,2,3} which do not end in 3. (E.g., at n=3, we have 111, 112, 121, 122, 132, 211, 212, 221, 222, 232, 321, 322 and 332.) See A028859. - Jon Perry, Aug 04 2003
Appears to give all solutions > 1 to the equation: x^2 = ceiling(x*r*floor(x/r)) where r=phi=(1+sqrt(5))/2. - Benoit Cloitre, Feb 24 2004
a(1) = 1, a(2) = 2, then the least number such that the square of any term is just less than the geometric mean of its neighbors. a(n+1)*a(n-1) > a(n)^2. - Amarnath Murthy, Apr 06 2004
All positive integer solutions of Pell equation b(n)^2 - 5*a(n+1)^2 = -4 together with b(n)=A002878(n), n >= 0. - Wolfdieter Lang, Aug 31 2004
Essentially same as Pisot sequence E(2,5).
Number of permutations of [n+1] avoiding 321 and 3412. E.g., a(3) = 13 because the permutations of [4] avoiding 321 and 3412 are 1234, 2134, 1324, 1243, 3124, 2314, 2143, 1423, 1342, 4123, 3142, 2413, 2341. - Bridget Tenner, Aug 15 2005
Number of 1324-avoiding circular permutations on [n+1].
A subset of the Markoff numbers (A002559). - Robert G. Wilson v, Oct 05 2005
(x,y) = (a(n), a(n+1)) are the solutions of x/(yz) + y/(xz) + z/(xy) = 3 with z=1. - Floor van Lamoen, Nov 29 2001
Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 1. - Herbert Kociemba, Jun 10 2004
With interpolated zeros, counts closed walks of length n at the start or end node of P_4. a(n) counts closed walks of length 2n at the start or end node of P_4. The sequence 0,1,0,2,0,5,... counts walks of length n between the start and second node of P_4. - Paul Barry, Jan 26 2005
a(n) is the number of ordered trees on n edges containing exactly one non-leaf vertex all of whose children are leaves (every ordered tree must contain at least one such vertex). For example, a(0) = 1 because the root of the tree with no edges is not considered to be a leaf and the condition "all children are leaves" is vacuously satisfied by the root and a(4) = 13 counts all 14 ordered trees on 4 edges (A000108) except (ignore dots)
  |..|
  .\/.
  which has two such vertices. - David Callan, Mar 02 2005
Number of directed column-convex polyominoes of area n. Example: a(2)=2 because we have the 1 X 2 and the 2 X 1 rectangles. - Emeric Deutsch, Jul 31 2006
Same as the number of Kekulé structures in polyphenanthrene in terms of the number of hexagons in extended (1,1)-nanotubes. See Table 1 on page 411 of I. Lukovits and D. Janezic. - Parthasarathy Nambi, Aug 22 2006
Number of free generators of degree n of symmetric polynomials in 3-noncommuting variables. - Mike Zabrocki, Oct 24 2006
Inverse: With phi = (sqrt(5) + 1)/2, log_phi((sqrt(5)*a(n) + sqrt(5*a(n)^2 - 4))/2) = n for n >= 1. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007
Consider a teacher who teaches one student, then he finds he can teach two students while the original student learns to teach a student. And so on with every generation an individual can teach one more student then he could before. a(n) starting at a(2) gives the total number of new students/teachers (see program). - Ben Paul Thurston, Apr 11 2007
The Diophantine equation a(n)=m has a solution (for m >= 1) iff ceiling(arcsinh(sqrt(5)*m/2)/log(phi)) != ceiling(arccosh(sqrt(5)*m/2)/log(phi)) where phi is the golden ratio. An equivalent condition is A130255(m)=A130256(m). - Hieronymus Fischer, May 24 2007
a(n+1) = B^(n)(1), n >= 0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 2=`0`, 5=`00`, 13=`000`, ..., in Wythoff code.
Bisection of the Fibonacci sequence into odd-indexed nonzero terms (1, 2, 5, 13, ...) and even-indexed terms (1, 3, 8, 21, ...) may be represented as row sums of companion triangles A140068 and A140069. - Gary W. Adamson, May 04 2008
a(n) is the number of partitions pi of [n] (in standard increasing form) such that Flatten[pi] is a (2-1-3)-avoiding permutation. Example: a(4)=13 counts all 15 partitions of [4] except 13/24 and 13/2/4. Here "standard increasing form" means the entries are increasing in each block and the blocks are arranged in increasing order of their first entries. Also number that avoid 3-1-2. - David Callan, Jul 22 2008
Let P be the partial sum operator, A000012: (1; 1,1; 1,1,1; ...) and A153463 = M, the partial sum & shift operator. It appears that beginning with any randomly taken sequence S(n), iterates of the operations M * S(n), -> M * ANS, -> P * ANS, etc. (or starting with P) will rapidly converge upon a two-sequence limit cycle of (1, 2, 5, 13, 34, ...) and (1, 1, 3, 8, 21, ...). - Gary W. Adamson, Dec 27 2008
Number of musical compositions of Rhythm-music over a time period of n-1 units. Example: a(4)=13; indeed, denoting by R a rest over a time period of 1 unit and by N[j] a note over a period of j units, we have (writing N for N[1]): NNN, NNR, NRN, RNN, NRR, RNR, RRN, RRR, N[2]R, RN[2], NN[2], N[2]N, N[3] (see the J. Groh reference, pp. 43-48). - Juergen K. Groh (juergen.groh(AT)lhsystems.com), Jan 17 2010
Given an infinite lower triangular matrix M with (1, 2, 3, ...) in every column but the leftmost column shifted upwards one row. Then (1, 2, 5, ...) = lim_{n->infinity} M^n. (Cf. A144257.) - Gary W. Adamson, Feb 18 2010
As a fraction: 8/71 = 0.112676 or 98/9701 = 0.010102051334... (fraction 9/71 or 99/9701 for sequence without initial term). 19/71 or 199/9701 for sequence in reverse. - Mark Dols, May 18 2010
For n >= 1, a(n) is the number of compositions (ordered integer partitions) of 2n-1 into an odd number of odd parts. O.g.f.: (x-x^3)/(1-3x^2+x^4) = A(A(x)) where A(x) = 1/(1-x)-1/(1-x^2).
For n > 0, determinant of the n X n tridiagonal matrix with 1's in the super and subdiagonals, (1,3,3,3,...) in the main diagonal, and the rest zeros. - Gary W. Adamson, Jun 27 2011
The Gi3 sums, see A180662, of the triangles A108299 and A065941 equal the terms of this sequence without a(0). - Johannes W. Meijer, Aug 14 2011
The number of permutations for which length equals reflection length. - Bridget Tenner, Feb 22 2012
Number of nonisomorphic graded posets with 0 and 1 and uniform Hasse graph of rank n+1, with exactly 2 elements of each rank between 0 and 1. (Uniform used in the sense of Retakh, Serconek and Wilson. Graded used in R. Stanley's sense that all maximal chains have the same length.)
HANKEL transform of sequence and the sequence omitting a(0) is the sequence A019590(n). This is the unique sequence with that property. - Michael Somos, May 03 2012
The number of Dyck paths of length 2n and height at most 3. - Ira M. Gessel, Aug 06 2012
Pisano period lengths: 1, 3, 4, 3, 10, 12, 8, 6, 12, 30, 5, 12, 14, 24, 20, 12, 18, 12, 9, 30, ... - R. J. Mathar, Aug 10 2012
Primes in the sequence are 2, 5, 13, 89, 233, 1597, 28657, ... (apparently A005478 without the 3). - R. J. Mathar, May 09 2013
a(n+1) is the sum of rising diagonal of the Pascal triangle written as a square - cf. comments in A085812. E.g., 13 = 1+5+6+1. - John Molokach, Sep 26 2013
a(n) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 1; 1, 1, 1; 0, 1, 1] or [1, 1, 1; 0, 1, 1; 1, 1, 1] or [1, 1, 0; 1, 1, 1; 1, 1, 1] or [1, 0, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
Except for the initial term, positive values of x (or y) satisfying x^2 - 3xy + y^2 + 1 = 0. - Colin Barker, Feb 04 2014
Except for the initial term, positive values of x (or y) satisfying x^2 - 18xy + y^2 + 64 = 0. - Colin Barker, Feb 16 2014
Positive values of x such that there is a y satisfying x^2 - xy - y^2 - 1 = 0. - Ralf Stephan, Jun 30 2014
a(n) is also the number of permutations simultaneously avoiding 231, 312 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl, Aug 07 2014
(1, a(n), a(n+1)), n >= 0, are Markoff triples (see A002559 and Robert G. Wilson v's Oct 05 2005 comment). In the Markoff tree they give one of the outer branches. Proof: a(n)*a(n+1) - 1 = A001906(2*n)^2 = (a(n+1) - a(n))^2 = a(n)^2 + a(n+1)^2 - 2*a(n)*a(n+1), thus 1^2 + a(n)^2 + a(n+1)^2 = 3*a(n)*a(n+1). - Wolfdieter Lang, Jan 30 2015
For n > 0, a(n) is the smallest positive integer not already in the sequence such that a(1) + a(2) + ... + a(n) is a Fibonacci number. - Derek Orr, Jun 01 2015
Number of vertices of degree n-2 (n >= 3) in all Fibonacci cubes, see Klavzar, Mollard, & Petkovsek. - Emeric Deutsch, Jun 22 2015
Except for the first term, this sequence can be generated by Corollary 1 (ii) of Azarian's paper in the references for this sequence. - Mohammad K. Azarian, Jul 02 2015
Precisely the numbers F(n)^k + F(n+1)^k that are also Fibonacci numbers with k > 1, see Luca & Oyono. - Charles R Greathouse IV, Aug 06 2015
a(n) = MA(n) - 2*(-1)^n where MA(n) is exactly the maximum area of a quadrilateral with lengths of sides in order L(n-2), L(n-2), F(n+1), F(n+1) for n > 1 and L(n)=A000032(n). - J. M. Bergot, Jan 28 2016
a(n) is the number of bargraphs of semiperimeter n+1 having no valleys (i.e., convex bargraphs). Equivalently, number of bargraphs of semiperimeter n+1 having exactly 1 peak. Example: a(5) = 34 because among the 35 (=A082582(6)) bargraphs of semiperimeter 6 only the one corresponding to the composition [2,1,2] has a valley. - Emeric Deutsch, Aug 12 2016
Integers k such that the fractional part of k*phi is less than 1/k. See Byszewski link p. 2. - Michel Marcus, Dec 10 2016
Number of words of length n-1 over {0,1,2,3} in which binary subwords appear in the form 10...0. - Milan Janjic, Jan 25 2017
With a(0) = 0 this is the Riordan transform with the Riordan matrix A097805 (of the associated type) of the Fibonacci sequence A000045. See a Feb 17 2017 comment on A097805. - Wolfdieter Lang, Feb 17 2017
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) < e(j) < e(k). [Martinez and Savage, 2.12] - Eric M. Schmidt, Jul 17 2017
Number of permutations of [n] that avoid the patterns 321 and 2341. - Colin Defant, May 11 2018
The sequence solves the following problem: find all the pairs (i,j) such that i divides 1+j^2 and j divides 1+i^2. In fact, the pairs (a(n), a(n+1)), n > 0, are all the solutions. - Tomohiro Yamada, Dec 23 2018
Number of permutations in S_n whose principal order ideals in the Bruhat order are lattices (equivalently, modular, distributive, Boolean lattices). - Bridget Tenner, Jan 16 2020
From Wolfdieter Lang, Mar 30 2020: (Start)
a(n) is the upper left entry of the n-th power of the 2 X 2 tridiagonal matrix M_2 = Matrix([1,1], [1,2]) from A322602: a(n) = ((M_2)^n)[1,1].
Proof: (M_2)^2 = 3*M + 1_2 (with the 2 X 2 unit matrix 1_2) from the characteristic polynomial of M_2 (see a comment in A322602) and the Cayley-Hamilton theorem. The recurrence M^n = M*M^(n-1) leads to (M_n)^n = S(n, 3)*1_2 + S(n-a, 3)*(M - 3*1_2), for n >= 0, with S(n, 3) = F(2(n+1)) = A001906(n+1). Hence ((M_2)^n)[1,1] = S(n, 3) - 2*S(n-1, 3) = a(n) = F(2*n-1) = (1/(2*r+1))*r^(2*n-1)*(1 + (1/r^2)^(2*n-1)), with r = rho(5) = A001622 (golden ratio) (see the first Aug 31 2004 formula, using the recurrence of S(n, 3), and the Michael Somos Oct 28 2002 formula). This proves a conjecture of Gary W. Adamson in A322602.
The ratio a(n)/a(n-1) converges to r^2 = rho(5)^2 = A104457 for n -> infinity (see the a(n) formula in terms of r), which is one of the statements by Gary W. Adamson in A322602. (End)
a(n) is the number of ways to stack coins with a bottom row of n coins such that any coin not on the bottom row touches exactly two coins in the row below, and all the coins on any row are contiguous [Wilf, 2.12]. - Greg Dresden, Jun 29 2020
a(n) is the upper left entry of the (2*n)-th power of the 4 X 4 Jacobi matrix L with L(i,j)=1 if |i-j| = 1 and L(i,j)=0 otherwise. - Michael Shmoish, Aug 29 2020
All positive solutions of the indefinite binary quadratic F(1, -3, 1) := x^2 - 3*x*y + y^2, of discriminant 5, representing -1 (special Markov triples (1, y=x, z=y) if y <= z) are [x(n), y(n)] = [abs(F(2*n+1)), abs(F(2*n-1))], for n = -infinity..+infinity. (F(-n) = (-1)^(n+1)*F(n)). There is only this single family of proper solutions, and there are no improper solutions. [See also the Floor van Lamoen Nov 29 2001 comment, which uses this negative n, and my Jan 30 2015 comment.] - Wolfdieter Lang, Sep 23 2020
These are the denominators of the lower convergents to the golden ratio, tau; they are also the numerators of the upper convergents (viz. 1/1 < 3/2 < 8/5 < 21/13 < ... < tau < ... 13/8 < 5/3 < 2/1). - Clark Kimberling, Jan 02 2022
a(n+1) is the number of subgraphs of the path graph on n vertices. - Leen Droogendijk, Jun 17 2023
For n > 4, a(n+2) is the number of ways to tile this 3 x n "double-box" shape with squares and dominos (reflections or rotations are counted as distinct tilings). The double-box shape is made up of two horizontal strips of length n, connected by three vertical columns of length 3, and the center column can be located anywhere not touching the two outside columns.
     _   _   _   _  
  |||_|||_|||_|||_|||
  || _  |_|  _   _ ||
  |||_|||_|||_|||_|||. - Greg Dresden and Ruishan Wu, Aug 25 2024
a(n+1) is the number of integer sequences a_1, ..., a_n such that for any number 1 <= k <= n, (a_1 + ... + a_k)^2 = a_1^3 + ... + a_k^3. - Yifan Xie, Dec 07 2024

			a(3) = 13: there are 14 ordered trees with 4 edges; all of them, except for the path with 4 edges, have height at most 3.

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Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (3,-1).
Index entries for "core" sequences
Index entries for two-way infinite sequences

Fibonacci A000045 = union of this sequence and A001906.
Cf. A001653, A055105, A055106, A055107, A074664, A101368, A124292, A124293, A124294, A124295, A140069, A153463, A153266, A153267, A144257, A211216, A002559, A082582.
a(n)= A060920(n, 0).
Row 3 of array A094954.
Equals A001654(n+1) - A001654(n-1), n > 0.
A122367 is another version. Inverse sequences A130255 and A130256. Row sums of A140068, A152251, A153342, A179806, A179745, A213948.
Cf. A001622, A001906, A104457, A153387, A322602.

a:=[1,1];; for n in [3..10^2] do a[n]:=3*a[n-1]-a[n-2]; od; a; # Muniru A Asiru, Sep 27 2017

a001519 n = a001519_list !! n
a001519_list = 1 : zipWith (-) (tail a001906_list) a001906_list
-- Reinhard Zumkeller, Jan 11 2012
a001519_list = 1 : f a000045_list where f (_:x:xs) = x : f xs
-- Reinhard Zumkeller, Aug 09 2013

[1] cat [(Lucas(2*n) - Fibonacci(2*n))/2: n in [1..50]]; // Vincenzo Librandi, Jul 02 2014

A001519:=-(-1+z)/(1-3*z+z**2); # Simon Plouffe in his 1992 dissertation; gives sequence without an initial 1
A001519 := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n>=2 then 3*procname(n-1)-procname(n-2) fi: end: seq(A001519(n), n=0..28); # Johannes W. Meijer, Aug 14 2011

Fibonacci /@ (2Range[29] - 1) (* Robert G. Wilson v, Oct 05 2005 *)
LinearRecurrence[{3, -1}, {1, 1}, 29] (* Robert G. Wilson v, Jun 28 2012 *)
a[ n_] := With[{c = Sqrt[5]/2}, ChebyshevT[2 n - 1, c]/c]; (* Michael Somos, Jul 08 2014 *)
CoefficientList[ Series[(1 - 2x)/(1 - 3x + x^2), {x, 0, 30}], x] (* Robert G. Wilson v, Feb 01 2015 *)

a[0]:1$ a[1]:1$ a[n]:=3*a[n-1]-a[n-2]$ makelist(a[n],n,0,30); /* Martin Ettl, Nov 15 2012 */

{a(n) = fibonacci(2*n - 1)}; /* Michael Somos, Jul 19 2003 */

{a(n) = real( quadgen(5) ^ (2*n))}; /* Michael Somos, Jul 19 2003 */

{a(n) = subst( poltchebi(n) + poltchebi(n - 1), x, 3/2) * 2/5}; /* Michael Somos, Jul 19 2003 */

[lucas_number1(n,3,1)-lucas_number1(n-1,3,1) for n in range(30)] # Zerinvary Lajos, Apr 29 2009

G.f.: (1-2*x)/(1-3*x+x^2).
G.f.: 1 / (1 - x / (1 - x / (1 - x))). - Michael Somos, May 03 2012
a(n) = A001906(n+1) - 2*A001906(n).
a(n) = a(1-n) for all n in Z.
a(n+2) = (a(n+1)^2+1)/a(n) with a(1)=1, a(2)=2. - Benoit Cloitre, Aug 29 2002
a(n) = (phi^(2*n-1) + phi^(1-2*n))/sqrt(5) where phi=(1+sqrt(5))/2. - Michael Somos, Oct 28 2002
a(n) = A007598(n-1) + A007598(n) = A000045(n-1)^2 + A000045(n)^2 = F(n)^2 + F(n+1)^2. - Henry Bottomley, Feb 09 2001
a(n) = Sum_{k=0..n} binomial(n+k, 2*k). - Len Smiley, Dec 09 2001
a(n) ~ (1/5)*sqrt(5)*phi^(2*n+1). - Joe Keane (jgk(AT)jgk.org), May 15 2002
a(n) = Sum_{k=0..n} C(n, k)*F(k+1). - Benoit Cloitre, Sep 03 2002
Let q(n, x) = Sum_{i=0..n} x^(n-i)*binomial(2*n-i, i); then q(n, 1)=a(n) (this comment is essentially the same as that of L. Smiley). - Benoit Cloitre, Nov 10 2002
a(n) = (1/2)*(3*a(n-1) + sqrt(5*a(n-1)^2-4)). - Benoit Cloitre, Apr 12 2003
Main diagonal of array defined by T(i, 1) = T(1, j) = 1, T(i, j) = max(T(i-1, j) + T(i-1, j-1); T(i-1, j-1) + T(i, j-1)). - Benoit Cloitre, Aug 05 2003
Hankel transform of A002212. E.g., Det([1, 1, 3;1, 3, 10;3, 10, 36]) = 5. - Philippe Deléham, Jan 25 2004
Solutions x > 0 to equation floor(x*r*floor(x/r)) = floor(x/r*floor(x*r)) when r=phi. - Benoit Cloitre, Feb 15 2004
a(n) = Sum_{i=0..n} binomial(n+i, n-i). - Jon Perry, Mar 08 2004
a(n) = S(n-1, 3) - S(n-2, 3) = T(2*n-1, sqrt(5)/2)/(sqrt(5)/2) with S(n, x) = U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first kind. See triangle A049310, resp. A053120. - Wolfdieter Lang, Aug 31 2004
a(n) = ((-1)^(n-1))*S(2*(n-1), i), with the imaginary unit i and S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. - Wolfdieter Lang, Aug 31 2004
a(n) = Sum_{0<=i_1<=i_2<=n} binomial(i_2, i_1)*binomial(n, i_1+i_2). - Benoit Cloitre, Oct 14 2004
a(n) = L(n,3), where L is defined as in A108299; see also A002878 for L(n,-3). - Reinhard Zumkeller, Jun 01 2005
a(n) = a(n-1) + Sum_{i=0..n-1} a(i)*a(n) = F(2*n+1)*Sum_{i=0..n-1} a(i) = F(2*n). - Andras Erszegi (erszegi.andras(AT)chello.hu), Jun 28 2005
The i-th term of the sequence is the entry (1, 1) of the i-th power of the 2 X 2 matrix M = ((1, 1), (1, 2)). - Simone Severini, Oct 15 2005
a(n-1) = (1/n)*Sum_{k=0..n} B(2*k)*F(2*n-2*k)*binomial(2*n, 2*k) where B(2*k) is the (2*k)-th Bernoulli number. - Benoit Cloitre, Nov 02 2005
a(n) = A055105(n,1) + A055105(n,2) + A055105(n,3) = A055106(n,1) + A055106(n,2). - Mike Zabrocki, Oct 24 2006
a(n) = (2/sqrt(5))*cosh((2n-1)*psi), where psi=log(phi) and phi=(1+sqrt(5))/2. - Hieronymus Fischer, Apr 24 2007
a(n) = (phi+1)^n - phi*A001906(n) with phi=(1+sqrt(5))/2. - Reinhard Zumkeller, Nov 22 2007
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3); a(n) = ((sqrt(5) + 5)/10)*(3/2 + sqrt(5)/2)^(n-2) + ((-sqrt(5) + 5)/10)*(3/2 - sqrt(5)/2)^(n-2). - Antonio Alberto Olivares, Mar 21 2008
a(n) = A147703(n,0). - Philippe Deléham, Nov 29 2008
Sum_{n>=0} atan(1/a(n)) = (3/4)*Pi. - Jaume Oliver Lafont, Feb 27 2009
With X,Y defined as X = ( F(n) F(n+1) ), Y = ( F(n+2) F(n+3) ), where F(n) is the n-th Fibonacci number (A000045), it follows a(n+2) = X.Y', where Y' is the transpose of Y (n >= 0). - K.V.Iyer, Apr 24 2009
From Gary Detlefs, Nov 22 2010: (Start)
a(n) = Fibonacci(2*n+2) mod Fibonacci(2*n), n > 1.
a(n) = (Fibonacci(n-1)^2 + Fibonacci(n)^2 + Fibonacci(2*n-1))/2. (End)
INVERT transform is A166444. First difference is A001906. Partial sums is A055588. Binomial transform is A093129. Binomial transform of A000045(n-1). - Michael Somos, May 03 2012
a(n) = 2^n*f(n;1/2), where f(n;d), n=0,1,...,d, denote the so-called delta-Fibonacci numbers (see Witula et al. papers and comments in A000045). - Roman Witula, Jul 12 2012
a(n) = (Fibonacci(n+2)^2 + Fibonacci(n-3)^2)/5. - Gary Detlefs, Dec 14 2012
G.f.: 1 + x/( Q(0) - x ) where Q(k) = 1 - x/(x*k + 1 )/Q(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 23 2013
G.f.: (1-2*x)*G(0)/(2-3*x), where G(k) = 1 + 1/( 1 - x*(5*k-9)/(x*(5*k-4) - 6/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 19 2013
G.f.: 1 + x*(1-x^2)*Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 + 2*x - x^2)/( x*(4*k+4 + 2*x - x^2 ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 11 2013
G.f.: Q(0,u), where u=x/(1-x), Q(k,u) = 1 + u^2 + (k+2)*u - u*(k+1 + u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
Sum_{n>=2} 1/(a(n) - 1/a(n)) = 1. Compare with A001906, A007805 and A097843. - Peter Bala, Nov 29 2013
Let F(n) be the n-th Fibonacci number, A000045(n), and L(n) be the n-th Lucas number, A000032(n). Then for n > 0, a(n) = F(n)*L(n-1) + (-1)^n. - Charlie Marion, Jan 01 2014
a(n) = A238731(n,0). - Philippe Deléham, Mar 05 2014
1 = a(n)*a(n+2) - a(n+1)*a(n+1) for all n in Z. - Michael Somos, Jul 08 2014
a(n) = (L(2*n+4) + L(2*n-6))/25 for L(n)=A000032(n). - J. M. Bergot, Dec 30 2014
a(n) = (L(n-1)^2 + L(n)^2)/5 with L(n)=A000032(n). - J. M. Bergot, Dec 31 2014
a(n) = (L(n-2)^2 + L(n+1)^2)/10 with L(n)=A000032(n). - J. M. Bergot, Oct 23 2015
a(n) = 3*F(n-1)^2 + F(n-3)*F(n) - 2*(-1)^n. - J. M. Bergot, Feb 17 2016
a(n) = (F(n-1)*L(n) + F(n)*L(n-1))/2 = (A081714(n-1) + A128534(n))/2. - J. M. Bergot, Mar 22 2016
E.g.f.: (2*exp(sqrt(5)*x) + 3 + sqrt(5))*exp(-x*(sqrt(5)-3)/2)/(5 + sqrt(5)). - Ilya Gutkovskiy, Jul 04 2016
a(n) = ((M_2)^n)[1,1] = S(n, 3) - 2*S(n-1, 3), with the 2 X 2 tridiagonal matrix M_2 = Matrix([1,1], [1,2]) from A322602. For a proof see the Mar 30 2020 comment above. - Wolfdieter Lang, Mar 30 2020
Sum_{n>=1} 1/a(n) = A153387. - Amiram Eldar, Oct 05 2020
a(n+1) = Product_{k=1..n} (1 + 4*cos(2*Pi*k/(2*n + 1))^2). Special case of A099390. - Greg Dresden, Oct 16 2021
a(n+1) = 4^(n+1)*Sum_{k >= n} binomial(2*k,2*n)*(1/5)^(k+1). Cf. A102591. - Peter Bala, Nov 29 2021
a(n) = cosh((2*n-1)*arcsinh(1/2))/sqrt(5/4). - Peter Luschny, May 21 2022
From J. M. Bergot, May 27 2022: (Start)
a(n) = F(n-1)*L(n) - (-1)^n where L(n)=A000032(n) and F(n)=A000045(n).
a(n) = (L(n-1)^2 + L(n-1)*L(n+1))/5 + (-1)^n.
a(n) = 2*(area of a triangle with vertices at (L(n-2), L(n-1)), (F(n), F(n-1)), (L(n), L(n+1))) + 5*(-1)^n for n > 2. (End)
a(n) = A059929(n-1)+A059929(n-2), n>1. - R. J. Mathar, Jul 09 2024

Entry revised by N. J. A. Sloane, Aug 24 2006, May 13 2008

A006012 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - 2a(n-2), n >= 2.

Original entry on oeis.org

1, 2, 6, 20, 68, 232, 792, 2704, 9232, 31520, 107616, 367424, 1254464, 4283008, 14623104, 49926400, 170459392, 581984768, 1987020288, 6784111616, 23162405888, 79081400320, 270000789504, 921840357376, 3147359850496

Offset: 0

N. J. A. Sloane

Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 8 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = 4, s(2n) = 4. - Herbert Kociemba, Jun 12 2004
a(n-1) counts permutations pi on [n] for which the pairs {i, pi(i)} with i < pi(i), considered as closed intervals [i+1,pi(i)], do not overlap; equivalently, for each i in [n] there is at most one j <= i with pi(j) > i. Counting these permutations by the position of n yields the recurrence relation. - David Callan, Sep 02 2003
a(n) is the sum of (n+1)-th row terms of triangle A140070. - Gary W. Adamson, May 04 2008
The binomial transform is in A083878, the Catalan transform in A084868. - R. J. Mathar, Nov 23 2008
Equals row sums of triangle A152252. - Gary W. Adamson, Nov 30 2008
Counts all paths of length (2*n), n >= 0, starting at the initial node on the path graph P_7, see the second Maple program. - Johannes W. Meijer, May 29 2010
From L. Edson Jeffery, Apr 04 2011: (Start)
Let U_1 and U_3 be the unit-primitive matrices (see [Jeffery])
U_1 = U_(8,1) = [(0,1,0,0); (1,0,1,0); (0,1,0,1); (0,0,2,0)] and
U_3 = U_(8,3) = [(0,0,0,1); (0,0,2,0); (0,2,0,1); (2,0,2,0)]. Then a(n) = (1/4) * Trace(U_1^(2*n)) = (1/2^(n+2)) * Trace(U_3^(2*n)). (See also A084130, A001333.) (End)
Pisano period lengths: 1, 1, 8, 1, 24, 8, 6, 1, 24, 24, 120, 8, 168, 6, 24, 1, 8, 24, 360, 24, ... - R. J. Mathar, Aug 10 2012
a(n) is the first superdiagonal of array A228405. - Richard R. Forberg, Sep 02 2013
Conjecture: With offset 1, a(n) is the number of permutations on [n] with no subsequence abcd such that (i) bc are adjacent in position and (ii) max(a,c) < min(b,d). For example, the 4 permutations of [4] not counted by a(4) are 1324, 1423, 2314, 2413. - David Callan, Aug 27 2014
The conjecture of David Callan above is correct - with offset 1, a(n) is the number of permutations on [n] with no subsequence abcd such that (i) bc are adjacent in position and (ii) max(a,c) < min(b,d). - Yonah Biers-Ariel, Jun 27 2017
From Gary W. Adamson, Jul 22 2016: (Start)
A production matrix for the sequence is M =
  1, 1, 0, 0, 0, 0, ...
  1, 0, 3, 0, 0, 0, ...
  1, 0, 0, 3, 0, 0, ...
  1, 0, 0, 0, 3, 0, ...
  1, 0, 0, 0, 0, 3, ...
  ...
Take powers of M, extracting the upper left terms; getting the sequence starting: (1, 1, 2, 6, 20, 68, ...). (End)
From Gary W. Adamson, Jul 24 2016: (Start)
The sequence is the INVERT transform of the powers of 3 prefaced with a "1": (1, 1, 3, 9, 27, ...) and is N=3 in an infinite of analogous sequences starting:
   N=1 (A000079):  1, 2, 4,  8,  16,   32, ...
   N=2 (A001519):  1, 2, 5, 13,  34,   89, ...
   N=3 (A006012):  1, 2, 6, 20,  68,  232, ...
   N=4 (A052961):  1, 2, 7, 29, 124,  533, ...
   N=5 (A154626):  1, 2, 8, 40, 208, 1088, ...
   N=6:            1, 2, 9, 53, 326, 2017, ...
   ... (End)
Number of permutations of length n > 0 avoiding the partially ordered pattern (POP) {1>2, 1>3, 4>2, 4>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first and fourth elements are larger than the second and third elements. - Sergey Kitaev, Dec 08 2020
a(n-1) is the number of permutations of [n] that can be obtained by placing n points on an X-shape (two crossing lines with slopes 1 and -1), labeling them 1,2,...,n by increasing y-coordinate, and then reading the labels by increasing x-coordinate. - Sergi Elizalde, Sep 27 2021
Consider a stack of pancakes of height n, where the only allowed operation is reversing the top portion of the stack. First, perform a series of reversals of decreasing sizes, followed by a series of reversals of increasing sizes. The number of distinct permutations of the initial stack that can be reached through these operations is a(n). - Thomas Baruchel, May 12 2025
Number of permutations of [n] that are correctly sorted after performing one left-to-right pass and one right-to-left pass of the cocktail sort. - Thomas Baruchel, May 16 2025

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms. Birkhäuser, Boston, 3rd edition, 1990, p. 86.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 5.4.8 Answer to Exer. 8.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Andrei Asinowski and Toufik Mansour, Separable d-Permutations and Guillotine Partitions, arXiv 0803.3414 [math.CO], 2008. Annals of Combinatorics 14 (1) pp.17-43 Springer, 2010; Abstract
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 155
George Balla, Ghislain Fourier, and Kunda Kambaso, PBW filtration and monomial bases for Demazure modules in types A and C, arXiv:2205.01747 [math.RT], 2022.
M. Barnabei, F. Bonetti, and M. Silimbani, Two permutation classes related to the Bubble Sort operator, Electronic Journal of Combinatorics 19(3) (2012), #P25. - From _N. J. A. Sloane_, Dec 25 2012
Yonah Biers-Ariel, A New Quantity Counted by OEIS Sequence A006012, arXiv:1706.07064 [math.CO], 2017.
Yonah Biers-Ariel, The Number of Permutations Avoiding a Set of Generalized Permutation Patterns, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.3.
Rocco Chirivì, Xin Fang, and Ghislain Fourier, Degenerate Schubert varieties in type A, Transformation Groups (2020).
CombOS - Combinatorial Object Server, Generate pattern-avoiding permutations
Sergi Elizalde, The X-class and almost-increasing permutations, arXiv:0710.5168 [math:CO], 2007. Ann. Comb. 15 (2011), 51-68.
S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Luca Ferrari, Enhancing the connections between patterns in permutations and forbidden configurations in restricted elections, arXiv:1906.10553 [math.CO], 2019.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
L. E. Jeffery, Unit-primitive matrices
Donghyun Kim and Lauren Williams, Schubert polynomials, the inhomogeneous TASEP, and evil-avoiding permutations, arXiv:2106.13378 [math.CO], 2021.
Sergey Kitaev and Artem Pyatkin, On permutations avoiding partially ordered patterns defined by bipartite graphs, arXiv:2204.08936 [math.CO], 2022.
Joshua Marsh and Nathan Williams, Nesting Nonpartitions, J. Int. Seq., Vol. 25 (2022), Article 22.8.8.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Joris Nieuwveld, Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding, Master's Thesis, arXiv:2108.11382 [math.NT], 2021.
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
J. Riordan, The distribution of crossings of chords joining pairs of 2n points on a circle, Math. Comp., 29 (1975), 215-222.
Markus Saers, Dekai Wu, and Chris Quirk, On the Expressivity of Linear Transductions.
Yi-dong Sun and Cang-zhi Jia, Counting Dyck paths with strictly increasing peak sequences, J. Math. Res. Expos. 27 (2) (2007) 253, Theorem 3.11.
Index entries for linear recurrences with constant coefficients, signature (4,-2).

Cf. A000079, A000129, A001333, A001519, A002203, A003480, A007052, A007070, A024175, A030436, A052961, A056236, A083978, A084130, A084868, A094803, A098158, A140070, A147703, A152252, A154626, A201730, A228405, A265185.

a006012 n = a006012_list !! n
a006012_list = 1 : 2 : zipWith (-) (tail $ map (* 4) a006012_list)
(map (* 2) a006012_list)
-- Reinhard Zumkeller, Oct 03 2011

[n le 2 select n else 4*Self(n-1)- 2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 05 2011

A006012:=-(-1+2*z)/(1-4*z+2*z**2); # Simon Plouffe in his 1992 dissertation
with(GraphTheory): G:=PathGraph(7): A:= AdjacencyMatrix(G): nmax:=24; n2:=2*nmax: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[1,k],k=1..7); od: seq(a(2*n),n=0..nmax); # Johannes W. Meijer, May 29 2010

LinearRecurrence[{4,-2},{1,2},50] (* or *) With[{c=Sqrt[2]}, Simplify[ Table[((2+c)^n+(3+2c)(2-c)^n)/(2(2+c)),{n,50}]]] (* Harvey P. Dale, Aug 29 2011 *)

{a(n) = real(((2 + quadgen(8))^n))}; /* Michael Somos, Feb 12 2004 */

{a(n) = if( n<0, 2^n, 1) * polsym(x^2 - 4*x + 2, abs(n))[abs(n)+1] / 2}; /* Michael Somos, Feb 12 2004 */

Vec((1-2*x)/(1-4*x+2*x^2) + O(x^100)) \\ Altug Alkan, Dec 05 2015

l = [1, 2]
for n in range(2, 101): l.append(4 * l[n - 1] - 2 * l[n - 2])
print(l)  # Indranil Ghosh, Jul 02 2017

A006012=BinaryRecurrenceSequence(4,-2,1,2)
print([A006012(n) for n in range(41)]) # G. C. Greubel, Aug 27 2025

G.f.: (1-2*x)/(1 - 4*x + 2*x^2).
a(n) = 2*A007052(n-1) = A056236(n)/2.
Limit_{n -> oo} a(n)/a(n-1) = 2 + sqrt(2). - Zak Seidov, Oct 12 2002
From Paul Barry, May 08 2003: (Start)
Binomial transform of A001333.
E.g.f.: exp(2*x)*cosh(sqrt(2)*x). (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*2^(n-k) = Sum_{k=0..n} binomial(n, k)*2^(n-k/2)(1+(-1)^k)/2. - Paul Barry, Nov 22 2003 (typo corrected by Manfred Scheucher, Jan 17 2023)
a(n) = ((2+sqrt(2))^n + (2-sqrt(2))^n)/2.
a(n) = [M^n]{2,2}, where M is the 3 X 3 matrix: M = [1,1,1;1,2,1;1,1,1]. - _Simone Severini, Jun 11 2006
a(n) = Sum_{k=0..n} 2^k*A098158(n,k). - Philippe Deléham, Dec 04 2006
a(n) = A007070(n) - 2*A007070(n-1). - R. J. Mathar, Nov 16 2007
a(n) = Sum_{k=0..n} A147703(n,k). - Philippe Deléham, Nov 29 2008
a(n) = Sum_{k=0..n} A201730(n,k). - Philippe Deléham, Dec 05 2011
G.f.: G(0) where G(k)= 1 + 2*x/((1-2*x) - 2*x*(1-2*x)/(2*x + (1-2*x)*2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 10 2012
G.f.: G(0)*(1-2*x)/2, where G(k) = 1 + 1/(1 - 2*x*(4*k+2-x)/( 2*x*(4*k+4-x) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 27 2014
a(-n) = a(n) / 2^n for all n in Z. - Michael Somos, Aug 24 2014
a(n) = A265185(n) / 4, connecting this sequence to the simple Lie algebra B_4. - Tom Copeland, Dec 04 2015
From G. C. Greubel, Aug 27 2025: (Start)
a(n) = 2^((n-2)/2)*( (n+1 mod 2)*A002203(n) + 2*sqrt(2)*(n mod 2)*A000129(n) ).
a(n) = 2^(n/2)*ChebyshevT(n, sqrt(2)). (End)

A152166 a(2n) = 2^n; a(2n+1) = -(2^(n+1)).

Original entry on oeis.org

1, -2, 2, -4, 4, -8, 8, -16, 16, -32, 32, -64, 64, -128, 128, -256, 256, -512, 512, -1024, 1024, -2048, 2048, -4096, 4096, -8192, 8192, -16384, 16384, -32768, 32768, -65536, 65536, -131072, 131072, -262144, 262144, -524288, 524288, -1048576, 1048576

Offset: 0

Philippe Deléham, Nov 27 2008

Ratios of successive terms are -2,-1,-2,-1,-2,-1,-2,-1,... - Philippe Deléham, Dec 12 2008

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079, A016116, A147703.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

LinearRecurrence[{0, 2}, {1, -2}, 50] (* Paolo Xausa, Jul 19 2024 *)

G.f.: (1 - 2*x)/(1 - 2*x^2).
a(n) = 2*a(n-2); a(0)=1, a(1)=-2.
a(n) = Sum_{k=0..n} A147703(n,k)*(-3)^k.
E.g.f.: cosh(sqrt(2)*x) - sqrt(2)*sinh(sqrt(2)*x). - Stefano Spezia, Feb 05 2023

A052984 a(n) = 5a(n-1) - 2a(n-2) for n>1, with a(0) = 1, a(1) = 3.

Original entry on oeis.org

1, 3, 13, 59, 269, 1227, 5597, 25531, 116461, 531243, 2423293, 11053979, 50423309, 230008587, 1049196317, 4785964411, 21831429421, 99585218283, 454263232573, 2072145726299, 9452202166349, 43116719379147, 196679192563037, 897162524056891, 4092454235158381

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

a(n) = A020698(n) - 4*A020698(n-1) + 4*A020698(n-2) (n>=2). Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).
Stanley, Richard P. "Some Linear Recurrences Motivated by Stern’s Diatomic Array." The American Mathematical Monthly 127.2 (2020): 99-111.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1058
Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. See p. 3.
Zeying Xu, Graphical zonotopes with the same face vector, arXiv:1809.08764 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (5,-2).

Cf. A005824, A020698, A107839, A147703.

a:=[1,3];; for n in [3..30] do a[n]:=5*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Oct 23 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-5*x+2*x^2) )); // G. C. Greubel, Feb 10 2019

a:=[1,3]; [n le 2 select a[n] else 5*Self(n-1)-2*Self(n-2):n in [1..25]]; // Marius A. Burtea, Oct 23 2019

spec:= [S,{S=Sequence(Union(Prod(Sequence(Union(Z,Z)),Union(Z,Z)),Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
a[0]:=1: a[1]:=3: for n from 2 to 25 do a[n]:=5*a[n-1]-2*a[n-2] od: seq(a[n],n=0..25); # Emeric Deutsch

a[0]=1; a[1]=3; a[n_]:= a[n] = 5a[n-1]-2a[n-2]; Table[ a[n], {n, 0, 30}]
LinearRecurrence[{5,-2},{1,3},30] (* Harvey P. Dale, Apr 08 2014 *)
CoefficientList[Series[(1-2x)/(1-5x+2x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 09 2014 *)

Vec((1-2*x)/(1-5*x+2*x^2)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2011

def A052984_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x)/(1-5*x+2*x^2) ).list()
A052984_list(30) # G. C. Greubel, Feb 10 2019

a(n) = A005824(2n).
G.f.: (1-2*x)/(1-5*x+2*x^2).
a(n) = Sum_{alpha=RootOf(1-5*z+2*z^2)} (1 + 6*alpha)*alpha^(-1-n)/17.
a(n) = [M^(n+1)]2,2, where M is the 3 X 3 matrix defined as follows: M = [2,1,2; 1,1,1; 2,1,2]. - _Simone Severini, Jun 12 2006
a(n-1) = Sum_{k=0..n} A147703(n,k)*(-1)^k*2^(n-k), n>1. - Philippe Deléham, Nov 29 2008
a(n) = (a(n-1)^2 + 2^n)/a(n-2). - Irene Sermon, Oct 29 2013
a(n) = A107839(n) - 2*A107839(n-1). - R. J. Mathar, Feb 27 2019
E.g.f.: exp(5*x/2)*(sqrt(17)*cosh(sqrt(17)*x/2) + sinh(sqrt(17)*x/2))/sqrt(17). - Stefano Spezia, Jun 17 2025

Edited by Robert G. Wilson v, Dec 29 2002

A059502 a(n) = (3nF(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045.

Original entry on oeis.org

0, 1, 3, 9, 27, 80, 234, 677, 1941, 5523, 15615, 43906, 122868, 342409, 950727, 2631165, 7260579, 19982612, 54865566, 150316973, 411015705, 1121818311, 3056773383, 8316416134, 22593883752, 61301547025, 166118284299, 449639574897, 1215751720491, 3283883157848

Offset: 0

Floor van Lamoen, Jan 19 2001

Substituting x(1-x)/(1-2x) into x/(1-x)^2 yields g.f. of sequence.

Variation of A059216 (and of Boustrophedon transform) applied to 1,2,3,4,...: fill an array by diagonals, each time in the same direction, say the 'up' direction. The first column is 1,2,3,4,... For the next element of a diagonal, add to the previous element the elements of the row the new element is in. The first row gives a(n).

			The array (see A059503) begins
  1 3  9 27 80 ...
  2 5 14 40 ...
  3 7 19 ...
  4 9  5 ...

Harry J. Smith, Table of n, a(n) for n = 0..200
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 17, 19.
Index entries for two-way infinite sequences
Index entries for sequences related to boustrophedon transform
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000045, A000667, A059216, A059219, A027994, A059512, A059503.

[(3*n*Fibonacci(2*n-1)+(3-n)*Fibonacci(2*n))/5: n in [0..100]]; // Vincenzo Librandi, Apr 23 2011

Table[(3n Fibonacci[2n-1]+(3-n)Fibonacci[2n])/5,{n,0,30}] (* or *) CoefficientList[Series[x(1-x)(1-2x)/(1-3x+x^2)^2,{x,0,30}],x] (* Harvey P. Dale, Apr 23 2011 *)

a(n)=(3*n*fibonacci(2*n-1)+(3-n)*fibonacci(2*n))/5

a(n) = 2*a(n-1) + Sum{m<=n-2} a(m) + A001519(n-2).
G.f.: x*(1 - x)*(1 - 2*x)/(1 - 3*x + x^2)^2. - Emeric Deutsch, Oct 07 2002
a(n) = A147703(n,1). - Philippe Deléham, Nov 29 2008
a(n) = A001871(n-1) - 3*A001871(n-2) + 2*A001871(n-3). - R. J. Mathar, Apr 09 2019
E.g.f.: 2*exp(3*x/2)*(5*x*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Mar 04 2025

A082761 Trinomial transform of the Fibonacci numbers (A000045).

Original entry on oeis.org

1, 4, 20, 104, 544, 2848, 14912, 78080, 408832, 2140672, 11208704, 58689536, 307302400, 1609056256, 8425127936, 44114542592, 230986743808, 1209462292480, 6332826779648, 33159111507968, 173623361929216, 909103725543424, 4760128905543680, 24924358531088384, 130505635564355584

Offset: 0

Emanuele Munarini, May 21 2003

Hankel transform of Sum_{k=0..n} (-1)^k*C(2k, k) (see A054108). - Paul Barry, Jan 13 2009
Hankel transform of A046748. - Paul Barry, Apr 14 2010
For positive n, a(n) equals the permanent of the (2n) X (2n) tridiagonal matrix with sqrt(2)'s along the three central diagonals. - John M. Campbell, Jul 12 2011
The limiting ratio is: Lim_{n -> oo} a(n)/a(n-1) = 1 + phi^3. - Bob Selcoe, Mar 18 2014
Invert transform of A052984. Invert transform is A083066. Binomial transform of A033887. Binomial transform is A163073. - Michael Somos, May 26 2014

			a(5) = 2848 = 5*(544) + 4 + 20 + 104.
G.f. = 1 + 4*x + 20*x^2 + 104*x^3 + 544*x^4 + 2848*x^5 + 14912*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..150
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (6,-4).

Cf. A000045, A001519, A027907, A033887, A046748, A052984, A054108, A069429, A083066, A147703, A163073.

[2^n * Fibonacci(2*n+1): n in [0..40]]; // Vincenzo Librandi, Jul 15 2011

a[ n_] := 2^n Fibonacci[ 2 n + 1]; (* Michael Somos, May 26 2014 *)
a[ n_] := If[ n < 0, SeriesCoefficient[ (2 - x) / (4 - 6 x + x^2), {x, 0, -1 - n}], SeriesCoefficient[ (1 - 2 x) / (1 - 6 x + 4 x^2), {x, 0, n}]]; (* Michael Somos, Oct 22 2017 *)
LinearRecurrence[{6,-4},{1,4},30] (* Harvey P. Dale, Jul 11 2014 *)

a(n)=fibonacci(2*n+1)<Charles R Greathouse IV, Jul 15 2011

{a(n) = if( n<0, n = -1 - n; 2^(-1-2*n), 1) * polcoeff( (1 - 2*x) / (1 - 6*x + 4*x^2) + x * O(x^n), n)}; /* Michael Somos, Oct 22 2017 */

[2^n*fibonacci(2*n+1) for n in range(41)] # G. C. Greubel, Jul 28 2024

a(n) = Sum_{k=0..2*n} A027907(n, k)*A000045(k+1).
From Paul Barry, Jul 16 2003: (Start)
Third binomial transform of (1, 1, 5, 5, 25, 25, ....).
a(n) = ((1+sqrt(5))(3+sqrt(5))^n-(1-sqrt(5))*(3-sqrt(5))^n)/(2*sqrt(5)). (End)
From R. J. Mathar, Nov 04 2008: (Start)
G.f.: (1-2*x)/(1-6*x+4*x^2).
a(n) = 6*a(n-1) - 4*a(n-2). (End)
a(n) = Sum_{k=0..n} A147703(n,k)*3^k. - Philippe Deléham, Nov 14 2008
For n>=2: a(n) = 5*a(n-1) + Sum_{i=1..n-2} a(i). - Bob Selcoe, Mar 18 2014
a(n) = a(-1-n) * 2^(2*n+1) for all n in Z. - Michael Somos, Mar 18 2014
a(n) = 2^n*Fibonacci(2*n+1), or 2^n*A001519(n+1). - Bob Selcoe, May 25 2014
From Michael Somos, May 26 2014: (Start)
a(n) - a(n-1) = A069429(n).
a(n+1) * a(n-1) - a(n)^2 = 4^n.
G.f.: 1 / (1 - 4*x / (1 - x / (1 - x))). (End)
E.g.f.: exp(3*x)*(5*cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, May 24 2024

A152163 a(n) = a(n-1)+a(n-2), n>1 ; a(0)=1, a(1)=-1.

Original entry on oeis.org

1, -1, 0, -1, -1, -2, -3, -5, -8, -13, -21, -34, -55, -89, -144, -233, -377, -610, -987, -1597, -2584, -4181, -6765, -10946, -17711, -28657, -46368, -75025, -121393, -196418, -317811, -514229, -832040, -1346269, -2178309, -3524578, -5702887

Offset: 0

Philippe Deléham, Nov 27 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000045.

I:=[1, -1]; [n le 2 select I[n] else Self(n-1)+Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 23 2013

LinearRecurrence[{1,1},{1,-1},40] (* Harvey P. Dale, Oct 09 2012 *)

a(n)=-fibonacci(n-2) \\ M. F. Hasler, May 10 2017

def A152163():
    a, b = True, False
    x, y = 1, 1
    while True:
        yield x if a else -x
        x, y = y, x - y
        a, b = b, a
a = A152163()
print([next(a) for  in range(50)]) # _Peter Luschny, Mar 19 2020

G.f.: (1-2*x)/(1-x-x^2).
a(n) = Sum_{k, 0<=k<=n}A147703(n,k)*(-2)^k.
a(n) = -Fibonacci(n-2) for n >= 2, and for all n if A000045 is extended in the natural way to negative indices; see also A039834. [Extended by M. F. Hasler, May 10 2017]
a(n) = (-1)^n*A039834(n-2). - R. J. Mathar, Mar 22 2011
G.f.: (1/(1-Q(0))-1)*(1-2*x)/x where Q(k)=1 - x^k/(1 - x/( x - x^k/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 23 2013
G.f.: 2 - 2/(Q(0)+1) where Q(k) = 1 - 2*x/(1 - x/(x - 1/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 05 2013
a(n) = A000045(n+1)-2*A000045(n). - R. J. Mathar, Jun 26 2013
G.f.: 1 - x - x^3*Q(0)/2, where Q(k) = 1 + 1/(1 - x*(6*k+1 + x)/(x*(6*k+4 + x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 02 2014
G.f.: 1+1/x - x - Q(0)/x, where Q(k) = 1 + x^2 - x^3 - k*x*(1+x^2) - x^2*( x*(k+2)-1)*( k*x -1)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jan 13 2014
E.g.f.: exp(x/2)*(5*cosh(sqrt(5)*x/2) - 3*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Mar 09 2025

A062110 A(n,k) is the coefficient of x^k in (1-x)^n/(1-2*x)^n for n, k >= 0; Table A read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 5, 3, 1, 0, 8, 12, 9, 4, 1, 0, 16, 28, 25, 14, 5, 1, 0, 32, 64, 66, 44, 20, 6, 1, 0, 64, 144, 168, 129, 70, 27, 7, 1, 0, 128, 320, 416, 360, 225, 104, 35, 8, 1, 0, 256, 704, 1008, 968, 681, 363, 147, 44, 9, 1, 0, 512, 1536, 2400, 2528, 1970

Offset: 0

Henry Bottomley, May 30 2001

The triangular version of this square array is defined by T(n,k) = A(k,n-k) for 0 <= k <= n. Conversely, A(n,k) = T(n+k,n) for n,k >= 0. We have [o.g.f of T](x,y) = [o.g.f. of A](x*y, x) and [o.g.f. of A](x,y) = [o.g.f. of T](y,x/y). - Petros Hadjicostas, Feb 11 2021
From Paul Barry, Nov 10 2008: (Start)
As number triangle, Riordan array (1, x(1-x)/(1-2x)). A062110*A007318 is A147703.
[0,1,1,0,0,0,....] DELTA [1,0,0,0,.....]. (Philippe Deléham's DELTA is defined in A084938.) (End)
Modulo 2, this triangle T becomes triangle A106344. - Philippe Deléham, Dec 18 2008

			Table A(n,k) (with rows n >= 0 and columns k >= 0) begins:
  1, 0,  0,   0,   0,    0,    0,     0,     0,     0, ...
  1, 1,  2,   4,   8,   16,   32,    64,   128,   256, ...
  1, 2,  5,  12,  28,   64,  144,   320,   704,  1536, ...
  1, 3,  9,  25,  66,  168,  416,  1008,  2400,  5632, ...
  1, 4, 14,  44, 129,  360,  968,  2528,  6448, 16128, ...
  1, 5, 20,  70, 225,  681, 1970,  5500, 14920, 39520, ...
  1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, ...
  ... - _Petros Hadjicostas_, Feb 15 2021
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,   1;
  0,   4,   5,   3,   1;
  0,   8,  12,   9,   4,   1;
  0,  16,  28,  25,  14,   5,   1;
  0,  32,  64,  66,  44,  20,   6,   1;
  0,  64, 144, 168, 129,  70,  27,   7,   1;
  0, 128, 320, 416, 360, 225, 104,  35,   8,   1;
  ... - _Philippe Deléham_, Nov 30 2008

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, and Juana Sendra, Upper Hessenberg and Toeplitz Bohemians, arXiv:1907.10677 [cs.SC], 2019.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq., 21 (2018), #18.1.4.

Rows of A include A000007, A011782, A045623, A058396, A062109, A169792, A169793, A169794, A169795, A169796, A169797.
Columns of A include A000012, A001477, A000096, A000297.
Main diagonal of A is A002002.
Table A(n, k) is a multiple of 2^(k-n); dividing by this gives a table similar to A050143 except at the edges.
Essentially the same array as A105306, A160232.

t[n_, n_] = 1; t[n_, k_] := 2^(n-2*k)*k*Hypergeometric2F1[1-k, n-k+1, 2, -1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 30 2013, after Philippe Deléham + symbolic sum *)

a(i,j)=if(i<0 || j<0,0,polcoeff(((1-x)/(1-2*x)+x*O(x^j))^i,j))

Formulas for the square array (A(n,k): n,k >= 0):
A(n, k) = A(n-1, k) + Sum_{0 <= j < k} A(n, j) for n >= 1 and k >= 0 with A(0, k) = 0^k for k >= 0.
G.f.: 1/(1-x*(1-y)/(1-2*y)) = Sum_{i, j >= 0} A(i, j) x^i*y^j.
From Petros Hadjicostas, Feb 15 2021: (Start)
A(n,k) = 2^(k-n)*n*hypergeom([1-n, k+1], [2], -1) for n >= 0 and k >= 1.
A(n,k) = 2*A(n,k-1) + A(n-1,k) - A(n-1,k-1) for n,k >= 1 with A(n,0) = 1 for n >= 0 and A(0,k) = 0 for k >= 1. (End)
Formulas for the triangle (T(n,k): 0 <= k <= n):
From Philippe Deléham, Aug 01 2006: (Start)
T(n,k) = A121462(n+1,k+1)*2^(n-2*k) for 0 <= k < n.
T(n,k) = 2^(n-2*k)*k*hypergeom([1-k, n-k+1], [2], -1) for 0 <= k < n. (End)
Sum_{k=0..n} T(n,k)*x^k = A152239(n), A152223(n), A152185(n), A152174(n), A152167(n), A152166(n), A152163(n), A000007(n), A001519(n), A006012(n), A081704(n), A082761(n), A147837(n), A147838(n), A147839(n), A147840(n), A147841(n), for x = -7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively. - Philippe Deléham, Dec 09 2008
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 1 <= k <= n-1 with T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, and T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013
G.f.: Sum_{n.k>=0} T(n,k)*x^n*y^k = (1 - 2*x)/(x^2*y - x*y - 2*x + 1). - Petros Hadjicostas, Feb 15 2021

Various sections edited by Petros Hadjicostas, Feb 15 2021

A081704 Let f(0)=1, f(1)=t, f(n+1) = (f(n)^2+t^n)/f(n-1). f(t) is a polynomial with integer coefficients. Then a(n) = f(n) when t=3.

Original entry on oeis.org

1, 3, 12, 51, 219, 942, 4053, 17439, 75036, 322863, 1389207, 5977446, 25719609, 110665707, 476169708, 2048851419, 8815747971, 37932185598, 163213684077, 702271863591, 3021718265724, 13001775737847, 55943723892063, 240713292246774, 1035735289557681

Offset: 0

Victor Ufnarovski (ufn(AT)maths.lth.se), Apr 02 2003

f satisfies the linear recursion f(n+1) = (t+2)*f(n)-t*f(n-1). For t=3 this gives a(n+1) = 5*a(n)-3*a(n-1).

Given the 3 X 3 matrix [1,1,1; 1,1,2; 1,1,3] = M, a(n) = term (1,1) in M^(n+1). - Gary W. Adamson, Aug 06 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-3).

Cf. A006012, A001519.

Equals 3*A018902(n-1) for n>0.

f := proc(n) if n=0 then 1 elif n=1 then t else sort(simplify((f(n-1)^2+t^(n-1))/f(n-2)),t) fi end; a := i->subs(t=3,f(i));

a[0]=1; a[1]=3; a[n_] := a[n]=5a[n-1]-3a[n-2]; Array[a,25,0]
LinearRecurrence[{5,-3},{1,3},30] (* Harvey P. Dale, Jul 28 2013 *)

Vec((1-2*x)/(1-5*x+3*x^2) + O(x^30)) \\ Colin Barker, Nov 26 2016

a(n+1) = (a(n)^2 + 3^n) / a(n-1).
From Philippe Deléham, Nov 14 2008: (Start)
G.f.: (1-2*x)/(1-5*x+3*x^2).
a(n) = Sum_{k, 0<=k<=n} A147703(n,k)*2^k. (End)
a(n) = (2^(-1-n)*((5-sqrt(13))^n*(-1+sqrt(13)) + (1+sqrt(13))*(5+sqrt(13))^n))/sqrt(13). - Colin Barker, Nov 26 2016
E.g.f.: exp(5*x/2)*(sqrt(13)*cosh(sqrt(13)*x/2) + sinh(sqrt(13)*x/2))/sqrt(13). - Stefano Spezia, Jul 09 2022

A147704 Diagonal sums of Riordan array ((1-2x)/(1 - 3x + x^2),x(1-x)/(1 - 3x + x^2)).

Original entry on oeis.org

1, 1, 3, 8, 23, 66, 190, 547, 1575, 4535, 13058, 37599, 108262, 311728, 897585, 2584493, 7441751, 21427668, 61698511, 177653782, 511533678, 1472902523, 4241053787, 12211627683, 35161980526, 101244887791, 291523035690, 839407126544, 2416976491841, 6959406439833

Offset: 0

Paul Barry, Nov 10 2008

Diagonal sums of A147703.
Hankel transform is := 1,2,3,0,0,0,0,0,0,0,... - Philippe Deléham, Dec 15 2008
For n -> infinity, a(n+1)/a(n) -> 2.87938... = 1/A130880 = the largest diagonal of a nonagon (9-gon) with side 1 (see Redondo & Huylebrouck); compare to the F(n+1)/F(n) -> 1.618... = A001622 = the golden section or diagonal of a pentagon with side 1, where F is the Fibonacci sequence A000045. - Dirk Huylebrouck, Feb 15 2015

A. Redondo Buitrago and D. Huylebrouck, Nonagons in the Hagia Sophia and the Selimiye Mosque, Nexus Network Journal on Architecture and Mathematics, January 2015.
Index entries for linear recurrences with constant coefficients, signature (3,0,-1).

Cf. A000045, A001622, A130880.

LinearRecurrence[{3,0,-1},{1,1,3},30] (* Harvey P. Dale, May 24 2016 *)

Vec((1-x^2)/(1-3*x+x^3) + O(x^20)) \\ Michel Marcus, Feb 16 2015

G.f.: (1-x^2)/(1 - 3x + x^3).
a(n) = 3*a(n-1) - a(n-3), n>2 ; a(0)=1, a(1)=1, a(2)=3. - Philippe Deléham, Dec 15 2008
a(n) = (floor(A^n)+1)/3 for n>=1 where A = 2.8793... is the largest root of x^3-3x^2+1. - Stephen Bartell, Aug 15 2024

cp's OEIS Frontend

A001519 a(n) = 3*a(n-1) - a(n-2) for n >= 2, with a(0) = a(1) = 1.

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A006012 a(0) = 1, a(1) = 2, a(n) = 4*a(n-1) - 2*a(n-2), n >= 2.

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A152166 a(2*n) = 2^n; a(2*n+1) = -(2^(n+1)).

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A052984 a(n) = 5*a(n-1) - 2*a(n-2) for n>1, with a(0) = 1, a(1) = 3.

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A059502 a(n) = (3*n*F(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045.

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A006012 a(0) = 1, a(1) = 2, a(n) = 4a(n-1) - 2a(n-2), n >= 2.

A152166 a(2n) = 2^n; a(2n+1) = -(2^(n+1)).

A052984 a(n) = 5a(n-1) - 2a(n-2) for n>1, with a(0) = 1, a(1) = 3.

A059502 a(n) = (3nF(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045.