A073387 -id:A073387 - cp's OEIS frontend

A002605 a(n) = 2*(a(n-1) + a(n-2)), a(0) = 0, a(1) = 1.

0, 1, 2, 6, 16, 44, 120, 328, 896, 2448, 6688, 18272, 49920, 136384, 372608, 1017984, 2781184, 7598336, 20759040, 56714752, 154947584, 423324672, 1156544512, 3159738368, 8632565760, 23584608256, 64434348032, 176037912576, 480944521216, 1313964867584

Offset: 0

Colin Mallows

Individually, both this sequence and A028859 are convergents to 1 + sqrt(3). Mutually, both sequences are convergents to 2 + sqrt(3) and 1 + sqrt(3)/2. - Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Nov 04 2001
The number of (s(0), s(1), ..., s(n+1)) such that 0 < s(i) < 6 and |s(i) - s(i-1)| <= 1 for i = 1, 2, ..., n + 1, s(0) = 2, s(n+1) = 3. - Herbert Kociemba, Jun 02 2004
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators 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 sqrt(4). - Cino Hilliard, Sep 25 2005
The Hankel transform of this sequence is [1, 2, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Nov 21 2007
[1, 3; 1, 1]^n *[1, 0] = [A026150(n), a(n)]. - Gary W. Adamson, Mar 21 2008
(1 + sqrt(3))^n = A026150(n) + a(n)*sqrt(3). - Gary W. Adamson, Mar 21 2008
a(n+1) is the number of ways to tile a board of length n using red and blue tiles of length one and two. - Geoffrey Critzer, Feb 07 2009
Starting with offset 1 = INVERT transform of the Jacobsthal sequence, A001045: (1, 1, 3, 5, 11, 21, ...). - Gary W. Adamson, May 12 2009
Starting with "1" = INVERTi transform of A007482: (1, 3, 11, 39, 139, ...). - Gary W. Adamson, Aug 06 2010
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 85, 277, 337 and 340, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A026150, without the first leading 1. - Johannes W. Meijer, Aug 15 2010
The sequence 0, 1, -2, 6, -16, 44, -120, 328, -896, ... (with alternating signs) is the Lucas U(-2,-2)-sequence. - R. J. Mathar, Jan 08 2013
a(n+1) counts n-walks (closed) on the graph G(1-vertex;1-loop,1-loop,2-loop,2-loop). - David Neil McGrath, Dec 11 2014
Number of binary strings of length 2*n - 2 in the regular language (00+11+0101+1010)*. - Jeffrey Shallit, Dec 14 2015
For n >= 1, a(n) equals the number of words of length n - 1  over {0, 1, 2, 3} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Dec 17 2015
a(n+1) is the number of compositions of n into parts 1 and 2, both of two kinds. - Gregory L. Simay, Sep 20 2017
Number of associative, quasitrivial, and order-preserving binary operations on the n-element set {1, ..., n} that have neutral elements. - J. Devillet, Sep 28 2017
(1 + sqrt(3))^n = A026150(n) + a(n)*sqrt(3), for n >= 0; integers in the real quadratic number field Q(sqrt(3)). - Wolfdieter Lang, Feb 10 2018
Starting with 1, 2, 6, 16, ..., number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>3, 1>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the third and fourth elements. - Sergey Kitaev, Dec 09 2020
a(n) is the number of tilings of a 2 X n board missing one corner cell, with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares). Compare to A127864. - Greg Dresden and Yilin Zhu, Jul 17 2025

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 8, 29.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017.
M. Diepenbroek, M. Maus, and A. Stoll, Pattern Avoidance in Reverse Double Lists, Preprint 2015. See Table 3.
Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.
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.
Dale Gerdemann Bird Flock, Youtube video, 2011.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=q=2.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 476
D. Jhala, G. P. S. Rathore, and K. Sisodiya, Some Properties of k-Jacobsthal Numbers with Arithmetic Indexes, Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 4, 119-124.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (39), (41) and (45), lhs, m=2.
D. H. Lehmer, On Lucas's test for the primality of Mersenne's numbers, Journal of the London Mathematical Society 1.3 (1935): 162-165. See U_n.
Toufik Mansour and Mark Shattuck, Pattern avoidance in flattened derangements, Disc. Math. Lett. (2025) Vol. 15, 67-74. See p. 74.
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
Alan Prince, Counting parses, Rutgers Optimality Archive, 2010.
Index entries for linear recurrences with constant coefficients, signature (2,2).
Index entries for sequences related to Chebyshev polynomials.
Index entries for Lucas sequences.

First differences are given by A026150.
a(n) = A073387(n, 0), n>=0 (first column of triangle).
Equals (1/3) A083337. First differences of A077846. Pairwise sums of A028860 and abs(A077917).
a(n) = A028860(n)/2 apart from the initial terms.
Row sums of A081577 and row sums of triangle A156710.
The following sequences (and others) belong to the same family: A001333, A000129, A026150, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
Cf. A080953, A052948, A080040, A028859, A030195, A106435, A108898, A125145, A265106, A265107, A265278, A270810, A293005, A293006, A293007.
Cf. A175289 (Pisano periods).
Cf. A002530.
Cf. A127864.

a002605 n = a002605_list !! n
a002605_list =
   0 : 1 : map (* 2) (zipWith (+) a002605_list (tail a002605_list))
-- Reinhard Zumkeller, Oct 15 2011

[Floor(((1 + Sqrt(3))^n - (1 - Sqrt(3))^n)/(2*Sqrt(3))): n in [0..30]]; // Vincenzo Librandi, Aug 18 2011

[n le 2 select n-1 else 2*Self(n-1) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 07 2018

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+2*a[n-2]od: seq(a[n], n=0..33); # Zerinvary Lajos, Dec 15 2008
a := n -> `if`(n<3, n, 2^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -2));
seq(simplify(a(n)), n=0..29); # Peter Luschny, Dec 16 2015

Expand[Table[((1 + Sqrt[3])^n - (1 - Sqrt[3])^n)/(2Sqrt[3]), {n, 0, 30}]] (* Artur Jasinski, Dec 10 2006 *)
a[n_]:=(MatrixPower[{{1,3},{1,1}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{2, 2}, {0, 1}, 30] (* Robert G. Wilson v, Apr 13 2013 *)
Round@Table[Fibonacci[n, Sqrt[2]] 2^((n - 1)/2), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2016 *)
nxt[{a_,b_}]:={b,2(a+b)}; NestList[nxt,{0,1},30][[All,1]] (* Harvey P. Dale, Sep 17 2022 *)

Vec(x/(1-2*x-2*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jun 10 2011

A002605(n)=([2,2;1,0]^n)[2,1] \\ M. F. Hasler, Aug 06 2018

[lucas_number1(n,2,-2) for n in range(0, 30)] # Zerinvary Lajos, Apr 22 2009

a = BinaryRecurrenceSequence(2,2)
print([a(n) for n in (0..29)])  # Peter Luschny, Aug 29 2016

a(n) = (-I*sqrt(2))^(n-1)*U(n-1, I/sqrt(2)) where U(n, x) is the Chebyshev U-polynomial. - Wolfdieter Lang
G.f.: x/(1 - 2*x - 2*x^2).
From Paul Barry, Sep 17 2003: (Start)
E.g.f.: x*exp(x)*(sinh(sqrt(3)*x)/sqrt(3) + cosh(sqrt(3)*x)).
a(n) = (1 + sqrt(3))^(n-1)*(1/2 + sqrt(3)/6) + (1 - sqrt(3))^(n-1)*(1/2 - sqrt(3)/6), for n>0.
Binomial transform of 1, 1, 3, 3, 9, 9, ... Binomial transform is A079935. (End)
a(n) = Sum_{k=0..floor(n/2)} binomial(n - k, k)*2^(n - k). - Paul Barry, Jul 13 2004
a(n) = A080040(n) - A028860(n+1). - Creighton Dement, Jan 19 2005
a(n) = Sum_{k=0..n} A112899(n,k). - Philippe Deléham, Nov 21 2007
a(n) = Sum_{k=0..n} A063967(n,k). - Philippe Deléham, Nov 03 2006
a(n) = ((1 + sqrt(3))^n - (1 - sqrt(3))^n)/(2*sqrt(3)).
a(n) = Sum_{k=0..n} binomial(n, 2*k + 1) * 3^k.
Binomial transform of expansion of sinh(sqrt(3)x)/sqrt(3) (0, 1, 0, 3, 0, 9, ...). E.g.f.: exp(x)*sinh(sqrt(3)*x)/sqrt(3). - Paul Barry, May 09 2003
a(n) = (1/3)*Sum_{k=1..5} sin(Pi*k/2)*sin(2*Pi*k/3)*(1 + 2*cos(Pi*k/6))^n, n >= 1. - Herbert Kociemba, Jun 02 2004
a(n+1) = ((3 + sqrt(3))*(1 + sqrt(3))^n + (3 - sqrt(3))*(1 - sqrt(3))^n)/6. - Al Hakanson (hawkuu(AT)gmail.com), Jun 29 2009
Antidiagonals sums of A081577. - J. M. Bergot, Dec 15 2012
G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k + 2 + 2*x)/(x*(4*k + 4 + 2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013
a(n) = 2^(n - 1)*hypergeom([1 - n/2, (1 - n)/2], [1 - n], -2) for n >= 3. - Peter Luschny, Dec 16 2015
Sum_{k=0..n} a(k)*2^(n-k) = a(n+2)/2 - 2^n. - Greg Dresden, Feb 11 2022
a(n) = 2^floor(n/2) * A002530(n). - Gregory L. Simay, Sep 22 2022
From Peter Bala, May 08 2024: (Start)
G.f.: x/(1 - 2*x - 2*x^2) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (k + 2*x + 1)/(1 + k*x) )
Also x/(1 - 2*x - 2*x^2) = Sum_{n >= 0} (2*x)^n *( x*Product_{k = 1..n} (m*k + 2 - m + x)/(1 + 2*m*k*x) ) for arbitrary m (both series are telescoping). (End)
a(n) = A127864(n-1) + A127864(n-2). - Greg Dresden and Yilin Zhu, Jul 17 2025

Edited by N. J. A. Sloane, Apr 15 2009

A007482 a(n) is the number of subsequences of [ 1, ..., 2n ] in which each odd number has an even neighbor.

Original entry on oeis.org

1, 3, 11, 39, 139, 495, 1763, 6279, 22363, 79647, 283667, 1010295, 3598219, 12815247, 45642179, 162557031, 578955451, 2061980415, 7343852147, 26155517271, 93154256107, 331773802863, 1181629920803, 4208437368135

Offset: 0

N. J. A. Sloane

The even neighbor must differ from the odd number by exactly one.
If we defined this sequence by the recurrence (a(n) = 3*a(n-1) + 2*a(n-2)) that it satisfies, we could prefix it with an initial 0.
a(n) equals term (1,2) in M^n, M = the 3 X 3 matrix [1,1,2; 1,0,1; 2,1,1]. - Gary W. Adamson, Mar 12 2009
a(n) equals term (2,2) in M^n, M = the 3 X 3 matrix [0,1,0; 1,3,1; 0,1,0]. - Paul Barry, Sep 18 2009
From Gary W. Adamson, Aug 06 2010: (Start)
Starting with "1" = INVERT transform of A002605: (1, 2, 6, 16, 44, ...).
Example: a(3) = 39 = (16, 6, 2, 1) dot (1, 1, 3, 11) = (16 + 6 + 6 + 11). (End)
Pisano periods: 1, 1, 4, 1, 24, 4, 48, 2, 12, 24, 30, 4, 12, 48, 24, 4,272, 12, 18, 24, ... . - R. J. Mathar, Aug 10 2012
A007482 is also the number of ways of tiling a 3 X n rectangle with 1 X 1 squares, 2 X 2 squares and 2 X 1 (vertical) dominoes. - R. K. Guy, May 20 2015
With offset 1 (a(0) = 0, a(1) = 1) this is a divisibility sequence. - Michael Somos, Jun 03 2015
Number of elements of size 2^(-n) in a fractal generated by the second-order reversible cellular automaton, rule 150R (see the reference and the link). - Yuriy Sibirmovsky, Oct 04 2016
a(n) is the number of compositions (ordered partitions) of n into parts 1 (of three kinds) and 2 (of two kinds). - Joerg Arndt, Oct 05 2016
a(n) equals the number of words of length n over {0,1,2,3,4} in which 0 and 1 avoid runs of odd lengths. - Milan Janjic, Jan 08 2017
Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 2 X 2 cells and remove the cells that have two '1's in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.833. - Peter Karpov, Apr 20 2017
This is the Lucas sequence U(P=3,Q=-2), and hence for n>=0, a(n+2)/a(n+1) equals the continued fraction 3 + 2/(3 + 2/(3 + 2/(3 + ... + 2/3))) with n 2's. - Greg Dresden, Oct 06 2019

			G.f. = 1 + 3*x + 11*x^2 + 39*x^3 + 139*x^4 + 495*x^5 + 1763*x^6 + ...
From _M. F. Hasler_, Jun 16 2019: (Start)
For n = 0, (1, ..., 2n) = () is the empty sequence, which is equal to its only subsequence, which satisfies the condition voidly, whence a(0) = 1.
For n = 1, (1, ..., 2n) = (1, 2); among the four subsequences {(), (1), (2), (1,2)} only (1) does not satisfy the condition, whence a(1) = 3.
For n = 2, (1, ..., 2n) = (1, 2, 3, 4); among the sixteen subsequences {(), ..., (1,2,3,4)}, the 5 subsequences (1), (3), (1,3), (2,3,4) and (1,2,3,4) do not satisfy the condition, whence a(2) = 16 - 5 = 11.
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 439.

T. D. Noe, Table of n, a(n) for n = 0..200
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 13, 29.
Alexander Burstein and Opel Jones, Enumeration of Dumont permutations avoiding certain four-letter patterns, arXiv:2002.12189 [math.CO], 2020.
R. K. Guy and William O. J. Moser, Numbers of subsequences without isolated odd members, Fibonacci Quarterly, 34, No. 2, 152-155 (1996). Math. Rev. 97d:11017.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 442
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..8)
Yuriy Sibirmovsky, A fractal with number of elements described by a(n)
Index entries for linear recurrences with constant coefficients, signature (3,2).

Row sums of triangle A073387.
Cf. A000045, A000129, A001045, A007455, A007481, A007483, A007484, A015518, A201000 (prime subsequence), A052913 (binomial transform), A026597 (inverse binomial transform).
Cf. A206776.

a007482 n = a007482_list !! (n-1)
a007482_list = 1 : 3 : zipWith (+)
               (map (* 3) $ tail a007482_list) (map (* 2) a007482_list)
-- Reinhard Zumkeller, Oct 21 2015

I:=[1,3]; [n le 2 select I[n] else 3*Self(n-1) + 2*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

a := n -> `if`(n=0, 1, 3^n*hypergeom([(1-n)/2,-n/2], [-n], -8/9)):
seq(simplify(a(n)), n = 0..23); # Peter Luschny, Jun 28 2017

a[n_]:=(MatrixPower[{{1,4},{1,2}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{3,2},{1,3},30] (* Harvey P. Dale, May 25 2013 *)
a[ n_] := Module[ {m = n + 1, s = 1}, If[ m < 0, {m, s} = -{m, (-2)^m}]; s SeriesCoefficient[ x / (1 - 3 x - 2 x^2), {x, 0, m}]]; (* Michael Somos, Jun 03 2015 *)
a[ n_] := With[{m = n + 1}, If[ m < 0, (-2)^m a[ -m], Expand[((3 + Sqrt[17])/2)^m - ((3 - Sqrt[17])/2)^m ] / Sqrt[17]]]; (* Michael Somos, Oct 13 2016 *)

a(n) := if n=0 then 1 elseif n=1 then 3 else 3*a(n-1)+2*a(n-2);
makelist(a(n),n,0,12); /* Emanuele Munarini, Jun 28 2017 */

{a(n) = 2*imag(( (3 + quadgen(68)) / 2)^(n+1))}; /* Michael Somos, Jun 03 2015 */

[lucas_number1(n,3,-2) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

G.f.: 1/(1-3*x-2*x^2).
a(n) = 3*a(n-1) + 2*a(n-2).
a(n) = (ap^(n+1)-am^(n+1))/(ap-am), where ap = (3+sqrt(17))/2 and am = (3-sqrt(17))/2.
Let b(0) = 1, b(k) = floor(b(k-1)) + 2/b(k-1); then, for n>0, b(n) = a(n)/a(n-1). - Benoit Cloitre, Sep 09 2002
The Hankel transform of this sequence is [1,2,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)2^k*3^(n-2k). - Paul Barry, Apr 23 2005
a(n) = Sum_{k=0..n} A112906(n,k). - Philippe Deléham, Nov 21 2007
a(n) = - a(-2-n) * (-2)^(n+1) for all n in Z. - Michael Somos, Jun 03 2015
If c = (3 + sqrt(17))/2, then c^n = (A206776(n) + sqrt(17)*a(n-1)) / 2. - Michael Somos, Oct 13 2016
a(n) = 3^n*hypergeom([(1-n)/2,-n/2], [-n], -8/9) for n>=1. - Peter Luschny, Jun 28 2017
a(n) = round(((sqrt(17) + 3)/2)^(n+1)/sqrt(17)). The distance of the argument from the nearest integer is about 1/2^(n+3). - M. F. Hasler, Jun 16 2019
E.g.f.: (1/17)*exp(3*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2)). - Stefano Spezia, Oct 07 2019
a(n) = (sqrt(2)*i)^n * ChebyshevU(n, -3*i/(2*sqrt(2))). - G. C. Greubel, Dec 24 2021
G.f.: 1/(1 - 3*x - 2*x^2) = Sum_{n >= 0} x^n * Product_{k = 1..n} (k + 2*x + 2)/(1 + k*x) (a telescoping series). Cf. A015518. - Peter Bala, May 08 2024

A073388 Convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

Original entry on oeis.org

1, 4, 16, 56, 188, 608, 1920, 5952, 18192, 54976, 164608, 489088, 1443776, 4238336, 12382208, 36022272, 104407296, 301618176, 868765696, 2495715328, 7152286720, 20452548608, 58369409024

Offset: 0

Wolfdieter Lang, Aug 02 2002

Muniru A Asiru, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4,0,-8,-4).

Second (m=1) column of triangle A073387.

Cf. A002605.

List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+1,1)*Binomial(n-k,k)*(1/2)^k)); # Muniru A Asiru, Jun 12 2018

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^2 )); // G. C. Greubel, Oct 03 2022

CoefficientList[Series[1/(1-2*x-2*x^2)^2, {x,0,40}], x] (* G. C. Greubel, Oct 03 2022 *)

taylor( 1/(1-2*x-2*x^2)^2, x, 0, 24).list() # Zerinvary Lajos, Jun 03 2009; modified by G. C. Greubel, Oct 03 2022

a(n) = Sum_{k=0..n} b(k)*b(n-k), with b(k) = A002605(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+1, 1)*binomial(n-k, k)*2^(n-k).
a(n) = ((n+1)*U(n+1) + 2*(n+2)*U(n))/6, with U(n) = A002605(n), n >= 0.
G.f.: 1/(1-2*x*(1+x))^2.
a(n) = Sum_{k=0..floor((n+2)/2)} k*binomial(n-k+2, k)2^(n-k+1). - Paul Barry, Oct 15 2004

A073389 Second convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

Original entry on oeis.org

1, 6, 30, 128, 504, 1872, 6672, 23040, 77616, 256288, 832416, 2666496, 8441600, 26454528, 82174464, 253280256, 775316736, 2358812160, 7137023488, 21487386624, 64401106944, 192229535744, 571630694400, 1693996941312, 5004131659776, 14738997288960, 43293528760320

Offset: 0

Wolfdieter Lang, Aug 02 2002

Muniru A Asiru, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,-6,-16,12,24,8).

Third (m=2) column of triangle A073387, A073388.

Cf. A002605.

List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+2,2)*Binomial(n-k,k)*(1/2)^k)); # Muniru A Asiru, Jun 12 2018

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-2*x-2*x^2)^3 )); // G. C. Greubel, Oct 03 2022

CoefficientList[Series[1/(1-2x(1+x))^3,{x,0,25}],x]  (* Harvey P. Dale, Mar 14 2011 *)

taylor( 1/(1-2*x-2*x^2)^3, x, 0, 25).list() # Zerinvary Lajos, Jun 03 2009; modified by G. C. Greubel, Oct 03 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k) and c(k) = A073388(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+2, 2)*binomial(n-k, k)*2^(n-k).
a(n) = (n+3)*((n+1)*U(n+1) + (n+2)*U(n))/12, with U(n) = A002605(n), n >= 0.
G.f.: 1/(1-2*x*(1+x))^3.

A073405 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n >= 0, (generalized (2,2)-Fibonacci). Companion triangle is A073406.

Original entry on oeis.org

1, 36, 12, 1536, 888, 120, 80448, 62592, 15168, 1152, 5068800, 4813056, 1600704, 222336, 10944, 375598080, 413351424, 169917696, 32811264, 2992896, 103680, 32103751680, 39661608960, 19066503168, 4592982528

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are p(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of U0(n) := A002605(n), n >= 0, ((2,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073387(n+k,k) = 2*(p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*U0(n))/(k!*12^k)), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073406(k,m).

			k=2: U2(n)=2*((36+12*n)*(n+1)*U0(n+1)+(36+12*n)*(n+2)*U0(n))/(2!*12^2), cf. A073389.
Triangle begins:
  1;
  36, 12;
  1536, 888, 120;
  ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

W. Lang, First 7 rows.

Cf. A002605, A073387, A073406, A073403.

Recursion for row polynomials defined in the comments: p(k, n)= 2*(2*(n+2)*p(k-1, n+1)+2*(n+2*(k+1))*p(k-1, n)+(n+3)*q(k-1, n+1)); q(k, n)= 4*((n+1)*p(k-1, n+1)+(n+2*(k+1))*q(k-1, n)), k >= 1. [Corrected by Sean A. Irvine, Nov 25 2024]

A073403 Coefficient triangle of polynomials (falling powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073404.

Original entry on oeis.org

1, 12, 36, 120, 888, 1536, 1152, 15168, 62592, 80448, 10944, 222336, 1600704, 4813056, 5068800, 103680, 2992896, 32811264, 169917696, 413351424, 375598080, 981504, 38112768, 587976192, 4592982528

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are p(k,x) := sum(a(k,m)*x^(k-m),m=0..k), k=0,1,2,..

The k-th convolution of U0(n) := A002605(n), n>= 0, ((2,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073387(n+k,k) = 2*(p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*U0(n))/(k!*12^k), k=1,2,..., where the companion polynomials q(k,n) := sum(b(k,m)*n^(k-m),m=0..k) are the row polynomials of triangle b(k,m)= A073404(k,m).

			k=2: U2(n)=(2*(36+12*n)*(n+1)*U0(n+1)+2*(36+12*n)*(n+2)*U0(n))/(2!*12^2), cf. A073389.
1; 12,36; 120,888,1536; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

W. Lang, First 7 rows.

Cf. A002605, A073387, A073404.

Recursion for row polynomials defined in the comments: see A073405.

A073390 Third convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

Original entry on oeis.org

1, 8, 48, 240, 1080, 4512, 17856, 67776, 248880, 889600, 3109376, 10664448, 35989248, 119761920, 393676800, 1280157696, 4122985728, 13165099008, 41713192960, 131243970560, 410315433984, 1275348344832

Offset: 0

Wolfdieter Lang, Aug 02 2002

Muniru A Asiru, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-16,-16,56,32,-64,-64,-16).

Fourth (m=3) column of triangle A073387.

Cf. A002605, A073389.

List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+3,3)*Binomial(n-k,k)*(1/2)^k)); # Muniru A Asiru, Jun 12 2018

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^4 )); // G. C. Greubel, Oct 03 2022

CoefficientList[Series[1/(1-2*x-2*x^2)^4, {x,0,40}], x] (* G. C. Greubel, Oct 03 2022 *)

def A073390_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^4 ).list()
A073390_list(40) # G. C. Greubel, Oct 03 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k) and c(k) = A073389(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+3, 3)*binomial(n-k, k)*2^(n-k).
a(n) = ((64 + 37*n + 5*n^2)*(n+1)*U(n+1) + 4*(11 + 7*n + n^2)*(n+2)*U(n))/(6^3), with U(n) = A002605(n), n >= 0.
G.f.: 1/(1-2*x*(1+x))^4.

A073391 Fourth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

Original entry on oeis.org

1, 10, 70, 400, 2020, 9352, 40600, 167680, 665440, 2555840, 9551936, 34880000, 124853120, 439228160, 1521839360, 5202292736, 17571249920, 58712184320, 194280061440, 637228462080, 2073332481024, 6696470231040

Offset: 0

Wolfdieter Lang, Aug 02 2002

Muniru A Asiru, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (10,-30,0,120,-48,-240,0,240,160,32).

Fifth (m=4) column of triangle A073387.

Cf. A002605, A073390.

List([0..25], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+4,4)*Binomial(n-k,k)*(1/2)^k)); # Muniru A Asiru, Jun 12 2018

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^5 )); // G. C. Greubel, Oct 04 2022

CoefficientList[Series[1/(1-2*x-2*x^2)^5, {x,0,40}], x] (* G. C. Greubel, Oct 04 2022 *)

def A073391_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2)^5 ).list()
A073391_list(40) # G. C. Greubel, Oct 04 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073390(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+4, 4)*binomial(n-k, k)*2^(n-k).
a(n) = (2*(419 + 326*n + 79*n^2 + 6*n^3)*(n+1)*U(n+1) + (458 + 421*n + 112*n^2 + 9*n^3)*(n+2)*U(n))/(2^5*3^4), with U(n) = A002605(n), n >= 0.
G.f.: 1/(1-2*x*(1+x))^5.

A073392 Fifth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

Original entry on oeis.org

1, 12, 96, 616, 3444, 17472, 82432, 367488, 1565280, 6421376, 25525248, 98773248, 373450112, 1383674880, 5036089344, 18041821184, 63727070976, 222249968640, 766234140672, 2614196680704, 8834194123776

Offset: 0

Wolfdieter Lang, Aug 02 2002

			x^6 + 12*x^7 + 96*x^8 + 616*x^9 + 3444*x^10 + ... + 222249968640*x^23 + 766234140672*x^24 + 2614196680704*x^25 + 8834194123776*x^26 + ... - _Zerinvary Lajos_, Jun 03 2009

Muniru A Asiru, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (12,-48,40,180,-288,-384,576,720,-320,-768,-384,-64).

Sixth (m=5) column of triangle A073387.

Cf. A002605, A073391.

List([0..30], n->2^n*Sum([0..Int(n/2)],k->Binomial(n-k+5,5)*Binomial(n-k,k)*(1/2)^k)); # Muniru A Asiru, Jun 12 2018

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^6 )); // G. C. Greubel, Oct 04 2022

CoefficientList[Series[1/(1-2*x*(1+x))^6, {x,0,30}],x] (* Harvey P. Dale, May 12 2018 *)

taylor( 1/(1-2*x-2*x^2)^6, x, 0, 30).list() # Zerinvary Lajos, Jun 03 2009; modified by G. C. Greubel, Oct 04 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A002605(k) and c(k) = A073391(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+5, 5)*binomial(n-k, k)*2^(n-k).
a(n) = (n+4)*(n+8)*((19*n^2 + 158*n + 275)*(n+1)*U(n+1) + 2*(7*n^2 + 52*n + 65)*(n+2)*U(n))/(2^6*3^4*5), with U(n) = A002605(n), n >= 0.
G.f.: 1/(1-2*x*(1+x))^6.

A073393 Sixth convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

Original entry on oeis.org

1, 14, 126, 896, 5488, 30240, 153888, 735744, 3344544, 14581952, 61378240, 250693632, 997593856, 3880249856, 14791776768, 55385874432, 204082373376, 741186464256, 2656771815936, 9410113241088

Offset: 0

Wolfdieter Lang, Aug 02 2002

			x^7 + 14*x^8 + 126*x^9 + 896*x^10 + 5488*x^11 + ... + 204082373376*x^23 + 741186464256*x^24 + 2656771815936*x^25 + 9410113241088*x^26 + ... - _Zerinvary Lajos_, Jun 03 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-70,112,196,-728,-168,1920,336,-2912,-1568, 1792,2240,896,128).

Seventh (m=6) column of triangle A073387.

Cf. A002605, A073392.

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 1/(1-2*x-2*x^2)^7 )); // G. C. Greubel, Oct 05 2022

CoefficientList[Series[1/(1-2x(1+x))^7,{x,0,30}],x] (* or *)
LinearRecurrence[{14,-70,112,196,-728,-168,1920,336,-2912,-1568,1792,2240,896,128},{1,14,126,896,5488,30240,153888,735744,3344544,14581952,61378240,250693632, 997593856,3880249856},30](* Harvey P. Dale, Jan 24 2013 *)

taylor( 1/(1-2*x-2*x^2)^7, x, 0, 26).list() # Zerinvary Lajos, Jun 03 2009; modified by G. C. Greubel, Oct 05 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k) with b(k) = A002605(k) and c(k) = A073392(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+6, 6)*binomial(n-k, k)*2^(n-k).
a(n) = ((54340 + 59802*n + 24583*n^2 + 4747*n^3 + 433*n^4 + 15*n^5)*(n+1)*U(n+1) + (23420 + 32768*n + 15333*n^2 + 3201*n^3 + 307*n^4 + 11*n^5)*(n+2)*U(n))/(2^7*3^5*5), with U(n) := A002605(n), n >= 0.
G.f.: 1/(1-2*x*(1+x))^7.
a(0)=1, a(1)=14, a(2)=126, a(3)=896, a(4)=5488, a(5)=30240, a(6)=153888, a(7)=735744, a(8)=3344544, a(9)=14581952, a(10)=61378240, a(11)=250693632, a(12)=997593856, a(13)=3880249856, a(n) = 14*a(n-1) - 70*a(n-2) + 112*a(n-3) + 196*a(n-4) - 728*a(n-5) - 168*a(n-6) + 1920*a(n-7) + 336*a(n-8) - 2912*a(n-9) - 1568*a(n-10) + 1792*a(n-11) + 2240*a(n-12) + 896*a(n-13) + 128*a(n-14). - Harvey P. Dale, Jan 24 2013

cp's OEIS Frontend

A002605 a(n) = 2*(a(n-1) + a(n-2)), a(0) = 0, a(1) = 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Magma

Maple

Mathematica

PARI

PARI

Sage

Sage

Formula

Extensions

A007482 a(n) is the number of subsequences of [ 1, ..., 2n ] in which each odd number has an even neighbor.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

Maxima

PARI

Sage

Formula

A073388 Convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Mathematica

Sage

Formula

A073389 Second convolution of A002605(n) (generalized (2,2)-Fibonacci), n >= 0, with itself.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Mathematica

Sage

Formula

A073405 Coefficient triangle of polynomials (rising powers) related to convolutions of A002605(n), n >= 0, (generalized (2,2)-Fibonacci). Companion triangle is A073406.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A073403 Coefficient triangle of polynomials (falling powers) related to convolutions of A002605(n), n>=0, (generalized (2,2)-Fibonacci). Companion triangle is A073404.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs