A069403 -id:A069403 - cp's OEIS frontend

A085327 Duplicate of A069403.

1, 3, 9, 25, 67, 177, 465, 1219, 3193, 8361, 21891, 57313, 150049, 392835, 1028457

Offset: 0

dead

A128588 Expansion of g.f. x*(1+x+x^2)/(1-x-x^2).

1, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338

Offset: 1

Gary W. Adamson, Mar 11 2007

Previous name was: A007318 * A128587.
a(n)/a(n-1) tends to phi, 1.618... = A001622.
Regardless of initial two terms, any linearly recurring sequence with signature (1,1) will yield an a(n)/a(n+1) ratio tending to phi (the Golden Ratio). - Harvey P. Dale, Mar 29 2017
Apart from the initial term, double the Fibonacci numbers. O.g.f.: x*(1+x+x^2)/(1-x-x^2). a(n) gives the number of binary strings of length n-1 avoiding the substrings 000 and 111. a(n) also gives the number of binary strings of length n-1 avoiding the substrings 010 and 101. - Peter Bala, Jan 22 2008
Row lengths of triangle A232642. - Reinhard Zumkeller, May 14 2015
a(n) is the number of binary strings of length n-1 avoiding the substrings 000 and 111. - Allan C. Wechsler, Feb 13 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..2500
J.-P. Allouche and T. Johnson, Narayana's Cows and Delayed Morphisms.
Andrei Asinowski and Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
Elena Barcucci, Antonio Bernini, Stefano Bilotta, and Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016. See 1st column of Table 2 p. 11.
Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009, page 52.
B. Winterfjord, Binary strings and substring avoidance.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000045, A001622, A128587, A128586, A007318.
Cf. A006355, A055389.
Cf. A014217, A069403, A153819.
Cf. A232642, A242593.
Cf. A038754, A068922.
Main diagonal of A303696.

Concatenation([1], List([2..40], n-> 2*Fibonacci(n))); # G. C. Greubel, Jul 10 2019

a128588 n = a128588_list !! (n-1)
a128588_list = 1 : cows where
                   cows = 2 : 4 : zipWith (+) cows (tail cows)
-- Reinhard Zumkeller, May 14 2015

[1] cat [2*Fibonacci(n): n in [2..40]]; // G. C. Greubel, Jul 10 2019

a:= n-> `if`(n<2, n, 2*(<<0|1>, <1|1>>^n)[1,2]):
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 28 2018

nn=40; a=(1-x^3)/(1-x); b=x*(1-x^2)/(1-x); CoefficientList[Series[a^2 /(1-b^2), {x,0,nn}], x]  (* Geoffrey Critzer, Sep 01 2012 *)
LinearRecurrence[{1,1}, {1,2,4}, 40] (* Harvey P. Dale, Mar 29 2017 *)
Join[{1}, 2*Fibonacci[Range[2,40]]] (* G. C. Greubel, Jul 10 2019 *)

{a(n) = if( n<2, n==1, 2 * fibonacci(n))}; /* Michael Somos, Jul 18 2015 */

[1]+[2*fibonacci(n) for n in (2..40)] # G. C. Greubel, Jul 10 2019

G.f.: x*(1+x+x^2)/(1-x-x^2).
Binomial transform of A128587; a(n+2) = a(n+1) + a(n), n>3.
a(n) = A068922(n-1), n>2. - R. J. Mathar, Jun 14 2008
For n > 1: a(n+1) = a(n) + if a(n) odd then max{a(n),a(n-1)} else min{a(n),a(n-1)}, see also A038754. - Reinhard Zumkeller, Oct 19 2015
E.g.f.: 4*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5) - x. - Stefano Spezia, Feb 19 2023

New name from Joerg Arndt, Feb 16 2024

A052995 Expansion of 2x(1 - x)/(1 - 3*x + x^2).

Original entry on oeis.org

0, 2, 4, 10, 26, 68, 178, 466, 1220, 3194, 8362, 21892, 57314, 150050, 392836, 1028458, 2692538, 7049156, 18454930, 48315634, 126491972, 331160282, 866988874, 2269806340, 5942430146, 15557484098, 40730022148, 106632582346, 279167724890, 730870592324

Offset: 0

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

Terms >=4 give solutions x to floor(phi^2*x^2) - floor(phi*x)^2 = 5, where phi =(1 + sqrt(5))/2. - Benoit Cloitre, Mar 16 2003
Except for the first term, positive values of x (or y) satisfying x^2 - 18*x*y + y^2 + 256 = 0. - Colin Barker, Feb 14 2014
a(n+1) is the square of the distance AB, where A is the point (F(n), F(n+1)), B is the 90-degree rotation of A about the origin, and F(n)=A000045(n) are the Fibonacci numbers. - Burak Muslu, Mar 24 2021

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 30.
B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 60-61.

Colin Barker, Table of n, a(n) for n = 0..1000
Younseok Choo, Some Results on the Infinite Sums of Reciprocal Generalized Fibonacci Numbers, International Journal of Mathematical Analysis, 12(12) (2018), 621-629.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1072.
Youngwoo Kwon, Binomial transforms of the modified k-Fibonacci-like sequence, arXiv:1804.08119 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (3,-1).

Bisection of A006355.
First differences of A025169.
Cf. A000032, A000045, A055819, A006355, A025169.

spec := [S, S=Prod(Sequence(Union(Prod(Sequence(Z),Z),Z)),Union(Z,Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

LinearRecurrence[{3, -1}, {0, 2, 4}, 30] (* or *)
Nest[Append[#, 3 #[[-1]] - #[[-2]]] &, {0, 2, 4}, 27] (* or *)
CoefficientList[Series[-2 x (-1 + x)/(1 - 3 x + x^2), {x, 0, 29}], x] (* Michael De Vlieger, Jul 18 2018 *)

concat(0, Vec(2*x*(1-x)/(1-3*x+x^2) + O(x^50))) \\ Colin Barker, Mar 30 2016

a(n) = fibonacci(max(0,2*n-1))<<1; \\ Kevin Ryde, Mar 25 2021

G.f.: -2*x*(-1 + x)/(1 - 3*x + x^2).
a(0) = 0, a(1) = 2, a(2) = 4; for n > 0, a(n) - 3*a(n+1) + a(n+2) = 0.
a(n) = A069403(n-1)+1.
a(n) = Sum(2/5*(-1 + 4*_alpha)*_alpha^(-1-n), _alpha = RootOf(_Z^2 - 3*_Z + 1)).
a(n) = 2*Fibonacci(2*n-1) = 2*A001519(n) for n > 0. - Vladeta Jovovic, Mar 19 2003
a(n+2) = F(n)^2 + F(n+3)^2 = 2*F(n+1)^2 + 2*F(n+2)^2, where F = A000045. -  N. J. A. Sloane, Feb 20 2005
a(n) = 1/2*(F(2*n+8) mod F(2*n+2)) for n > 2. - Gary Detlefs, Nov 22 2010
a(n) = F(n-3)*F(n-1) + F(n)*F(n+2) for n > 0, F(-2) = -1, F(-1) = 1. - Bruno Berselli, Nov 03 2015
a(n) = (2^(-n)*((3 - sqrt(5))^n*(1 + sqrt(5)) + (-1 + sqrt(5))*(3 + sqrt(5))^n))/sqrt(5) for n > 0. - Colin Barker, Mar 30 2016
a(n) = Fibonacci(2*n-2) + Lucas(2*n-2) for n > 0. - Bruno Berselli, Oct 13 2017
a(n) = Lucas(2*n) - Fibonacci(2*n) for n > 0. - Diego Rattaggi, Mar 08 2023

More terms from James Sellers, Jun 05 2000

A192904 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 0, 1, 5, 16, 49, 153, 480, 1505, 4717, 14784, 46337, 145233, 455200, 1426721, 4471733, 14015632, 43928817, 137684905, 431542080, 1352570689, 4239325789, 13287204352, 41645725825, 130529073953, 409113752000, 1282274186177

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x.  The resulting sequence typifies a general class which we shall describe here. Suppose that u,v,a,b,c,d,e,f are numbers used to define these polynomials:
...
q(x) = x^2
s(x) = u*x + v
p(0,x) = a, p(1,x) = b*x + c
p(n,x) = d*(x^2)*p(n-1,x) + e*x*p(n-2,x) + f.
...
We shall assume that u is not 0 and that {d,e} is not {0}.  The reduction of p(n,x) by the repeated substitution q(x) -> s(x), as defined and described at A192232 and A192744, has the form h(n) + k(n)*x.  The numerical sequences h and k are linear recurrence sequences, formally of order 5.  The Mathematica program below, with first line deleted, shows initial terms and recurrence coefficients, which imply these properties:
(1)  the recurrence coefficients depend only on u,v,d,e; the parameters a,b,c,f affect only the initial terms.
(2)  if e=0 or v=0, the order of recurrence is <= 3;
(3)  if e=0 and v=0, the recurrence coefficients are 1+d*u^2 and -d*u^2 (cf. similar results at A192872).
...
Examples:
u v a b c d e f... seq h.....seq k
1 1 1 1 1 1 0 0... A001906..A001519
1 1 1 1 0 0 1 0... A103609..A193609
1 1 1 1 0 1 1 0... A192904..A192905
1 1 1 1 1 1 0 0... A001519..A001906
1 1 1 1 1 1 1 0... A192907..A192907
1 1 1 1 1 1 0 1... A192908..A069403
1 1 1 1 1 1 1 1... A192909..A192910
The terms of these sequences involve Fibonacci numbers, F(n)=A000045(n); e.g.,
A001906: even-indexed F(n)
A001519: odd-indexed F(n)
A103609: (1,1,1,1,2,2,3,3,5,5,8,8,...)

			The first six polynomials and reductions:
1 -> 1
x -> x
x + x^3 -> 1 + 3*x
x^2 + x^3 + x^5 -> 5 + 8*x
x^2 + 2*x^4 + x^5 + x^7 -> 16 + 25*x
x^3 + 2*x^4 + 3*x^6 + x^7 + x^9 -> 49 + 79*x, so that
A192904 = (1,0,1,5,16,49,...) and
A192905 = (0,1,3,8,25,79,...)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,1,1).

Cf. A192232, A192744, A192905, A192872.

a:=[1,0,1,5];; for n in [5..40] do a[n]:=3*a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 10 2019

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4) )); // G. C. Greubel, Jan 10 2019

(* To obtain general results, delete the next line. *)
u = 1; v = 1; a = 1; b = 1; c = 0; d = 1; e = 1; f = 0;
q = x^2; s = u*x + v; z = 24;
p[0, x_] := a; p[1, x_] := b*x + c;
p[n_, x_] :=  d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192904 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192905 *)
Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
LinearRecurrence[{3,0,1,1}, {1,0,1,5}, 40] (* G. C. Greubel, Jan 10 2019 *)

my(x='x+O('x^40)); Vec((1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4)) \\ G. C. Greubel, Jan 10 2019

((1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jan 10 2019

a(n) = 3*a(n-1) + a(n-3) + a(n-4).

G.f.: (1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4). - Colin Barker, Aug 31 2012

A165278 Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-indexed Fibonacci numbers in its Zeckendorf representation.

Original entry on oeis.org

2, 5, 1, 7, 3, 4, 13, 6, 9, 12, 15, 8, 11, 25, 33, 18, 10, 17, 30, 67, 88, 20, 14, 22, 32, 80, 177, 232, 34, 16, 24, 46, 85, 211, 465, 609, 36, 19, 27, 59, 87, 224, 554, 1219, 1596, 39, 21, 29, 64, 122, 229, 588, 1452, 3193, 4180, 41, 23, 31, 66, 156, 231, 601

Offset: 1

Clark Kimberling, Sep 13 2009

For n>=0, row n is the monotonic sequence of positive integers m such that the number of even-indexed Fibonacci numbers in the Zeckendorf representation of m is n.
We begin the indexing at 2; that is, 1=F(2), 2=F(3), 3=F(4), 5=F(5),...
Every positive integer occurs exactly once in the array, so that as a sequence it is a permutation of the positive integers.
For counts of odd-indexed Fibonacci numbers, see A165279.
Essentially, (row 0)=A062879, (column 1)=A027941, (column 2)=A069403.

			Northwest corner:
2....5....7...13...15...18...20...34...36...
1....3....6....8...10...14...16...19...20...
4....9...11...17...22...24...27...29...31...
12..25...30...32...46...59...64...66...72...
Examples:
20=13+5+2=F(7)+F(5)+F(3), zero evens, so 20 is in row 0.
19=13+5+1=F(7)+F(5)+F(2), one even, so 19 is in row 1.
22=21+1=F(8)+F(2), two evens, so 22 is in row 2.

Cf. A165276, A165277, A165279.

f[n_] := Module[{i = Ceiling[Log[GoldenRatio, Sqrt[5]*n]], v = {}, m = n}, While[i > 1, If[Fibonacci[i] <= m, AppendTo[v, 1]; m -= Fibonacci[i], If[v != {}, AppendTo[v, 0]]]; i--]; Total[Reverse[v][[1 ;; -1 ;; 2]]]]; T = GatherBy[SortBy[ Range[10^4], f], f]; Table[Table[T[[n - k + 1, k]], {k, n, 1, -1}], {n, 1, Length[T]}] // Flatten (* Amiram Eldar, Feb 04 2020 *)

More terms from Amiram Eldar, Feb 04 2020

A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 1, 3, 7, 17, 43, 111, 289, 755, 1975, 5169, 13531, 35423, 92737, 242787, 635623, 1664081, 4356619, 11405775, 29860705, 78176339, 204668311, 535828593, 1402817467, 3672623807, 9615053953, 25172538051, 65902560199

Offset: 0

Clark Kimberling, Jul 12 2011

The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x + 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A192232, A192744, A192872, A192904.
Cf. A000045; A052995: 2*Fibonacci(2*n-1) for n>0.
CF. A055588, A097136.

Concatenation([1], List([1..30], n -> 1+2*Fibonacci(2*(n-1)))); # G. C. Greubel, Jan 11 2019

[1] cat [1+2*Fibonacci(2*(n-1)): n in [1..30]]; // G. C. Greubel, Jan 11 2019

u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 0; f = 1;
q = x^2; s = u*x + v; z = 26;
p[0, x_] := a;  p[1, x_] := b*x + c
p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]    (* A192908 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]    (* A069403 *)
Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
LinearRecurrence[{4,-4,1}, {1,1,3,7}, 30] (* G. C. Greubel, Jan 11 2019 *)

vector(30, n, n--; if(n==0,1,1+2*fibonacci(2*n-2))) \\ G. C. Greubel, Jan 11 2019

[1]+[1+2*fibonacci(2*(n-1)) for n in (1..30)] # G. C. Greubel, Jan 11 2019

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n>3.
G.f.: 1 + x*(1 - x - x^2)/((1 - x)*(1 - 3*x + x^2)). - R. J. Mathar, Jul 13 2011
a(n) = 2*Fibonacci(2*n-2) + 1 for n>0, a(0)=1. - Bruno Berselli, Dec 27 2016
a(n) = -1 + 3*a(n-1) - a(n-2) with a(1) = 1 and a(2) = 3. Cf. A055588 and A097136. - Peter Bala, Nov 12 2017

A084707 a(n) = 3a(n-1) - 3a(n-3) + a(n-4) for n > 3, with a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 27.

Original entry on oeis.org

1, 3, 9, 27, 73, 195, 513, 1347, 3529, 9243, 24201, 63363, 165889, 434307, 1137033, 2976795, 7793353, 20403267, 53416449, 139846083, 366121801, 958519323, 2509436169, 6569789187, 17199931393, 45030004995, 117890083593, 308640245787, 808030653769

Offset: 0

N. J. A. Sloane, Jul 06 2003

Define f(x, y) := 9 - x - 3*y + x^2 - 3*x*y + y^2. Then f(x, y) = f(-4-y, -4-x). All of the integer solutions of 0 = f(x, y) with x>=0 are given by x = a(2*n) and y = a(2*n+1) for all n in Z. - Michael Somos, Aug 19 2023

			G.f. = 1 + 3*x + 9*x^2 + 27*x^3 + 73*x^4 + 195*x^5 + 513*x^6 + ... - _Michael Somos_, Aug 19 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Hietarinta and C.-M. Viallet, Singularity confinement and chaos in discrete systems, Physical Review Letters 81 (1998), pp. 326-328.
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A005248, A069403.

A084707:=[1,3,9,27]; [n le 4 select A084707[n] else 3*Self(n-1)-3*Self(n-3)+Self(n-4): n in [1..30]]; // Wesley Ivan Hurt, Aug 15 2016

[(8*Lucas(2*n) -(-1)^n)/5 -2: n in [0..40]]; // G. C. Greubel, Apr 15 2023

a:=proc(n) option remember; if n=0 then 1 elif n=1 then 3 elif n=2 then 9 elif n=3 then 27 else 3*a(n-1)-3*a(n-3)+a(n-4); fi; end: seq(a(n), n=0..40); # Wesley Ivan Hurt, Aug 15 2016

a[n_]:=a[n]=3a[n-1] -3a[n-3] +a[n-4]; a[0]=1; a[1]=3; a[2]=9; a[3]=27;
Table[ a[n], {n, 0, 27}]
Transpose[NestList[Join[Rest[#],ListCorrelate[{1,-3,0,3},#]]&, {1,3,9,27},30]][[1]]
CoefficientList[Series[(1+3 x^3)/(1-3 x+3 x^3-x^4),{x,0,30}],x]  (* Harvey P. Dale, Mar 14 2011 *)
a[ n_] := Floor[(LucasL[2*n] - 1)*8/5]; (* Michael Somos, Aug 19 2023 *)

{a(n) = my(w=quadgen(5)); (real((1+w)^n*(2+w))-1)*8\5}; /* Michael Somos, Aug 19 2023 */

[(8*lucas_number2(2*n,1,-1) -(-1)^n)/5 -2 for n in range(41)] # G. C. Greubel, Apr 15 2023

G.f.: (1+3*x^3)/(1-3*x+3*x^3-x^4). - Harvey P. Dale, Mar 14 2011
a(n) = (8*LucasL(2*n) - (-1)^n - 10)/5. - G. C. Greubel, Apr 15 2023
a(n) = a(-n) = 4 + 2*a(n-1) + 2*a(n-2) - a(n-3) for all n in Z. - Michael Somos, Aug 19 2023

More terms from Ray Chandler, Jul 07 2003

A069423 Number of n X 3 binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row.

Original entry on oeis.org

1, 19, 147, 907, 5149, 28159, 151331, 806463, 4280141, 22670199, 119955523, 634412111, 3354416225, 17734138519, 93751306327, 495600489243, 2619870168133, 13849199313235, 73209594712831, 386999645945759, 2045750897807901, 10814208630415211, 57165847109895879

Offset: 1

R. H. Hardin, Mar 22 2002

Index entries for linear recurrences with constant coefficients, signature (11,-41,64,-35,-13,25,-12,2).

Cf. 2 X n A001047, n X 2 A034182, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

a(n) = A069378(n) - 2 * A069403(n). - Sean A. Irvine, Aug 23 2024

G.f.: x*(2*x^7-10*x^6+23*x^5-18*x^4-5*x^3+21*x^2-8*x-1) / ((x-1)*(x^2-3*x+1)*(2*x^5-4*x^4+x^3+9*x^2-7*x+1)). - Alois P. Heinz, Aug 23 2024

More terms from Sean A. Irvine, Aug 23 2024

A069441 Half the number of n X 3 binary arrays with no path of adjacent 1's or adjacent 0's from top row to bottom row.

Original entry on oeis.org

0, 4, 84, 1074, 11058, 102448, 896026, 7578952, 62820362, 514178822, 4174954460, 33725176208, 271523097884, 2181288088576, 17498432045552, 140241880816930, 1123280018219562, 8993350007112124, 71984384316723134, 576073752326317448, 4609640266662591130

Offset: 1

R. H. Hardin, Mar 22 2002

Cf. 2 X n A000079, n X 1 A000225, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

a(n) = 2^(3*n-1) - A069378(n) + A069403(n). - Sean A. Irvine, Aug 23 2024

More terms from Sean A. Irvine, Aug 23 2024

A128544 A007318 * A128540.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 9, 9, 3, 5, 16, 24, 17, 5, 6, 25, 50, 55, 33, 8, 7, 36, 90, 135, 123, 61, 13, 8, 49, 147, 280, 343, 259, 112, 21, 9, 64, 224, 518, 798, 812, 532, 202, 34, 10, 81, 324, 882, 1638, 2100, 1848, 1062, 361, 55

Offset: 0

Gary W. Adamson, Mar 10 2007

Row sums = A069403: (1, 3, 9, 25, 67, ...).

			First few rows of the triangle:
  1;
  2,  1;
  3,  4,  2;
  4,  9,  9,   3;
  5, 16, 24,  17,   5;
  6, 25, 50,  55,  33,  8;
  7, 36, 90, 135, 123, 61, 13;
  ...

Cf. A007318, A128540, A069403.

A007318 * A128540 as infinite lower triangular matrices; (binomial transform of A128540).

cp's OEIS Frontend

A085327 Duplicate of A069403.

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A128588 Expansion of g.f. x*(1+x+x^2)/(1-x-x^2).

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A052995 Expansion of 2*x*(1 - x)/(1 - 3*x + x^2).

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A192904 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

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A165278 Table read by antidiagonals: T(n, k) is the k-th number with n-1 even-indexed Fibonacci numbers in its Zeckendorf representation.

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A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

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A084707 a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4) for n > 3, with a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 27.

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A052995 Expansion of 2x(1 - x)/(1 - 3*x + x^2).

A084707 a(n) = 3a(n-1) - 3a(n-3) + a(n-4) for n > 3, with a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 27.