A095342 -id:A095342 - cp's OEIS frontend

A095344 Length of n-th string generated by a Kolakoski(9,1) rule starting with a(1)=1.

1, 1, 9, 9, 49, 81, 281, 601, 1729, 4129, 11049, 27561, 71761, 182001, 469049, 1197049, 3073249, 7861441, 20154441, 51600201, 132217969, 338618769, 867490649, 2221965721, 5691928321, 14579791201, 37347504489, 95666669289, 245056687249, 627723364401

Offset: 1

Benoit Cloitre, Jun 03 2004

Each string is derived from the previous string using the Kolakoski(9,1) rule and the additional condition: "string begins with 1 if previous string ends with 9 and vice versa". The strings are 1 -> 9 -> 111111111 -> 919191919 -> 11111111191111111119... -> ... and each one contains 1,1,9,9,31,... elements.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,4).

Cf. A000002, A066983, A095342, A095343.

Cf. A123270, A090390.

a:=[1,1,9];; for n in [4..35] do a[n]:=5*a[n-2]+4*a[n-3]; od; a; # G. C. Greubel, Dec 26 2019

a095344 n = a095344_list !! (n-1)
a095344_list = tail xs where
   xs = 1 : 1 : 1 : zipWith (-) (map (* 5) $ zipWith (+) (tail xs) xs) xs
-- Reinhard Zumkeller, Aug 16 2013

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( x*(1+x+ 4*x^2)/((1+x)*(1-x-4*x^2)) )); // G. C. Greubel, Dec 26 2019

seq(simplify(2*(-1)^n -(2/I)^n*(ChebyshevU(n, I/4) -2*I*ChebyshevU(n-1, I/4)) ), n = 1..35); # G. C. Greubel, Dec 26 2019

Table[2*(-1)^n - 2^n*(Fibonacci[n+1, 1/2] - 2*Fibonacci[n, 1/2]), {n,35}] (* G. C. Greubel, Dec 26 2019 *)
LinearRecurrence[{0,5,4},{1,1,9},40] (* Harvey P. Dale, Oct 12 2022 *)

Vec(x*(1+x+4*x^2)/((1+x)*(1-x-4*x^2)) + O(x^50)) \\ Colin Barker, Apr 20 2016

vector(35, n, round( 2*(-1)^n - (2/I)^n*(polchebyshev(n, 2, I/4) -2*I*polchebyshev(n-1, 2, I/4)) )) \\ G. C. Greubel, Dec 26 2019

def A095344_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x+4*x^2)/((1+x)*(1-x-4*x^2)) ).list()
a=A095344_list(35); a[1:] # G. C. Greubel, Dec 26 2019

a(1) = a(2) = 1; for n>1, a(n) = a(n-1) + 4*a(n-2) - 4*(-1)^n.
G.f.: x*(1 + x + 4*x^2)/((1 + x)*(1 - x - 4*x^2)). - Colin Barker, Mar 25 2012
a(n) = 5*a(n-2) + 4*a(n-3). - Colin Barker, Mar 25 2012
a(n) = 2*(-1)^n + (2^(-1-n)*(-(-7+sqrt(17))*(1+sqrt(17))^n - (1-sqrt(17))^n*(7+sqrt(17))))/sqrt(17). - Colin Barker, Apr 20 2016
a(n) = 2*(-1)^n - 2^n*(Fibonacci(n+1, 1/2) - 2*Fibonacci(n, 1/2)) = 2*(-1)^n - (2/I)^n*(ChebyshevU(n, I/4) - 2*I*ChebyshevU(n-1, I/4)). - G. C. Greubel, Dec 26 2019

A095343 Length of n-th string generated by a Kolakoski(7,1) rule starting with a(1)=1.

Original entry on oeis.org

1, 1, 7, 7, 31, 49, 145, 289, 727, 1591, 3775, 8545, 19873, 45505, 105127, 241639, 557023, 1281937, 2953009, 6798817, 15657847, 36054295, 83027839, 191190721, 440274241, 1013846401, 2334669127, 5376208327, 12380215711, 28508840689

Offset: 1

Benoit Cloitre, Jun 03 2004

nonn

Each string is derived from the previous string using the Kolakoski(7,1) rule and the additional condition: "string begins with 1 if previous string ends with 5 and vice versa". The strings are 1 -> 7 -> 1111111 -> 7171717 -> 11111117111111171111111711111117 -> ... and each one contains 1,1,7,7,31,... elements.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 4, 3).

Cf. A000002, A066983, A095342, A095344.

a:=[1,1];; for n in [3..35] do a[n]:=a[n-1]-3*a[n-2]-3*(-1)^n; od; a; # G. C. Greubel, Dec 26 2019

I:=[1,1]; [n le 2 select I[n] else Self(n-1) + 3*Self(n-2) - 3*(-1)^n: n in [1..35]]; // G. C. Greubel, Dec 26 2019

seq(coeff(series(x*(1+x+3*x^2)/((1+x)*(1-x-3*x^2)), x, n+1), x, n), n = 0..35); # G. C. Greubel, Dec 26 2019

Table[ 3*(-1)^n + 2*Sqrt[3]^n*(Sqrt[3]*Fibonacci[n, 1/Sqrt[3]] - Fibonacci[n+1, 1/Sqrt[3]]), {n,35}]//FullSimplify (* G. C. Greubel, Dec 26 2019 *)

vector(35, n, round(3*(-1)^n + 2*(sqrt(3)/I)^n*(sqrt(3)*I* polchebyshev(n-1, 2, I/(2*sqrt(3))) - polchebyshev(n, 2, I/(2*sqrt(3)))) )) \\ G. C. Greubel, Dec 26 2019

def A095343_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x+3*x^2)/((1+x)*(1-x-3*x^2)) ).list()
a=A095343_list(35); a[1:] # G. C. Greubel, Dec 26 2019

a(1) = a(2) = 1, a(n) = a(n-1) + 3*a(n-2) - 3*(-1)^n.
G.f.: x*(1+x+3*x^2)/((1+x)*(1-x-3*x^2)). - Colin Barker, Jul 02 2012
a(n) = 3*(-1)^n + 2*(sqrt(3)/i)^n*(sqrt(3)*i*ChebyshevU(n, i/(2*sqrt(3))) - ChebyshevU(n-1, i/(2*sqrt(3)))). - G. C. Greubel, Dec 26 2019

A025579 a(1)=1, a(2)=2, a(n) = 4*3^(n-3) for n >= 3.

Original entry on oeis.org

1, 2, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924

Offset: 1

Clark Kimberling

a(n) is the sum of the numbers in row n+1 of the array defined in A025564 (and of the array in A024996).
a(n) is the number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2; |s(i) - s(i-1)| <= 1 for i >= 3.
Equals binomial transform of A095342: (1, 1, 5, 5, 17, 25, 61, ...). - Gary W. Adamson, Mar 04 2010

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A003946, A024996, A025564, A095342.

Concatenation([1,2], List([3..30], n-> 4*3^(n-3) )); # G. C. Greubel, Dec 26 2019

[1,2] cat [4*3^(n-3): n in [3..30]]; // G. C. Greubel, Dec 26 2019

seq( `if`(n<3, n, 4*3^(n-3)), n=1..30); # G. C. Greubel, Dec 26 2019

Join[{1,2},4*3^Range[0,30]] (* or *) Join[{1,2},NestList[3#&,4,30]] (* Harvey P. Dale, Jun 27 2011 *)

a(n)=max(n,4*3^(n-3)) \\ Charles R Greathouse IV, Jun 28 2011

Vec(x*(1+x)*(1-2*x)/(1-3*x) + O(x^30)) \\ Colin Barker, Oct 29 2019

[1,2]+[4*3^(n-3) for n in (3..30)] # G. C. Greubel, Dec 26 2019

a(n) = A003946(n-2), n>2. - R. J. Mathar, May 28 2008
From Colin Barker, Oct 29 2019: (Start)
G.f.: x*(1 + x)*(1 - 2*x) / (1 - 3*x).
a(n) = 3*a(n-1) for n>3. (End)

Definition corrected by R. J. Mathar, May 28 2008

A103196 a(n) = (1/9)(2^(n+3)-(-1)^n(3n-1)).

Original entry on oeis.org

1, 2, 3, 8, 13, 30, 55, 116, 225, 458, 907, 1824, 3637, 7286, 14559, 29132, 58249, 116514, 233011, 466040, 932061, 1864142, 3728263, 7456548, 14913073, 29826170, 59652315, 119304656, 238609285, 477218598, 954437167

Offset: 0

Creighton Dement, Mar 18 2005

A floretion-generated sequence relating to the Jacobsthal sequence A001045 as well as to A095342 (Number of elements in n-th string generated by a Kolakoski(5,1) rule starting with a(1)=1). (a(n)) may be seen as the result of a certain transform of the natural numbers (see program code).

Floretion Algebra Multiplication Program, FAMP Code: 4jesleftforseq[A*B] with A = + 'i + 'j + i' + j' + 'ii' + 'jj' + 'ij' + 'ji' + e and B = - .25'i + .25'j + .25'k + .25i' - .25j' + .25k' - .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' - .25'ki' - .25'kj' - .25e; 1vesforseq[A*B](n) = n, ForType: 1A.

Index entries for linear recurrences with constant coefficients, signature (0,3,2).

Cf. A001045, A095342, A053088, A034299, A052953, A026644.

Table[(2^(n+3)-(-1)^n (3n-1))/9,{n,0,30}] (* or *) LinearRecurrence[ {0,3,2},{1,2,3},40] (* Harvey P. Dale, Jul 09 2018 *)

G.f. (2x+1)/((1-2x)(x+1)^2); Superseeker results: a(n) + a(n+1) = A001045(n+3); a(n+1) - a(n) = A095342(n+1); a(n+2) - a(n+1) - a(n) = A053088(n+1) = A034299(n+1) - A034299(n); a(n) + 2a(n+1) + a(n+2) = 2^(n+3); a(n+2) - 2a(n+1) + a(n) = A053088(n+1) - A053088(n); a(n+2) - a(n) = A001045(n+4) - A001045(n+3) = A052953(n+3) - A052953(n+2) = A026644(n+2) - A026644(n+1);

a(n)=sum{k=0..n+2, (-1)^(n-k)*C(n+2, k)phi(phi(3^k))}; a(n)=sum{k=0..n+2, (-1)^(n-k)*C(n+2, k)(2*3^k/9+C(1, k)/3+4*C(0, k)/9)}; a(n)=sum{k=0..n+2, J(n-k+3)((-1)^(k+1)-2C(1, k)+4C(0, k))} where J(n)=A001045(n); a(n)=A113954(n+2). - Paul Barry, Nov 09 2005

A124389 A square array of Kolakoski string lengths, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 7, 5, 7, 1, 1, 1, 1, 9, 17, 7, 9, 1, 1, 1, 1, 17, 25, 31, 9, 11, 1, 1, 1, 1, 25, 61, 49, 49, 11, 13, 1, 1, 1, 1, 43, 109, 145, 81, 71, 13, 15, 1, 1, 1

Offset: 0

Paul Barry, Oct 30 2006

Rows of square array include A066983,A095342,A095343,A095344. See A066983 for description of Kolakoski strings. Sums of antidiagonals is A124390.

			Square array begins
1, 1, 1, 1, 1, 1, 1...
1, 1, 1, 3, 3, 7, 9...
1, 1, 1, 5, 5, 17, 25...
1, 1, 1, 7, 7, 31, 49...
1, 1, 1, 9, 9, 49, 81...
1, 1, 1, 11, 11, 71, 121...
1, 1, 1, 13, 13, 97, 169...
As a number triangle, triangle begins
1,
1, 1,
1, 1, 1,
1, 1, 1, 1,
1, 3, 1, 1, 1,
1, 3, 5, 1, 1, 1,
1, 7, 5, 7, 1, 1, 1,
1, 9, 17, 7, 9, 1, 1, 1,
1, 17, 25, 31, 9, 11, 1, 1, 1

Row k of square array has g.f. (1+x-kx^2)/((1+x)(1-x-kx^2))

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A095344 Length of n-th string generated by a Kolakoski(9,1) rule starting with a(1)=1.

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A095343 Length of n-th string generated by a Kolakoski(7,1) rule starting with a(1)=1.

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A025579 a(1)=1, a(2)=2, a(n) = 4*3^(n-3) for n >= 3.

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A103196 a(n) = (1/9)(2^(n+3)-(-1)^n(3n-1)).

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A124389 A square array of Kolakoski string lengths, read by antidiagonals.

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