A025579 -id:A025579 - cp's OEIS frontend

A024996 Triangular array, read by rows: second differences in n,n direction of trinomial array A027907.

1, 1, 0, 1, 1, 0, 2, 0, 1, 1, 1, 3, 2, 3, 1, 1, 1, 2, 5, 6, 8, 6, 5, 2, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 1, 1, 4, 12, 24, 40, 52, 58, 52, 40, 24, 12, 4, 1, 1, 5, 17, 40, 76, 116, 150, 162, 150, 116, 76, 40, 17, 5, 1, 1, 6, 23, 62, 133, 232, 342, 428, 462, 428, 342, 232, 133, 62, 23, 6

Offset: 0

Clark Kimberling

For n > 2, T(n,k) is the number of integer strings s(0), ..., s(n) such that s(n) = n - k, s(0) = 0, |s(i) - s(i-1)| = 1 for i = 1,2 and <= 1 for i >= 3.

			                  1
               1  0  1
            1  0  2  0  1
         1  1  3  2  3  1  1
      1  2  5  6  8  6  5  2  1
   1  3  8 13 19 20 19 13  8  3  1

G. C. Greubel, Table of n, a(n) for n = 0..1874 [a(676) ff. corrected by _Georg Fischer_, Jun 24 2020]

First differences in n, n direction of array A025177.
Central column is essentially A024997, other columns are A024998, A026069, A026070, A026071. Row sums are in A025579.
Cf. A027907, A026552, A024072.

using Nemo
function A024996Expansion(prec)
    R, t = PolynomialRing(ZZ, "t")
    S, x = PowerSeriesRing(R, prec+1, "x")
    ser = divexact(x^2*t^3 + x^2*t + x*t - 1, x*t^2 + x*t + x - 1)
    L = zeros(ZZ, prec^2)
    for k ∈ 0:prec-1, n ∈ 0:2*k
        L[k^2+n+1] = coeff(coeff(ser, k), n)
    end
    L
end
A024996Expansion(8) |> println # Peter Luschny, Jun 25 2020

A024996 := proc(n,k)
    option remember;
    if n < 0 or k < 0 or k > 2*n then
        0 ;
    elif n <= 2 then
        if k = 2*n or k = 0 then
            1;
        elif k = 2*n-1 or k = 1 then
            0;
        elif k =2 then
            2;
        end if;
    else
        procname(n-1,k-1)+procname(n-1,k-2)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Jun 23 2013
seq(seq(A024996(n,k), k=0..2*n), n=0..11); # added by Georg Fischer, Jun 24 2020

nmax = 10; CoefficientList[CoefficientList[Series[y*x + (1 - y*x)^2/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* G. C. Greubel, May 22 2017; amended by Georg Fischer, Jun 24 2020 *)

T(n,k)=if(n<0||k<0||k>2*n,0,if(n==0,1,if(n==1,[1,0,1][k+1],if(n==2,[1,0,2,0,1][k+1],T(n-1,k-2)+T(n-1,k-1)+T(n-1,k))))) \\ Ralf Stephan, Jan 09 2004
nmax=8; for(n=0, nmax, for(k=0, 2*n, print1(T(n,k),","))) \\ added by _Georg Fischer, Jun 24 2020

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1], [1, 0, 2, 0, 1].

G.f.: y*z + (1-y*z)^2 / (1-z*(1+y+y^2)). - Ralf Stephan, Jan 09 2005 [corrected by Peter Luschny, Jun 25 2020]

Edited by Ralf Stephan, Jan 09 2004

Offset corrected by R. J. Mathar, Jun 23 2013

A073724 a(n) = (4^(n+1) + 6n + 5)/9.

Original entry on oeis.org

1, 3, 9, 31, 117, 459, 1825, 7287, 29133, 116515, 466041, 1864143, 7456549, 29826171, 119304657, 477218599, 1908874365, 7635497427, 30541989673, 122167958655, 488671834581, 1954687338283, 7818749353089, 31274997412311

Offset: 0

Wouter Meeussen, Sep 01 2002

a(n) is the number of times a disk is moved from peg 1 to peg 2 during a move of a tower of 2n or (2n-1) disks from peg 1 to peg 2 ("Tower of Hanoi" problem). Binomial transform of A025579.

An approximation to A091841.

			Moving a tower of 4 disks = 2^4 - 1 moves, coded {1,0,5,1,2,3,1,0,5,4,2,5,1,0,5}. The move from peg 1 to peg 2 has code "0" and this occurs 3 times. For 3 disks we also find 3 zeros in {0,1,3,0,4,5,0}. Hence a(2)=3. The coding corresponds to the rank of the permutation {'from peg' 1, 'to peg' 2, 'by peg' 3} or {1,2,3} with rank 0.

Vincenzo Librandi, Table of n, a(n) for n = 0..170
Hacène Belbachir and El-Mehdi Mehiri, Enumerating moves in the optimal solution of the Tower of Hanoi, arXiv:2210.08657 [math.CO], 2022.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence [pdf, ps].
Index entries for sequences related to Gijswijt's sequence
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).
Index entries for sequences related to Towers of Hanoi

Cf. A001045, A002450, A007583, A020988, A025579, A047849 (first differences), A090822, A091841.

[(4^(n+1)+6*n+5)/9: n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011

Mathematica
```
Table[(4^(n+1)+6n+5)/9, {n, 0, 24}]
```
PARI
```
a(n)=(4*4^n+6*n+5)/9
```

a(n)=polcoeff((1-3*x)/(1-4*x)/(1-x)^2+x*O(x^n),n)

G.f.: (1-3*x)/((1-4*x)*(1-x)^2).
a(n) = Sum_{k=0..n} A047849(k). - L. Edson Jeffery, May 01 2021
From Elmo R. Oliveira, Dec 11 2023: (Start)
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3) for n>2.
E.g.f.: (1/9)*(4*(exp(4*x)) + 6*x*exp(x) + 5*exp(x)). (End)

A095342 Number of elements in n-th string generated by a Kolakoski(5,1) rule starting with a(1)=1.

Original entry on oeis.org

1, 1, 5, 5, 17, 25, 61, 109, 233, 449, 917, 1813, 3649, 7273, 14573, 29117, 58265, 116497, 233029, 466021, 932081, 1864121, 3728285, 7456525, 14913097, 29826145, 59652341, 119304629, 238609313, 477218569, 954437197, 1908874333, 3817748729

Offset: 1

Benoit Cloitre, Jun 03 2004

nonn

Each string is derived from the previous string using the Kolakoski(5,1) rule and the additional condition: "string begins with 1 if previous string ends with 5 and vice versa". The strings are 1 -> 5 -> 11111 -> 51515 -> 11111511111511111 -> ... and each one contains 1,1,5,5,17,... elements.

Equals inverse binomial transform of A025579. - Gary W. Adamson, Mar 04 2010

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,2).

Cf. A000002, A066983, A095343, A095344.

Cf. A025579 . - Gary W. Adamson, Mar 04 2010

List([1..35], n-> (2^(n+2) + (-1)^n*(5-6*n))/9); # G. C. Greubel, Dec 26 2019

[(2^(n+2) + (-1)^n*(5-6*n))/9: n in [1..35]]; // G. C. Greubel, Dec 26 2019

seq( (2^(n+2) + (-1)^n*(5-6*n))/9, n=1..35); # G. C. Greubel, Dec 26 2019

Table[(2^(n+2) + (-1)^n*(5-6*n))/9, {n,35}] (* G. C. Greubel, Dec 26 2019 *)

vector(35, n, (2^(n+2) + (-1)^n*(5-6*n))/9) \\ G. C. Greubel, Dec 26 2019

[(2^(n+2) + (-1)^n*(5-6*n))/9 for n in (1..35)] # G. C. Greubel, Dec 26 2019

a(1) = a(2) = 1, a(n) = a(n-1) + 2*a(n-2) - 2*(-1)^n.
From R. J. Mathar, Apr 01 2010: (Start)
G.f.: x*(1+x+2*x^2)/((1-2*x)*(1+x)^2).
a(n) = (2^(n+2) + (-1)^n*(5-6*n))/9. (End)
E.g.f.: (exp(2*x) - 9 + (5+6*x)*exp(-x))/9. - G. C. Greubel, Dec 26 2019

A202337 Range of A062723.

Original entry on oeis.org

1, 2, 6, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924, 3389154437772, 10167463313316

Offset: 1

Reinhard Zumkeller, Dec 17 2011

nonn

Subsequence of A000792.

Apparently a(n) = A052156(n - 1) for n >= 4. - Georg Fischer, Mar 26 2019

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

Cf. A000792, A003946, A027327, A025579, A052156, A062723.

a202337 n = a202337_list !! (n-1)
a202337_list = f a062723_list where
   f (x:xs'@(x':xs)) = if x == x' then f xs' else x : f xs'

From Colin Barker, Mar 26 2019: (Start)
G.f.: x*(1 - x - 6*x^3) / (1 - 3*x).
a(n) = 4*3^(n-3) for n>3.
a(n) = 3*a(n-1) for n>4.
(End)

cp's OEIS Frontend

A024996 Triangular array, read by rows: second differences in n,n direction of trinomial array A027907.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Julia

Maple

Mathematica

PARI

Formula

Extensions

A073724 a(n) = (4^(n+1) + 6n + 5)/9.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A095342 Number of elements in n-th string generated by a Kolakoski(5,1) rule starting with a(1)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A202337 Range of A062723.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Formula