A130750 -id:A130750 - cp's OEIS frontend

A158745 a(3n)=A130750(n). a(3n+1)=A130752(n). a(3n+2)=A130755(n).

1, 2, 3, 3, 5, 4, 8, 9, 7, 17, 16, 15, 33, 31, 32, 64, 63, 65, 127, 128, 129, 255, 257, 256, 512, 513, 511, 1025, 1024, 1023, 2049, 2047, 2048, 4096, 4095, 4097, 8191, 8192, 8193, 16383, 16385, 16384, 32768, 32769, 32767, 65537, 65536, 65535, 131073, 131071, 131072, 262144

Offset: 0

Paul Curtz, Mar 25 2009

This mixes three sequences which are identical to their third differences.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 3, 0, 0, -3, 0, 0, 2).

Cf. A138635, A140430, A130785, A140495.

LinearRecurrence[{0,0,3,0,0,-3,0,0,2},{1,2,3,3,5,4,8,9,7},60] (* Harvey P. Dale, Mar 12 2023 *)

a(3n)+a(3n+1)+a(3n+2)= A007283(n+1).
a(18n) = A130750(6n)= 2^(6n+1)-1.
a(n) = 3*a(n-3)-3*a(n-6)+2*a(n-9). G.f.: -(1+2*x+3*x^2-x^4-5*x^5+2*x^6+4*x^8)/((2*x^3-1)*(x^6-x^3+1)). - R. J. Mathar, Jan 23 2009

Edited and extended by R. J. Mathar, Apr 09 2009

A130755 Binomial transform of periodic sequence (3, 1, 2).

Original entry on oeis.org

3, 4, 7, 15, 32, 65, 129, 256, 511, 1023, 2048, 4097, 8193, 16384, 32767, 65535, 131072, 262145, 524289, 1048576, 2097151, 4194303, 8388608, 16777217, 33554433, 67108864, 134217727, 268435455, 536870912, 1073741825, 2147483649

Offset: 0

Paul Curtz, Jul 13 2007

The third sequence of "less twisted numbers"; this sequence, A130750 and A130752 form a "suite en trio" (cf. reference, p. 130).
First differences of A130752, second differences of A130750.
Sequence equals its third differences:
  3     4     7    15    32    65   129   256   511  1023
     1     3     8    17    33    64   127   255   512
        2     5     9    16    31    63   128   257
           3     4     7    15    32    65   129

P. Curtz, Exercise Book, manuscript, 1995.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A010882 (periodic (1, 2, 3)), A128834 (periodic (0, 1, 1, 0, -1, -1)), A057079 (periodic (1, 2, 1, -1, -2, -1)), A130750 (first differences), A130752 (second differences).

m:=31; S:=[ [3, 1, 2][(n-1) mod 3 +1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; // Klaus Brockhaus, Aug 03 2007

I:=[3,4,7]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018

CoefficientList[Series[(3-5*x+4*x^2)/((1-2*x)*(1-x+x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{3,-3,2}, {3,4,7}, 30] (* G. C. Greubel, Jan 15 2018 *)

{m=31; v=vector(m); v[1]=3; v[2]=4; v[3]=7; for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} \\ Klaus Brockhaus, Aug 03 2007

{for(n=0, 30, print1(2^(n+1)+[1, 0, -1, -1, 0, 1][n%6+1], ","))} \\ Klaus Brockhaus, Aug 03 2007

G.f.: (3-5*x+4*x^2)/((1-2*x)*(1-x+x^2)).
a(0) = 3; a(1) = 4; a(2) = 7; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(n) = 2^(n+1) + A128834(n+2).
a(0) = 3; for n > 0, a(n) = 2*a(n-1) + A057079(n+3).

Edited and extended by Klaus Brockhaus, Aug 03 2007

A130752 Binomial transform of periodic sequence (2, 3, 1).

Original entry on oeis.org

2, 5, 9, 16, 31, 63, 128, 257, 513, 1024, 2047, 4095, 8192, 16385, 32769, 65536, 131071, 262143, 524288, 1048577, 2097153, 4194304, 8388607, 16777215, 33554432, 67108865, 134217729, 268435456, 536870911, 1073741823, 2147483648, 4294967297, 8589934593

Offset: 0

Paul Curtz, Jul 13 2007

The second sequence of "less twisted numbers"; this sequence, A130750 and A130755 form a "suite en trio" (cf. reference, p. 130).
First differences of A130750, second differences of A130755.
Sequence equals its third differences:
2.....5.....9....16....31....63...128...257...513..1024...
...3.....4.....7....15....32....65...129...256...511...
......1.....3.....8....17....33....64...127...255...
..........2.....5.....9....16....31....63...128...

P. Curtz, Exercise Book, manuscript, 1995.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A010882, A130755 (first differences), A130750 (second differences).

m:=31; S:=[ [2, 3, 1][(n-1) mod 3 +1]: n in [1..m] ]; [ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; /* Klaus Brockhaus, Aug 03 2007 */

a[n_] := 2^(n+1) + 2*Sin[n*Pi/3]/Sqrt[3]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 13 2012 *)
LinearRecurrence[{3,-3,2},{2,5,9},40] (* Harvey P. Dale, Jun 21 2017 *)

{m=31; v=vector(m); v[1]=2; v[2]=5; v[3]=9; for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} \\ Klaus Brockhaus, Aug 03 2007

{for(n=0, 30, print1(2^(n+1)+[0, 1, 1, 0, -1, -1][n%6+1], ","))} \\ Klaus Brockhaus, Aug 03 2007

Vec((2-x) / ((1-2*x)*(1-x+x^2)) + O(x^40)) \\ Colin Barker, Jan 20 2017

G.f.: (2 - x) / ((1 - 2*x)*(1 - x + x^2)).
a(0) = 2; a(1) = 5; a(2) = 9; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(n) = 2^(n+1) + A128834(n).
a(0) = 2; for n > 0, a(n) = 2*a(n-1) + A057079(n+1).
E.g.f.: 2*(sqrt(3)*exp(2*x) + sin(sqrt(3)*x/2)*exp(x/2))/sqrt(3). - Ilya Gutkovskiy, Jun 20 2016
a(n) = 2^(n+1) + (2*sin((Pi*n)/3))/sqrt(3). - Colin Barker, Jan 20 2017

Edited and extended by Klaus Brockhaus, Aug 03 2007

A130781 Sequence is identical to its third differences: a(n+3) = 3a(n+2) - 3a(n+1) + 2*a(n), with a(0)=a(1)=1, a(2)=2.

Original entry on oeis.org

1, 1, 2, 5, 11, 22, 43, 85, 170, 341, 683, 1366, 2731, 5461, 10922, 21845, 43691, 87382, 174763, 349525, 699050, 1398101, 2796203, 5592406, 11184811, 22369621, 44739242, 89478485, 178956971, 357913942, 715827883, 1431655765, 2863311530

Offset: 0

Paul Curtz, Jul 14 2007, Jul 18 2007

The inverse binomial transform is 1,0,1,... repeated with period 3, essentially A011655. - R. J. Mathar, Aug 28 2023

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

See A130750, A130752, A130755, A129339.

Essentially a duplicate of A024493.

a[n_] := a[n] = 3 a[n - 1] - 3 a[n - 2] + 2 a[n - 3]; a[0] = a[1] = 1; a[2] = 2; Table[a@n, {n, 0, 33}] (* Or *)
CoefficientList[ Series[(1 - 2 x + 2 x^2)/(1 - 3 x + 3 x^2 - 2 x^3), {x, 0, 33}], x] (* Robert G. Wilson v, Sep 08 2007 *)
LinearRecurrence[{3,-3,2},{1,1,2},40] (* Harvey P. Dale, Sep 17 2013 *)

3*a(n) = 2^(n+1) + A087204(n+1).
Also first differences of A024494.
G.f.: (1-2x+2x^2)/(1-3x+3x^2-2x^3).
Binomial transform of [1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...]; i.e., ones in positions 2, 5, 8, 11, ... and the rest zeros. [Corrected by Gary W. Adamson, Jan 07 2008]

Edited by N. J. A. Sloane, Jul 28 2007

A130785 Sequence identical to its third differences: a(n+3) = 3a(n+2)-3a(n+1)+2a(n), with a(0)=1, a(1)=4, a(2)=9.

Original entry on oeis.org

1, 4, 9, 17, 32, 63, 127, 256, 513, 1025, 2048, 4095, 8191, 16384, 32769, 65537, 131072, 262143, 524287, 1048576, 2097153, 4194305, 8388608, 16777215, 33554431, 67108864, 134217729, 268435457, 536870912, 1073741823, 2147483647, 4294967296, 8589934593

Offset: 0

Paul Curtz, Jul 15 2007

From R. J. Mathar, Nov 22 2007: (Start)
Sequences which equal the sequence of their d-th differences obey linear recurrences with constant binomial coefficients of the form Sum_{i=0..d} binomial(d,d-i)*(-1)^i*a(n-i) = a(n-d).
If d is even, this simplifies to Sum_{i=0..d-1} binomial(d,d-i)*(-1)^i*a(n-i) = 0.
This binding of d (d odd) or d-1 (d even) consecutive terms by the recurrences leaves d or d-1, respectively, free parameters to choose a(0),a(1),...,a(d) or a(0),a(1),...,a(d-1), respectively, which ultimately define the individual sequence.
The generating functions are
d=2: a(0)/(1-2*x).
d=3: (1/3)*(-a(0) + a(1) - a(2))/(-1+2*x) + (1/3)*(-4*a(0)*x - x*a(2) + 4*a(1)*x - a(2) + 2*a(0) + a(1))/(x^2-x+1).
d=4: (1/2)*(-2*a(0) + 2*a(1) - a(2))/(-1+2*x) + (1/2)*(2*a(1)*x - 4*a(0)*x - a(2) + 2*a(1))/(1 - 2*x + 2*x^2).
In the present sequence we have d=3 and g.f. = (x-1)/(x^2-x+1) - 2/(-1+2*x). (End)
Also binomial transform of A130784. a(n) = 2^(n+1) + A010892(n+4).
Recurrence in shorter form: a(n) = 2*a(n) + periodically extended [2, 1, -1, -2, -1, 1].
See A130750, A130752, A130755 for other examples of d=3 sequences, A130781 for an example of d=4.

			Triangle of sequence and 1st, 2nd, 3rd differences:
  1   4   9  17  32  63 127 256 513
    3   5   8  15  31  64 129 257
      2   3   7  16  33  65 128
        1   4   9  17  32  63 ... equal to first row

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

d = 3; nmax = 20; a[n_ /; n < d] := (n+1)^2; seq = Table[a[n], {n, 0, nmax}]; seq /. Solve[ Thread[ Take[seq, nmax - d + 1] == Differences[seq, d]]] // First (* Jean-François Alcover, Nov 07 2013 *)
LinearRecurrence[{3, -3, 2},{1, 4, 9},21] (* Ray Chandler, Sep 23 2015 *)
Table[2^(n + 1) - Cos[(2 n + 1) Pi/6] 2/Sqrt[3], {n, 0, 32}] (* Vladimir Reshetnikov, Oct 15 2017 *)

a(n) = 2^(n+1) - cos((2*n+1)*Pi/6) * 2/sqrt(3). - Vladimir Reshetnikov, Oct 15 2017

G.f.: (1+x)/((1-2*x)*(1-x+x^2)). - Joerg Arndt, Oct 16 2017

Edited and extended by R. J. Mathar, Nov 22 2007

A140430 Period 6: repeat [3, 2, 4, 1, 2, 0].

Original entry on oeis.org

3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2, 4, 1, 2, 0, 3, 2

Offset: 0

Paul Curtz, Jun 25 2008

Associate to sequence identical to half its p-th differences.
Corresponding n-th differences table:
    3,   2,   4,   1,   2,   0,   3;
   -1,   2,  -3,   1,  -2,   3,  -1;
    3,  -5,   4,  -3,   5,  -4,   3;
   -8,   9,  -7,   8,  -9,   7,  -8;
   17, -16,  15, -17,  16, -15,  17;
  -33,  31, -32,  33, -31,  32, -33;
   64, -63,  65, -64,  63, -65,  64;
Note that the main diagonal is 3 followed by A000079(n+1).
Note also the southeast diagonal 4, 1, 5, 7, 17 is 4 followed by A014551(n+1).
Note also 3*A001045(n+1), one signed and one unsigned, in two southeast diagonals.
Starting from second line, the first column is A130750 signed.
Starting from second line, the second column is A130752 signed.
Starting from second line, the third column is A130755 signed.

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A000079 (2^n), A001045 (Jacobsthal), A014551 (Jacobsthal-Lucas).

Cf. A130750, A130752, A130755.

[2 + ((-n-2) mod 3)*(-1)^n : n in [0..100]]; // Wesley Ivan Hurt, Aug 29 2014

A140430:=n->2+((-n-2) mod 3)*(-1)^n: seq(A140430(n), n=0..100); # Wesley Ivan Hurt, Aug 29 2014

CoefficientList[Series[(3 - x + 2 x^2)/((1 - x)*(1 + x^3)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 29 2014 *)
PadRight[{},120,{3,2,4,1,2,0}] (* Harvey P. Dale, Jan 21 2023 *)

a(n)=[3,2,4,1,2,0][n%6+1] \\ Charles R Greathouse IV, Jun 02 2011

From Wesley Ivan Hurt, Aug 29 2014: (Start)
G.f.: (3-x+2*x^2)/((1-x)*(1+x^3)).
a(n) = a(n-1)-a(n-3)+a(n-4);
a(n) = 2 + ((-n-2) mod 3) * (-1)^n. (End)
a(n) = (6 + 3*cos(n*Pi) + 2*sqrt(3)*sin(n*Pi/3))/3. - Wesley Ivan Hurt, Jun 20 2016

More terms from Wesley Ivan Hurt, Aug 29 2014

A281166 a(n) = 3a(n-1) - 3a(n-2) + 2*a(n-3) for n>2, a(0)=a(1)=1, a(2)=3.

Original entry on oeis.org

1, 1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, 4096, 8191, 16383, 32768, 65537, 131073, 262144, 524287, 1048575, 2097152, 4194305, 8388609, 16777216, 33554431, 67108863, 134217728, 268435457, 536870913, 1073741824, 2147483647, 4294967295, 8589934592

Offset: 0

Paul Curtz, Jan 16 2017

a(n) is the first sequence on three (with its first and second differences):
1, 1, 3, 8, 17, 33, 64, 127, ...;
0, 2, 5, 9, 16, 31, 63, 128, ..., that is 0 followed by A130752;
2, 3, 4, 7, 15, 32, 65, 129, ..., that is 2 followed by A130755;
1, 1, 3, 8, 17, 33, 64, 127, ..., this sequence.
The main diagonal is 2^n.
The sum of the first three lines is 3*2^n.
Alternated sum and subtraction of a(n) and its inverse binomial transform (period 3: repeat [1, 0, 2]) gives the autosequence of the first kind b(n):
   0,  1,  1,  9, 17, 35, 63, 127, ...
   1,  0,  8,  8, 18, 28, 64, 126, ...
  -1,  8,  0, 10, 10, 36, 62, 134, ...
   9, -8, 10,  0, 26, 26, 72, 118, ... .
The main diagonal is 0's. The first two upper diagonals are A259713.
The sum of the first three lines gives 9*A001045.
a(n) mod 9 gives a periodic sequence of length 6: repeat [1, 1, 3, 8, 8, 6].
a(n) = A130750(n-1) for n > 2. - Georg Fischer, Oct 23 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A000079, A001045, A005010, A007283, A014551 (a diagonal), A057079, A062510 (a diagonal), A128834, A130750, A130752, A130755, A153234, A153237, A259713.

I:=[1,1,3]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Jan 15 2018

LinearRecurrence[{3, -3, 2}, {1, 1, 3}, 30] (* Jean-François Alcover, Jan 16 2017 *)

Vec((1 - 2*x + 3*x^2) / ((1 - 2*x)*(1 - x + x^2)) + O(x^40)) \\ Colin Barker, Jan 16 2017

Binomial transform of the sequence of length 3: repeat [1, 0, 2].
a(n+3) = -a(n) + 9*2^n.
a(n) = 2^n - periodic 6: repeat [0, 1, 1, 0, -1, -1, 0].
a(n+6) = a(n) + 63*2^n.
a(n+1) = 2*a(n) - period 6: repeat [1, -1, -2, -1, 1, 2].
a(n) = 2^n - 2*sin(Pi*n/3)/sqrt(3). - Jean-François Alcover and Colin Barker, Jan 16 2017
G.f.: (1 - 2*x + 3*x^2)/((1 - 2*x)*(1 - x + x^2)). - Colin Barker, Jan 16 2017

cp's OEIS Frontend

A158745 a(3n)=A130750(n). a(3n+1)=A130752(n). a(3n+2)=A130755(n).

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A130755 Binomial transform of periodic sequence (3, 1, 2).

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A130752 Binomial transform of periodic sequence (2, 3, 1).

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A130781 Sequence is identical to its third differences: a(n+3) = 3*a(n+2) - 3*a(n+1) + 2*a(n), with a(0)=a(1)=1, a(2)=2.

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A130785 Sequence identical to its third differences: a(n+3) = 3a(n+2)-3a(n+1)+2a(n), with a(0)=1, a(1)=4, a(2)=9.

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A140430 Period 6: repeat [3, 2, 4, 1, 2, 0].

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A130781 Sequence is identical to its third differences: a(n+3) = 3a(n+2) - 3a(n+1) + 2*a(n), with a(0)=a(1)=1, a(2)=2.

A281166 a(n) = 3a(n-1) - 3a(n-2) + 2*a(n-3) for n>2, a(0)=a(1)=1, a(2)=3.