A063896 -id:A063896 - cp's OEIS frontend

A100703 (T(n-1) + T(n-2)) + T(n-1)*T(n-2) where T(0)=3, T(1)=5 and n >= 2.

3, 5, 23, 143, 3455, 497663, 1719926783, 855945643032575, 1472163837099830446915583, 1260092222195718836233500990239234064383, 1855062200927301576619335433612526767115692635401046316868960255

Offset: 0

Parthasarathy Nambi, Dec 09 2004

nonn

			(3 + 5) + 3*5 = 23

Similar to A063896 but with initial terms 3 and 5.

RecurrenceTable[{a[0]==3,a[1]==5,a[n]==a[n-1]+a[n-2]+a[n-1]a[n-2]},a,{n,10}] (* Harvey P. Dale, Sep 15 2014 *)

a(n) = (a(n-1) + a(n-2)) + a(n-1)*a(n-2) with a(0)=3 and a(1)=5

More terms from Harvey P. Dale, Sep 15 2014

A100704 a(n) = (a(n-1) + a(n-2)) * a(n-1) * a(n-2) with a(0)=1 and a(1)=1.

Original entry on oeis.org

1, 1, 2, 6, 96, 58752, 331914018816, 6472527787868799987640958976, 13905078437588927055776809046479521344793888146925785597768443625472

Offset: 0

Parthasarathy Nambi, Dec 09 2004

nonn

The next term is too large to include.

Essentially the same as A066091. - R. J. Mathar, Sep 02 2008

			(1 + 1) * 1*1 = 2.
(1 + 2) * 1*2 = 6.

Cf. A063896.

a[0]:=1: a[1]:=1: for n from 2 to 9 do a[n]:= (a[n-1]+a[n-2])*a[n-1]*a[n-2] od: seq(a[n],n=0..9); # Emeric Deutsch, Dec 23 2004

More terms from Emeric Deutsch, Dec 23 2004

A117077 Define binary strings S(0)=0, S(1)=1, S(n) = S(n-2)S(n-1); a(n) = S(n) converted to decimal.

Original entry on oeis.org

0, 1, 1, 5, 13, 173, 3501, 1420717, 7343549869, 24407739551034797, 264579267653248177273154989, 15107659029337673520218077770654501397966253, 5900314832748922900613950065282124787723453785544193308390237364661677

Offset: 0

Jordan Goldstein (jboymicro20X6(AT)aim.com), Apr 18 2006

Note that S(n) in general has leading zeros.

			S(3) = 01 (base 2) = 1 (base 10) so a(3) = 1.
S(4) = 101 (base 2) = 5 (base 10) so a(4) = 5.
S(5) = 01.101 = 01101 (base 2) = 13 (base 10) so a(5) = 13.
S(6) = 101.01101 = 10101101 (base 2) = 173 (base 10) so a(6) = 173.
S(7) = 01101.10101101 = 0110110101101 (base 2) = 3501 (base 10).

Cf. A063896.

a[1] = 0; a[2] = 1; a[n_] := a[n] = If[ OddQ@n, FromDigits[ Join[ IntegerDigits[ a[n - 2], 2], IntegerDigits[ a[n - 1], 2]], 2], FromDigits[ Join[ IntegerDigits[ a[n - 2], 2], {0}, IntegerDigits[ a[n - 1], 2]], 2]]; Array[a, 13] (* Robert G. Wilson v, Apr 20 2006 *)

S(0) = 0, S(1) = 1, so S(2) = 01, a(2) = 1.

Use the substitution system 0->1 and 1->01. The values generated from a(0)=0 are 1, 01, 101, 01101, which in base 10 give the sequence. - Jon Perry, Feb 06 2011

More terms from Robert G. Wilson v, Apr 20 2006

Edited by N. J. A. Sloane, Apr 23 2006

A228763 a(n) = 2^L(n) - 1, where L(n) is the n-th Lucas number (A000032).

Original entry on oeis.org

3, 1, 7, 15, 127, 2047, 262143, 536870911, 140737488355327, 75557863725914323419135, 10633823966279326983230456482242756607, 803469022129495137770981046170581301261101496891396417650687

Offset: 0

Yeshwant Shivrai Valaulikar, Sep 03 2013

nonn

Cf. A000032, A063896.

[2^Lucas(n)-1: n in [0..15]]; // Bruno Berselli, Sep 03 2013

Table[2^LucasL[n] - 1, {n, 0, 15}] (* Bruno Berselli, Sep 03 2013 *)

a(n)=2^(fibonacci(n-1)+fibonacci(n+1))-1 \\ Charles R Greathouse IV, Sep 03 2013

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

More terms from Bruno Berselli, Sep 03 2013

A228778 a(n) = 2^Fibonacci(n) + 1.

Original entry on oeis.org

2, 3, 3, 5, 9, 33, 257, 8193, 2097153, 17179869185, 36028797018963969, 618970019642690137449562113, 22300745198530623141535718272648361505980417, 13803492693581127574869511724554050904902217944340773110325048447598593

Offset: 0

Yeshwant Shivrai Valaulikar and M. Tamba, Sep 04 2013

nonn

Cf. A000045, A000301, A063896.

a:= n-> 1 + 2^(<<0|1>, <1|1>>^n)[1,2]:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 12 2017

Table[2^Fibonacci[n] + 1, {n, 0, 13}] (* T. D. Noe, Sep 07 2013 *)

a(n+2) = a(n+1)*a(n) - a(n) - a(n+1) + 2, a(0)=2, a(1)=3.
Binet type formula: log_2(a(n)-1) = (1/sqrt(5)) * (r^n - s^n), where r and s are the roots of x^2-x-1. (this is true by definition).
a(n) = A000301(n) + 1 = A063896(n) + 2. - Alois P. Heinz, Aug 12 2017

A258948 a(1)=1, a(2)=2; for n>2, a(n) = (1/2)a(n-1)a(n-2) + a(n-1) + a(n-2).

Original entry on oeis.org

1, 2, 4, 10, 34, 214, 3886, 419902, 816293374, 171382426877950, 69949169911638289022974, 5994029248777394614754727872037912574, 209638685189029793998133268981457005889853767752082771673086

Offset: 1

Morris Neene, Jun 15 2015

a(n) + 2 = (1/2)*(a(n-1) + 2)*(a(n-2) + 2), from which the general formula can be proved using the method shown in A063896.

			a(3) = (1/2)*2*1 + 2 + 1 = 4;
a(4) = (1/2)*4*2 + 4 + 2 = 10;
a(5) = (1/2)*10*4 + 10 + 4 = 34;
a(6) = 2*(3^3)(2^2) - 2 = 214.

Cf. A000045, A063896, A100701, A254132.

[n le 2 select n else Self(n-1)*Self(n-2)/2+Self(n-1)+Self(n-2): n in [1..13]];

[2*3^Fibonacci(n-2)*2^Fibonacci(n-3)-2: n in [1..20]]; // Vincenzo Librandi, Jun 17 2015

Table[2 3^Fibonacci[n-2] 2^Fibonacci[n-3] - 2, {n, 1, 20}] (* Vincenzo Librandi, Jun 17 2015 *)

a(n) = 2*(3^fibonacci(n-2))*(2^fibonacci(n-3)) - 2; \\ Michel Marcus, Jun 17 2015

a(n) = 2 * 3^A000045(n-2) * 2^A000045(n-3) - 2, where A000045(n) is the n-th Fibonacci number.

A342689 Square array read by antidiagonals (upwards): A(n,k) = (k^Fibonacci(n) - 1) / (k - 1) for k >= 0 and n >= 0 with lim_{k -> 1} A(n,k) = A(n,1) = Fibonacci(n).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 3, 1, 1, 0, 1, 5, 7, 4, 1, 1, 0, 1, 8, 31, 13, 5, 1, 1, 0, 1, 13, 255, 121, 21, 6, 1, 1, 0, 1, 21, 8191, 3280, 341, 31, 7, 1, 1, 0, 1, 34, 2097151, 797161, 21845, 781, 43, 8, 1, 1, 0, 1, 55, 17179869184, 5230176601, 22369621, 97656, 1555, 57, 9, 1, 1, 0

Offset: 0

Werner Schulte, May 18 2021

Replacing Fibonacci(n), A000045, with Lucas(n), A000032, you get another square array B(n,k). The terms satisfy the same recurrence equation B(n,k) = (k - 1) * B(n-1,k) * B(n-2,k) + B(n-1,k) + B(n-2,k) for k >= 0 and n > 1 with initial values B(0,k) = k+1 and B(1,k) = 1. Please take account of lim_{k -> 1} (k^Lucas(n) - 1) / (k - 1) = Lucas(n).

			The array A(n,k) for k >= 0 and n >= 0 begins:
n \ k: 0  1           2          3        4     5    6    7  8  9  10  11
=========================================================================
   0 : 0  0           0          0        0     0    0    0  0  0   0   0
   1 : 1  1           1          1        1     1    1    1  1  1   1   1
   2 : 1  1           1          1        1     1    1    1  1  1   1   1
   3 : 1  2           3          4        5     6    7    8  9 10  11  12
   4 : 1  3           7         13       21    31   43   57 73 91 111 133
   5 : 1  5          31        121      341   781 1555 2801
   6 : 1  8         255       3280    21845 97656
   7 : 1 13        8191     797161 22369621
   8 : 1 21     2097151 5230176601
   9 : 1 34 17179869184
  10 : 1 55
  11 : 1 89
  etc.

Cf. A011655 (column k = -1), A057427 (column 0), A000045 (column 1), A063896 (column 2), A000004 (row 0), A000012 (rows 1, 2), A000027 (row 3), A002061 (row 4), A053699 (row 5), A053717 (row 6), A060887 (row 7).

A(n,k) = (k - 1) * A(n-1,k) * A(n-2,k) + A(n-1,k) + A(n-2,k) for k >= 0 and n > 1 with initial values A(0,k) = 0 and A(1,k) = 1.

A367430 a(1)=a(2)=a(3)=1; for n > 3, a(n) = a(n-1)a(n-2)a(n-3) + a(n-1) + a(n-2) + a(n-3).

Original entry on oeis.org

1, 1, 1, 4, 10, 55, 2269, 1250284, 156030444388, 442641893445738546589, 86351628807475712878787449679327520349, 5963928033713995675288097975918984841058670353057557384156171322944994

Offset: 1

José María Grau Ribas, Nov 18 2023

nonn

Cf. A063896 (a(n) = a(n-1)*a(n-2) + a(n-1) + a(n-2)), A000073.

a[1] = 1; a[2] = 1; a[3] = 1;
a[n_] := a[n] = a[n - 1]*a[n - 2]*a[n - 3] + a[n - 1] + a[n - 2] + a[n - 3];
Table[a[n], {n, 1, 12}]

cp's OEIS Frontend

A100703 (T(n-1) + T(n-2)) + T(n-1)*T(n-2) where T(0)=3, T(1)=5 and n >= 2.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A100704 a(n) = (a(n-1) + a(n-2)) * a(n-1) * a(n-2) with a(0)=1 and a(1)=1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

A117077 Define binary strings S(0)=0, S(1)=1, S(n) = S(n-2)S(n-1); a(n) = S(n) converted to decimal.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A228763 a(n) = 2^L(n) - 1, where L(n) is the n-th Lucas number (A000032).

Views

Author

Keywords

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A228778 a(n) = 2^Fibonacci(n) + 1.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A258948 a(1)=1, a(2)=2; for n>2, a(n) = (1/2)*a(n-1)*a(n-2) + a(n-1) + a(n-2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Formula

A342689 Square array read by antidiagonals (upwards): A(n,k) = (k^Fibonacci(n) - 1) / (k - 1) for k >= 0 and n >= 0 with lim_{k -> 1} A(n,k) = A(n,1) = Fibonacci(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A367430 a(1)=a(2)=a(3)=1; for n > 3, a(n) = a(n-1)*a(n-2)*a(n-3) + a(n-1) + a(n-2) + a(n-3).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A258948 a(1)=1, a(2)=2; for n>2, a(n) = (1/2)a(n-1)a(n-2) + a(n-1) + a(n-2).

A367430 a(1)=a(2)=a(3)=1; for n > 3, a(n) = a(n-1)a(n-2)a(n-3) + a(n-1) + a(n-2) + a(n-3).