A109682 -id:A109682 - cp's OEIS frontend

A109684 Ternary equivalents of A109682.

2, 21, 220, 2222, 22211, 222210, 2222212, 22222201, 222222200, 2222222202, 22222222221, 222222222120, 2222222222122, 22222222222111, 222222222222110, 2222222222222112, 22222222222222101, 222222222222222100

Offset: 1

Philippe Deléham, Aug 08 2005

A109681 "Sloping ternary numbers": write numbers in ternary under each other (right-justified), read diagonals in upward direction, convert to decimal.

Original entry on oeis.org

0, 1, 5, 3, 4, 8, 6, 16, 11, 9, 10, 14, 12, 13, 17, 15, 25, 20, 18, 19, 23, 21, 22, 26, 51, 34, 29, 27, 28, 32, 30, 31, 35, 33, 43, 38, 36, 37, 41, 39, 40, 44, 42, 52, 47, 45, 46, 50, 48, 49, 53, 78, 61, 56, 54, 55, 59, 57, 58, 62, 60, 70, 65, 63, 64, 68, 66

Offset: 0

Philippe Deléham, Aug 08 2005

All terms are distinct, but certain terms (see A109682) are missing.

For the terms 3^k-1 (all 2's in ternary), the diagonal is not started at the leading 2, but at the leading 1 of the following term. - Georg Fischer, Mar 13 2020

			number diagonal decimal
    0      0     0
    1      1     1
    2     12     5
   10     10     3
   11     11     4
   12     22     8
   20     20     6
   21    121    16
   22    102    11
  100    100     9
  101    101    10
  102    112    14
  110    110    12
  11.    ...   ...
  1.
  .

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Georg Fischer, Perl program

Cf. A109682 (complement), A109683 (ternary version), A109684.
Cf. A102370 (base 2), A325644 (base 4), A325645 (base 5), A325692 (base 6), A325693 (base 7), A325805 (base 8), A325829 (base 9), A103205 (base 10).
Cf. A030341.

a109681 n = a109681_list !! n
a109681_list = map (foldr (\d v -> 3 * v + d) 0) $ f a030341_tabf where
   f vss = (g 0 vss) : f (tail vss)
   g k (ws:wss) = if k < length ws then ws !! k : g (k + 1) wss else []
-- Reinhard Zumkeller, Nov 19 2013

t:= (n, i)-> (d-> `if`(i=0, d, t(m, i-1)))(irem(n, 3, 'm')):
b:= (n, i)-> `if`(3^i>n, 0, t(n,i) +3*b(n+1, i+1)):
a:= n-> b(n, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 13 2020

Perl
```
Cf. link.
```

Conjectured g.f. and recurrence removed by Georg Fischer, Mar 13 2020

A215694 a(n) = 5a(n-1) - 6a(n-2) + a(n-3) with a(0)=1, a(1)=2, a(2)=7.

Original entry on oeis.org

1, 2, 7, 24, 80, 263, 859, 2797, 9094, 29547, 95968, 311652, 1011999, 3286051, 10669913, 34645258, 112492863, 365262680, 1186001480, 3850924183, 12503874715, 40599829957, 131826825678, 428039023363, 1389833992704, 4512762649020, 14652848312239, 47577499659779, 154483171074481, 501603705725970, 1628697001842743

Offset: 0

Roman Witula, Aug 21 2012

The Berndt-type sequence number 9 for the argument 2Pi/7 defined by the first trigonometric relation from section "Formula". For more connections with another sequences of trigonometric nature see comments to A215512 (a(n) is equal to the sequence b(n) in these comments) and Witula-Slota's reference (Section 3). We note that a(n)=A109682(n) for n=1,2,3,4. Moreover the following summation formula hold true: sum{k=3,..,n} a(k) = 5*a(n-1) - a(n-2) - 9, for every n=3,4,... - see comments to A215512.

The inverse binomial transform is 1,1, 4, 8, 19, 42, 95,... essentially a shifted, unsigned variant of A215112. - R. J. Mathar, Aug 22 2012

			We have 10*a(3) = 3*a(4), a(0)+a(1)+3*a(2) = a(3), a(0)+a(2)+3*a(3) = a(4), a(1)+3*a(2)+3*a(4) = a(5), and a(6) = 3*a(5)+3*a(4)-a(1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Roman Witula and Damian Slota, New Ramanujan-Type Formulas and Quasi-Fibonacci Numbers of Order 7, Journal of Integer Sequences, Vol. 10 (2007), Article 07.5.6
Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Cf. A215512, A215695.

I:=[1,2,7]; [n le 3 select I[n] else 5*Self(n-1) - 6*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 25 2018

Mathematica
```
LinearRecurrence[{5,-6,1}, {1,2,7}, 50]
```

Vec((1-3*x+3*x^2)/(1-5*x+6*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Oct 01 2012

sqrt(7)*a(n) = s(4)*c(1)^(2*n) + s(1)*c(2)^(2*n) + s(2)*c(4)^(2*n), where c(j):=2*cos(2*Pi*j/7) and s(j):=2*sin(2*Pi*j/7).
G.f.: (1-3*x+3*x^2)/(1-5*x+6*x^2-x^3).
a(n) = A005021(n)-3*A005021(n-1)+3*A005021(n-2). - R. J. Mathar, Aug 22 2012

A109683 Ternary equivalents of A109681.

Original entry on oeis.org

0, 1, 12, 10, 11, 22, 20, 121, 102, 100, 101, 112, 110, 111, 122, 120, 221, 202, 200, 201, 212, 210, 211, 222, 1220, 1021, 1002, 1000, 1001, 1012, 1010, 1011, 1022, 1020, 1121, 1102, 1100, 1101, 1112, 1110, 1111, 1122, 1120, 1221, 1202, 1200

Offset: 0

Philippe Deléham, Aug 08 2005

a(n) = A007089(A109681(n)). - Reinhard Zumkeller, Nov 19 2013

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A109681, A109682.

a109683 = a007089 . a109681  -- Reinhard Zumkeller, Nov 19 2013

cp's OEIS Frontend

A109684 Ternary equivalents of A109682.

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A109681 "Sloping ternary numbers": write numbers in ternary under each other (right-justified), read diagonals in upward direction, convert to decimal.

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A215694 a(n) = 5a(n-1) - 6a(n-2) + a(n-3) with a(0)=1, a(1)=2, a(2)=7.

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A109683 Ternary equivalents of A109681.

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cp's OEIS Frontend

A109684 Ternary equivalents of A109682.

Views

Author

Keywords

A109681 "Sloping ternary numbers": write numbers in ternary under each other (right-justified), read diagonals in upward direction, convert to decimal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Perl

Extensions

A215694 a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3) with a(0)=1, a(1)=2, a(2)=7.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A109683 Ternary equivalents of A109681.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A215694 a(n) = 5a(n-1) - 6a(n-2) + a(n-3) with a(0)=1, a(1)=2, a(2)=7.