A083068 -id:A083068 - cp's OEIS frontend

A083064 Square number array T(n,k) = (k*(k+2)^n+1)/(k+1) read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 14, 1, 1, 5, 19, 43, 41, 1, 1, 6, 29, 94, 171, 122, 1, 1, 7, 41, 173, 469, 683, 365, 1, 1, 8, 55, 286, 1037, 2344, 2731, 1094, 1, 1, 9, 71, 439, 2001, 6221, 11719, 10923, 3281, 1, 1, 10, 89, 638, 3511, 14006, 37325, 58594, 43691, 9842, 1

Offset: 0

Paul Barry, Apr 21 2003

			Rows begin:
1  1   1    1     1      1       1        1         1 ...
1  2   5   14    41    122     365     1094      3281 ...  A007051
1  3  11   43   171    683    2731    10923     43691 ...  A007583
1  4  19   94   469   2344   11719    58594    292969 ...  A083065
1  5  29  173  1037   6221   37325   223949   1343693 ...  A083066
1  6  41  286  2001  14006   98041   686286   4804001 ...  A083067
1  7  55  439  3511  28087  224695  1797559  14380471 ...  A083068
1  8  71  638  5741  51668  465011  4185098  37665881 ...  A187709
1  9  89  889  8889  88889  888889  8888889  88888889 ...  A059482
1 10 109 1198 13177 144946 1594405 17538454 192922993 ...  A199760, etc.
Column 2: A000027;
column 3: A028387;
column 4: A083074;
column 5: A125082;
column 6: A125083.
Diagonals:
1,  2,  11,   94,  1037,  14006, ... A083069;
1,  3,  19,  173,  2001,  28087, ... A083071;
1,  4,  29,  286,  3511,  51668, ... A083072;
1,  5,  41,  439,  5741,  88889, ... A083073;
1,  5,  43,  469,  6221,  98041, ... A083070;
1, 14, 171, 2344, 37325, 686286, ... A191690.
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 3, 5, 1;
1, 4, 11, 14, 1;
1, 5, 19, 43, 41, 1;
1, 6, 29, 94, 171, 122, 1; etc.

Cf. rows: A007051, A007583, A059482, A083065 - A083068, A187709, A199760; columns: A000027, A028387, A083074, A125082, A125083; diagonals: A083069 - A083073, A191690.

Edited by Bruno Berselli, Jun 21 2013

A187709 a(n) = (7*9^n + 1)/8.

Original entry on oeis.org

1, 8, 71, 638, 5741, 51668, 465011, 4185098, 37665881, 338992928, 3050936351, 27458427158, 247125844421, 2224132599788, 20017193398091, 180154740582818, 1621392665245361, 14592533987208248, 131332805884874231, 1181995252963868078, 10637957276674812701, 95741615490073314308, 861674539410659828771

Offset: 0

Sture Sjöstedt, Mar 30 2011

Case r=9 in a(n)=((r-2)*r^n+1)/(r-1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A007051, A083068, A270472 (first differences), A059482: cases r=3,8,10 in ((r-2)*r^n+1)/(r-1), respectively.

[(7*9^n+1)/8: n in [0..25]]; // Vincenzo Librandi, Mar 30 2011

(7*9^Range[0,30]+1)/8 (* or *) LinearRecurrence[{10,-9},{1,8},30] (* Harvey P. Dale, Jul 20 2012 *)

a(n)=(7*9^n+1)/8 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (7*9^n + 1)/8.
a(n) = +10*a(n-1) -9*a(n-2).
a(n) = 8*Sum_{i=0..n-1} a(i) -n + 1.
G.f.: (1-2*x)/((1-x)*(1-9*x)).
a(n) = 9^n - Sum_{i=0..n-1} 9^i for n>0. - Bruno Berselli, Jun 20 2013
E.g.f.: (7*exp(9*x) + exp(x))/8. - G. C. Greubel, Nov 06 2018

Additional formulas from Bruno Berselli

A083067 6th row of number array A083064.

Original entry on oeis.org

1, 6, 41, 286, 2001, 14006, 98041, 686286, 4804001, 33628006, 235396041, 1647772286, 11534406001, 80740842006, 565185894041, 3956301258286, 27694108808001, 193858761656006, 1357011331592041, 9499079321144286

Offset: 0

Paul Barry, Apr 21 2003

Binomial transform of A052934 - Paul Barry, Apr 30 2003

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=9, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). [From Milan Janjic, Feb 21 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A083066, A083068.

[(5*7^n+1)/6: n in [0..30]]; // Vincenzo Librandi, Nov 06 2011

f[n_]:=7^n; lst={}; Do[a=f[n]; Do[a-=f[m],{m,n-1,1,-1}]; AppendTo[lst,a/7],{n,1,30}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)

a(n) = (5*7^n+1)/6.
G.f.: (1-2*x)/((1-7*x)*(1-x)).
E.g.f.: (5*exp(7*x)+exp(x))/6.
a(n) = 7*a(n-1)-1 with a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 8*a(n-1)-7*a(n-2). - Vincenzo Librandi, Nov 06 2011
a(n) = 7^n - sum(7^i, i=0..n-1) for n>0. [Bruno Berselli, Jun 20 2013]

A125837 Numbers whose base 8 or octal representation is 6666666......6.

Original entry on oeis.org

0, 6, 54, 438, 3510, 28086, 224694, 1797558, 14380470, 115043766, 920350134, 7362801078, 58902408630, 471219269046, 3769754152374, 30158033218998, 241264265751990, 1930114126015926, 15440913008127414, 123527304065019318, 988218432520154550, 7905747460161236406

Offset: 1

Zerinvary Lajos, Feb 03 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A001018, A023001, A024088, A083068.

List([1..30], n-> 6*(8^(n-1)-1)/7); # G. C. Greubel, Aug 03 2019

[6*(8^(n-1)-1)/7: n in [1..30]]; // G. C. Greubel, Aug 03 2019

Maple
```
seq(6*(8^n-1)/7, n=0..30);
```

FromDigits[#,8]&/@Table[Table[6,{i}],{i,0,30}]  (* Harvey P. Dale, Mar 19 2011 *)
6*(8^(Range[30]-1) -1)/7 (* G. C. Greubel, Aug 03 2019 *)

vector(30, n, 6*(8^(n-1)-1)/7) \\ G. C. Greubel, Aug 03 2019

[6*(8^(n-1)-1)/7 for n in (1..30)] # G. C. Greubel, Aug 03 2019

a(n) = 6*(8^(n-1) -1)/7 = 6*A023001(n-1).
a(n) = 8*a(n-1) + 6 for n>1, a(1)=0. - Vincenzo Librandi, Oct 03 2010
G.f.: 6*x^2/( (1-x)*(1-8*x) ). - R. J. Mathar, Oct 07 2016
E.g.f.: 6*(exp(8*x) - exp(x))/7. - G. C. Greubel, Aug 03 2019
a(n) = -1 + A083068(n-1). - Alois P. Heinz, May 20 2023

A299913 a(n) = a(n-1) + 2a(n-2) if n even, or 3a(n-1) + 4*a(n-2) if n odd, starting with 0, 1.

Original entry on oeis.org

0, 1, 1, 7, 9, 55, 73, 439, 585, 3511, 4681, 28087, 37449, 224695, 299593, 1797559, 2396745, 14380471, 19173961, 115043767, 153391689, 920350135, 1227133513, 7362801079, 9817068105, 58902408631, 78536544841, 471219269047, 628292358729, 3769754152375, 5026338869833

Offset: 0

N. J. A. Sloane, Mar 10 2018

Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,8,8)

Bisections give A023001, A083068.

Cf. A299914, A299915, A299916.

a:= n-> (<<0|1|0>, <0|0|1>, <8|8|-1>>^n. <<0, 1, 1>>)[1,1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 10 2018

Fold[Append[#1, Inner[Times, 2 Boole[OddQ@ #2] + {1, 2}, {#1[[-1]], #1[[-2]]}, Plus]] &, {0, 1}, Range[2, 30]] (* or *)
CoefficientList[Series[-x (2 x + 1)/((x + 1) (8 x^2 - 1)), {x, 0, 30}], x] (* Michael De Vlieger, Mar 10 2018 *)
nxt[{n_,a_,b_}]:={n+1,b,If[OddQ[n],b+2a,3b+4a]}; NestList[nxt,{1,0,1},30][[;;,2]] (* Harvey P. Dale, Mar 02 2025 *)

concat(0, Vec(x*(1 + 2*x) / ((1 + x)*(1 - 8*x^2)) + O(x^40))) \\ Colin Barker, Mar 11 2018

G.f.: -x*(2*x+1)/((x+1)*(8*x^2-1)). - Alois P. Heinz, Mar 10 2018
From Colin Barker, Mar 11 2018: (Start)
a(n) = (2^(3*n/2) - 1) / 7 for n even.
a(n) = 3*2^((3*(n-1))/2+1)/7 + 1/7 for n odd.
a(n) = -a(n-1) + 8*a(n-2) + 8*a(n-3) for n>2.
(End)

More terms from Altug Alkan, Mar 10 2018

A363146 Triangle T(n,k) in which the n-th row encodes the inverse of a 3n X 3n Jacobi matrix, with 1's on the lower, main, and upper diagonals in GF(2), where the encoding consists of the decimal representations for the binary rows (n >= 1, 1 <= k <= 3n).

Original entry on oeis.org

3, 7, 6, 27, 59, 48, 3, 55, 54, 219, 475, 384, 27, 443, 432, 3, 439, 438, 1755, 3803, 3072, 219, 3547, 3456, 27, 3515, 3504, 3, 3511, 3510, 14043, 30427, 24576, 1755, 28379, 27648, 219, 28123, 28032, 27, 28091, 28080, 3, 28087, 28086, 112347, 243419, 196608, 14043, 227035, 221184, 1755, 224987, 224256, 219, 224731

Offset: 1

Nei Y. Soma, May 19 2023

Each term of the sequence encodes a line of the inverse of a Jacobi matrix that has 1s on its lower, main, and upper diagonals in GF(2). The associated inverse matrix column values come from the binary representation of that base-10 number, being a bit per column. These matrices have ascending and consecutive multiples of 3 sizes. If the binary number has fewer bits than the number of columns, it must be zero-padded to the left. To obtain the inverse matrices in real numbers instead of GF(2), alternate between + and - between the 1s in a row. If a row is a multiple of 3, alternate between - and +. The determinants of these 3m x 3m Jacobi matrices are 1 in GF(2), as proven by Sutner (1989), and alternate between -1 and 1 in R if m is odd or even, as proven by Melo (1987).
The recurrence, in line 3, uses the Iverson notation as presented in Graham, Knuth, and Patashnik (2002).
The proof of the correctness of that sequence of inverses is done by induction.

			For n = 1, the Jacobi 3 X 3 matrix has as rows
     1, 1, 0
     1, 1, 1
     0, 1, 1.
Its inverse has the rows
     0, 1, 1
     1, 1, 1
     1, 1, 0.
Representing these rows as decimal numbers the first three terms of the sequence are: 3, 7, and 6.
The next terms in the sequence occur for n = 2, given a sequence of six numbers. The Jacobi 6 X 6 matrix has as its rows:
      1, 1, 0, 0, 0, 0
      1, 1, 1, 0, 0, 0
      0, 1, 1, 1, 0, 0
      0, 0, 1, 1, 1, 0
      0, 0, 0, 1, 1, 1
      0, 0, 0, 0, 1, 1.
Its inverse has as rows:
      0, 1, 1, 0, 1, 1
      1, 1, 1, 0, 1, 1
      1, 1, 0, 0, 0, 0
      0, 0, 0, 0, 1, 1
      1, 1, 0, 1, 1, 1
      1, 1, 0, 1, 1, 0.
These 6 latter rows from binary to decimal give the next 6 terms of the sequence: 27, 49, 48, 3, 55, and 54.
Triangle T(n,k) begins:
     3,    7,    6;
    27,   59,   48,   3,   55,   54;
   219,  475,  384,  27,  443,  432,  3,  439,  438;
  1755, 3803, 3072, 219, 3547, 3456, 27, 3515, 3504, 3, 3511, 3510;
  ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Boston, 2nd Ed., 12th printing, 2002, pp. 24-25.
P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Boston, 1985, p. 35.
J. P. Melo, Reversibility of John von Neumann cellular automata, M.Sc. Thesis, Division of Computer Science, Instituto Tecnológico de Aeronáutica, 1997 (in Portuguese), p. 18.
K. Sutner, Linear Cellular Automata and the Garden-of-Eden, The Mathematical Intelligencer, 11(2), 1989, 49-53, p. 52.

Nei Y. Soma, Rows n = 1..30, flattened

Column k=1 gives A083713.
Column k=3 gives A083233.
T(n,3n) gives A125837(n+1).
T(n,3n-1) gives A083068.
T(n,3n-2) gives A010701.
Cf. A038184 one-dimensional cellular automaton (Rule 150) in a tape with 3n cells has as adjacency matrix the Jacobi matrices, 3n X 3n, with 1s on the lower, main and upper diagonals and the operations on it are in GF(2).

T:= n-> (M-> seq(add(abs(M[j, n*3-i])*2^i, i=0..n*3-1), j=1..n*3))
               (Matrix(n*3, (i, j)-> `if`(abs(i-j)<2, 1, 0))^(-1)):
seq(T(n), n=1..10);  # Alois P. Heinz, May 20 2023

sequence = {};
m = 6;
For[k = 1, k <= m, k++, {
  n = 3*k;
  J = ConstantArray[0, {n, n}];
  For[i = 1, i < n, i++,
   J[[i, i]] = J[[i + 1, i]] = J[[i, i + 1]] = 1];
  J[[1, 1]] = J[[n, n]] = 1;
  InvJ = Mod[Inverse[J], 2];
  For[i = 1, i <= n, i++, AppendTo[sequence, FromDigits[InvJ[[i]], 2]]]
  }
 ]
sequence

row(n)=my(m=3*n, A=lift(matrix(m, m, i, j, Mod(abs(i-j)<=1, 2))^(-1))); vector(m, i, fromdigits(A[i,], 2)) \\ Andrew Howroyd, May 20 2023

The recurrence has as its base: r(1, 1) = 3; r(2, 1) = 7 and r(3, 1) = 6;
For 2 <= k <= m, and i = 1, 2, ..., 3(k-1):
    r(i, k) = 8*r(i, k-1) + r(1,1) (i != 0 (mod 3)).
And r(3k-2, k) = r(1, 1);
    r(3k-1, k) = 8*r(3(k-1), k-1) + r(2,1);
    r(3k,   k) = 8*r(3(k-1), (k-1)) + r(3, 1).

A199554 6*8^n+1.

Original entry on oeis.org

7, 49, 385, 3073, 24577, 196609, 1572865, 12582913, 100663297, 805306369, 6442450945, 51539607553, 412316860417, 3298534883329, 26388279066625, 211106232532993, 1688849860263937, 13510798882111489, 108086391056891905

Offset: 0

Vincenzo Librandi, Nov 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Magma
```
[6*8^n+1: n in [0..30]];
```

6*8^Range[0,20]+1 (* or *) LinearRecurrence[{9,-8},{7,49},20] (* or *) NestList[8#-7&,7,20] (* Harvey P. Dale, Jul 14 2024 *)

a(n) = 7*A083068(n).
a(n) = 8*a(n-1)-7.
a(n) = 9*a(n-1)-8*a(n-2).
G.f.: 7*(1-2*x)/((1-x)*(1-8*x)).

cp's OEIS Frontend

A083064 Square number array T(n,k) = (k*(k+2)^n+1)/(k+1) read by antidiagonals.

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A187709 a(n) = (7*9^n + 1)/8.

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A083067 6th row of number array A083064.

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A125837 Numbers whose base 8 or octal representation is 6666666......6.

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GAP

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A299913 a(n) = a(n-1) + 2*a(n-2) if n even, or 3*a(n-1) + 4*a(n-2) if n odd, starting with 0, 1.

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Maple

Mathematica

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A363146 Triangle T(n,k) in which the n-th row encodes the inverse of a 3n X 3n Jacobi matrix, with 1's on the lower, main, and upper diagonals in GF(2), where the encoding consists of the decimal representations for the binary rows (n >= 1, 1 <= k <= 3n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A199554 6*8^n+1.

Views

Author

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Magma

A299913 a(n) = a(n-1) + 2a(n-2) if n even, or 3a(n-1) + 4*a(n-2) if n odd, starting with 0, 1.