A083067 -id:A083067 - cp's OEIS frontend

A083066 5th row of number array A083064.

1, 5, 29, 173, 1037, 6221, 37325, 223949, 1343693, 8062157, 48372941, 290237645, 1741425869, 10448555213, 62691331277, 376147987661, 2256887925965, 13541327555789, 81247965334733, 487487792008397, 2924926752050381

Offset: 0

Paul Barry, Apr 21 2003

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=8, (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). - Milan Janjic, Feb 21 2010

An Engel expansion of 3/2 to the base b := 6/5 as defined in A181565, with the associated series expansion 3/2 = b + b^2/5 + b^3/(5*29) + b^4/(5*29*173) + .... Cf. A007051. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: Counts derived from transfer matrices, arXiv:1406.7788 [math.CO], 2014, eq (43).
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A083064, A083065, A083067, A007051, A007583.

[(4*6^n+1)/5: n in [0..30]]; // Vincenzo Librandi, Nov 06 2011

f[n_]:=6^n; lst={}; Do[a=f[n]; Do[a-=f[m],{m,n-1,1,-1}]; AppendTo[lst,a/6],{n,1,30}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)

a(n) = (4*6^n+1)/5.
G.f.: (1-2*x)/((1-6*x)*(1-x)).
E.g.f.: (4*exp(6*x)+exp(x))/5.
a(n) = 6*a(n-1)-1 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 7*a(n-1)-6*a(n-2). - Vincenzo Librandi, Nov 04 2011
a(n) = 6^n - Sum_{i=0..n-1} 6^i for n>0. - Bruno Berselli, Jun 20 2013

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

Original entry on oeis.org

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

A083068 7th row of number array A083064.

Original entry on oeis.org

1, 7, 55, 439, 3511, 28087, 224695, 1797559, 14380471, 115043767, 920350135, 7362801079, 58902408631, 471219269047, 3769754152375, 30158033218999, 241264265751991, 1930114126015927, 15440913008127415, 123527304065019319, 988218432520154551, 7905747460161236407

Offset: 0

Paul Barry, Apr 21 2003

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=10, (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). - Milan Janjic, Feb 21 2010

Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A083067, A083066, A125837.

f[n_]:=8^n; lst={}; Do[a=f[n]; Do[a-=f[m],{m,n-1,1,-1}]; AppendTo[lst,a/8],{n,1,30}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 10 2010 *)
LinearRecurrence[{9,-8},{1,7},20] (* Harvey P. Dale, Jul 18 2019 *)

a(n)=(6*8^n+1)/7 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (6*8^n+1)/7.
G.f. (1-2*x)/((1-8*x)(1-x)).
E.g.f. (6*exp(8*x)+exp(x))/7.
a(n) = 8*a(n-1)-1 with n>0, a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = 8^n - sum(8^i, i=0..n-1) for n>0. - Bruno Berselli, Jun 20 2013
a(n) = 1 + A125837(n+1). - Alois P. Heinz, May 20 2023

A199422 5*7^n+1.

Original entry on oeis.org

6, 36, 246, 1716, 12006, 84036, 588246, 4117716, 28824006, 201768036, 1412376246, 9886633716, 69206436006, 484445052036, 3391115364246, 23737807549716, 166164652848006, 1163152569936036, 8142067989552246, 56994475926865716

Offset: 0

Vincenzo Librandi, Nov 07 2011

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

Magma
```
[5*7^n+1: n in [0..30]];
```

5*7^Range[0,20]+1 (* or *) LinearRecurrence[{8,-7},{6,36},20] (* Harvey P. Dale, Dec 23 2012 *)

a(n) = 6*A083067(n).
a(n) = 7*a(n-1)-6.
a(n) = 8*a(n-1)-7*a(n-2).
G.f.: 6*(1-2*x)/((1-x)*(1-7*x)).

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A083066 5th row of number array A083064.

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

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A083068 7th row of number array A083064.

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A199422 5*7^n+1.

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