A187709 -id:A187709 - 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

A199566 a(n) = (7*9^n + 1)/2.

Original entry on oeis.org

4, 32, 284, 2552, 22964, 206672, 1860044, 16740392, 150663524, 1355971712, 12203745404, 109833708632, 988503377684, 8896530399152, 80068773592364, 720618962331272, 6485570660981444, 58370135948832992, 525331223539496924, 4727981011855472312, 42551829106699250804

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 (10,-9).

Cf. A187709, A199567.

Magma
```
[(7*9^n+1)/2: n in [0..30]];
```

(7*9^Range[0,20]+1)/2 (* or *) LinearRecurrence[{10,-9},{4,32},20] (* Harvey P. Dale, Dec 08 2012 *)

a(n) = 4*A187709(n).
a(n) = 9*a(n-1) - 4 for n > 0.
a(n) = 10*a(n-1) - 9*a(n-2) for n > 1.
G.f.: 4*(1-2*x)/((1-x)*(1-9*x)).
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: exp(x)*(7*exp(8*x) + 1)/2.
a(n) = A199567(n)/2. (End)

A199567 a(n) = 7*9^n + 1.

Original entry on oeis.org

8, 64, 568, 5104, 45928, 413344, 3720088, 33480784, 301327048, 2711943424, 24407490808, 219667417264, 1977006755368, 17793060798304, 160137547184728, 1441237924662544, 12971141321962888, 116740271897665984, 1050662447078993848, 9455962023710944624, 85103658213398501608

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 (10,-9).

Cf. A096046, A187709, A199566.

Magma
```
[7*9^n+1: n in [0..30]];
```

A199567(n):=7*9^n+1$ makelist(A199567(n),n,0,30); /* Martin Ettl, Nov 13 2012 */

a(n) = 8*A187709(n).
a(n) = 9*a(n-1) - 8 for n > 0.
a(n) = 10*a(n-1) - 9*a(n-2) for n > 1.
G.f.: 8*(1-2*x)/((1-x)*(1-9*x)).
a(n) = 4*(A096046(n) + 1). - Martin Ettl, Nov 13 2012
From Elmo R. Oliveira, Aug 23 2024: (Start)
E.g.f.: exp(x)*(7*exp(8*x) + 1).
a(n) = 2*A199566(n). (End)

A270472 Expansion of g.f. (1-2x)/(1-9x).

Original entry on oeis.org

1, 7, 63, 567, 5103, 45927, 413343, 3720087, 33480783, 301327047, 2711943423, 24407490807, 219667417263, 1977006755367, 17793060798303, 160137547184727, 1441237924662543, 12971141321962887, 116740271897665983, 1050662447078993847, 9455962023710944623, 85103658213398501607

Offset: 0

Colin Barker, Mar 17 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A005032, A187709 (partial sums).

Cf. A055275: (1-x)/(1-9*x); A092810: (1-3*x)/(1-9*x).

CoefficientList[Series[(1 - 2 x)/(1 - 9 x), {x, 0, 20}], x] (* Michael De Vlieger, Mar 18 2016 *)

PARI
```
Vec((1-2*x)/(1-9*x) + O(x^30))
```

G.f.: (1-2*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 7*9^(n-1) for n>0.
a(n) = A005032(2*n-2). - R. J. Mathar, Jan 28 2025
E.g.f.: (7*exp(9*x) + 2)/9. - Elmo R. Oliveira, Mar 25 2025

A199565 a(n) = (7*9^n + 1)/4.

Original entry on oeis.org

2, 16, 142, 1276, 11482, 103336, 930022, 8370196, 75331762, 677985856, 6101872702, 54916854316, 494251688842, 4448265199576, 40034386796182, 360309481165636, 3242785330490722, 29185067974416496, 262665611769748462, 2363990505927736156, 21275914553349625402, 191483230980146628616

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 (10,-9).

Cf. A187709, A199566, A199567.

Magma
```
[(7*9^n+1)/4: n in [0..30]];
```

LinearRecurrence[{10,-9},{2,16},30] (* Harvey P. Dale, Mar 14 2015 *)

a(n) = 2*A187709(n).
a(n) = 9*a(n-1) - 2.
a(n) = 10*a(n-1) - 9*a(n-2).
G.f.: 2*(1-2*x)/((1-x)*(1-9*x)).
From Elmo R. Oliveira, Mar 02 2025: (Start)
E.g.f.: exp(x)*(7*exp(8*x) + 1)/4.
a(n) = A199566(n)/2 = A199567(n)/4. (End)

A120468 Expansion of -2x(-8-12x+9x^2) / ( (x-1)(3x-1)(3x+1)*(1+x) ).

Original entry on oeis.org

0, 16, 24, 142, 240, 1276, 2184, 11482, 19680, 103336, 177144, 930022, 1594320, 8370196, 14348904, 75331762, 129140160, 677985856, 1162261464, 6101872702, 10460353200, 54916854316, 94143178824, 494251688842, 847288609440

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jul 04 2006

Index entries for linear recurrences with constant coefficients, signature (0,10,0,-9).

LinearRecurrence[{0,10,0,-9},{0,16,24,142},30] (* Harvey P. Dale, Dec 06 2023 *)

8*a(n) = (-1)^(n+1) * (13+3^(n+2)) + 11*(3^(n+1) -1). - R. J. Mathar, Nov 07 2011

a(2n+1) = 2*A187709(n+1). - R. J. Mathar, Nov 07 2011

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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A199566 a(n) = (7*9^n + 1)/2.

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A199567 a(n) = 7*9^n + 1.

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A270472 Expansion of g.f. (1-2*x)/(1-9*x).

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PARI

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A199565 a(n) = (7*9^n + 1)/4.

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Magma

Mathematica

Formula

A120468 Expansion of -2*x*(-8-12*x+9*x^2) / ( (x-1)*(3*x-1)*(3*x+1)*(1+x) ).

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Mathematica

Formula

A270472 Expansion of g.f. (1-2x)/(1-9x).

A120468 Expansion of -2x(-8-12x+9x^2) / ( (x-1)(3x-1)(3x+1)*(1+x) ).