A198964 -id:A198964 - cp's OEIS frontend

A191681 a(n) = (9^n - 1)/2.

0, 4, 40, 364, 3280, 29524, 265720, 2391484, 21523360, 193710244, 1743392200, 15690529804, 141214768240, 1270932914164, 11438396227480, 102945566047324, 926510094425920, 8338590849833284, 75047317648499560, 675425858836496044

Offset: 0

Adi Dani, Jun 11 2011

Number of compositions of odd numbers into n parts < 9.
These are also the junctions of the Collatz trajectories of 2^(2k-1)-1 and 2^2k-1. - David Rabahy, Nov 01 2017
a(n) gives the number of turns in the n-th iteration of the Peano curve given by plotting (A163528, A163529) or by (Siromoney 1982). - Jason V. Morgan, Oct 08 2021

			a(2)=40: there are 40 compositions of odd numbers into 2 parts < 9:
1:  (0,1),(1,0);
3:  (0,3),(3,0),(1,2),(2,1);
5:  (0,5),(5,0),(1,4),(4,1),(2,3),(3,2);
7:  (0,7),(7,0),(1,6),(6,1),(2,5),(5,2),(3,4),(4,3);
9:  (1,8),(8,1),(2,7),(7,2),(3,6),(6,3),(4,5),(5,4);
11: (3,8),(8,3),(4,7),(7,4),(5,6),(6,5);
13: (5,8),(8,5),(6,7),(7,6);
15: (7,8),(8,7).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Adi Dani, Restricted compositions of natural numbers
Rani Siromoney and K. G. Subramanian, Space-filling curves and infinite graphs, International Workshop on Graph Grammars and Their Application to Computer Science, 1982.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A002452, A096053, A125857, A138894, A198964.

[(9^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Jun 16 2011

Table[(9^n - 1)/2, {n, 0, 19}]
LinearRecurrence[{10,-9},{0,4},30] (* Harvey P. Dale, Jun 19 2011 *)

a(n)=9^n\2 \\ Charles R Greathouse IV, Oct 16 2015

a(0)=0, a(1)=4, a(n) = 10*a(n-1) - 9*a(n-2). - Harvey P. Dale, Jun 19 2011
G.f.: 4*x / ((x-1)*(9*x-1)). - Colin Barker, May 16 2013
a(n) = 2 * A125857(n+1) = 4 * A002452(n). - Bernard Schott, Oct 29 2021

Example corrected by L. Edson Jeffery, Feb 13 2015

A198965 a(n) = 7*9^n - 1.

Original entry on oeis.org

6, 62, 566, 5102, 45926, 413342, 3720086, 33480782, 301327046, 2711943422, 24407490806, 219667417262, 1977006755366, 17793060798302, 160137547184726, 1441237924662542, 12971141321962886, 116740271897665982, 1050662447078993846, 9455962023710944622, 85103658213398501606

Offset: 0

Vincenzo Librandi, Nov 01 2011

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

Cf. A198964.

Magma
```
[7*9^n-1: n in [0..20]];
```

CoefficientList[Series[2 (3 + x)/(1 - 10 x + 9 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 03 2013 *)
NestList[9#+8&,6,20] (* or *) LinearRecurrence[{10,-9},{6,62},20] (* Harvey P. Dale, Feb 26 2015 *)

a(n)=7*9^n-1 \\ Charles R Greathouse IV, Jul 05 2024

a(n) = 9*a(n-1)+8.
a(n) = 10*a(n-1)-9*a(n-2), n>1.
G.f.: 2*(3 + x)/(1 - 10*x + 9*x^2). - Vincenzo Librandi, Jan 03 2013
From Elmo R. Oliveira, Aug 16 2024: (Start)
E.g.f.: exp(x)*(7*exp(8*x) - 1).
a(n) = 2*A198964(n). (End)

A255043 a(n) = (5*9^n - 1)/2.

Original entry on oeis.org

2, 22, 202, 1822, 16402, 147622, 1328602, 11957422, 107616802, 968551222, 8716961002, 78452649022, 706073841202, 6354664570822, 57191981137402, 514727830236622, 4632550472129602, 41692954249166422, 375236588242497802, 3377129294182480222

Offset: 0

L. Edson Jeffery, Feb 13 2015

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

Cf. A096053, A138894, A191681, A198964, A198969.

Row 2 of A255044.

[(5*9^n -1)/2: n in [0..20]]; // G. C. Greubel, Feb 07 2021

Table[(5*9^n - 1)/2, {n, 0, 19}]
LinearRecurrence[{10,-9},{2,22},20] (* Harvey P. Dale, Jun 15 2018 *)

[(5*9^n -1)/2 for n in (0..20)] # G. C. Greubel, Feb 07 2021

G.f.: 2*(1+x)/((1-x)*(1-9*x)).
Recurrence: a(n) = 10*a(n-1) - 9*a(n-2), n>=2, a(0) = 2, a(1) = 22.
a(n) = 2*A138894(n).
E.g.f.: (5*exp(9*x) - exp(x))/2. - G. C. Greubel, Feb 07 2021

A255044 Array A read by upward antidiagonals: A(n,k) = ((2n+1)9^k-1)/2, n,k >= 0.

Original entry on oeis.org

0, 1, 4, 2, 13, 40, 3, 22, 121, 364, 4, 31, 202, 1093, 3280, 5, 40, 283, 1822, 9841, 29524, 6, 49, 364, 2551, 16402, 88573, 265720, 7, 58, 445, 3280, 22963, 147622, 797161, 2391484, 8, 67, 526, 4009, 29524, 206671, 1328602, 7174453, 21523360

Offset: 0

L. Edson Jeffery, Feb 13 2015

			Array begins:
.   0   4   40   364   3280   29524   265720   2391484   21523360
.   1  13  121  1093   9841   88573   797161   7174453   64570081
.   2  22  202  1822  16402  147622  1328602  11957422  107616802
.   3  31  283  2551  22963  206671  1860043  16740391  150663523
.   4  40  364  3280  29524  265720  2391484  21523360  193710244
.   5  49  445  4009  36085  324769  2922925  26306329  236756965
.   6  58  526  4738  42646  383818  3454366  31089298  279803686
.   7  67  607  5467  49207  442867  3985807  35872267  322850407
.   8  76  688  6196  55768  501916  4517248  40655236  365897128

Cf. A191681, A096053, A255043, A198964, A198969 (rows 0-3 and 5).

Cf. A138894 (1/2 of row 2).

(* Array: *)
Grid[Table[((2*n + 1)*9^k - 1)/2, {n, 0, 8}, {k, 0, 8}]]
(* Array antidiagonals flattened: *)
Flatten[Table[((2*(n - k) + 1)*9^k - 1)/2, {n, 0, 8}, {k, 0, n}]]

G.f. for row n: (n+(4-n)*x)/((1-x)(1-9*x)).

Recurrence for row n: A(n,k) = 10*A(n,k-1)-9*A(n,k-2), k >= 2, A(n,0) = n, A(n,1) = 9*n+4.

cp's OEIS Frontend

A191681 a(n) = (9^n - 1)/2.

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A198965 a(n) = 7*9^n - 1.

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A255043 a(n) = (5*9^n - 1)/2.

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A255044 Array A read by upward antidiagonals: A(n,k) = ((2*n+1)*9^k-1)/2, n,k >= 0.

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Formula

A255044 Array A read by upward antidiagonals: A(n,k) = ((2n+1)9^k-1)/2, n,k >= 0.