A052909 -id:A052909 - cp's OEIS frontend

A182950 Joint-rank array of the numbers (3i+2)3^j, where i>=0, j>=0, by antidiagonals.

1, 3, 2, 9, 7, 4, 27, 22, 12, 5, 81, 67, 36, 16, 6, 243, 202, 108, 49, 20, 8, 729, 607, 324, 148, 62, 25, 10, 2187, 1822, 972, 445, 188, 76, 30, 11, 6561, 5467, 2916, 1336, 566, 229, 90, 34, 13, 19683, 16402, 8748, 4009, 1700, 688, 270, 103, 39, 14

Offset: 1

Clark Kimberling, Dec 15 2010

Joint-rank arrays are defined in the first comment at A182801.  As for any joint-rank array, A182950 is a permutation of the positive integers, but, a fortiori, A182950 is an interspersion: after initial terms every row is interspersed with all other rows.  The numbers (3*i+2)*3^j as an array comprise A182830; and sorted, possibly A026179.
(row 1)=A000244.
(row 2)=A060816.
(row 3)=A003946.
(row 4)=A052909.
(row 5)=A027107?

			Northwest corner:
1....3....9....27...
2....7...22....67...
4...12...36...108...
5...16...49...148...

Clark Kimberling, Interspersions and Dispersions.

Cf. A182801, A182830, A026179, A182949.

 M[i_,j_]:=j+Floor[Log[3*i/2+1]/Log[3]];
 T[i_,j_]:=Sum[Floor[1/3+(3*i+2)*3^(j-k-1)],{k,0,M[i,j]}];
 TableForm[Table[T[i,j],{i,0,9},{j,0,9}]]

A198646 a(n) = 11*3^n-1.

Original entry on oeis.org

10, 32, 98, 296, 890, 2672, 8018, 24056, 72170, 216512, 649538, 1948616, 5845850, 17537552, 52612658, 157837976, 473513930, 1420541792, 4261625378, 12784876136, 38354628410, 115063885232, 345191655698, 1035574967096, 3106724901290

Offset: 0

Vincenzo Librandi, Oct 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A027107, A048473, A171498.

Magma
```
[11*3^n-1: n in [0..30]]
```

11*3^Range[0, 30] - 1 (* Wesley Ivan Hurt, Oct 02 2021 *)

a(n) = 3*a(n-1)+2, a(0)=10.
G.f.: (10-8*x) / ((3*x-1)*(x-1)). - R. J. Mathar, Oct 30 2011
a(n) = 4*a(n-1)-3*a(n-2). - Wesley Ivan Hurt, Oct 02 2021
a(n) = 2*A052909(n+1). - R. J. Mathar, Apr 07 2022

A330246 a(n) = 4^(n+1) + (4^n-1)/3.

Original entry on oeis.org

4, 17, 69, 277, 1109, 4437, 17749, 70997, 283989, 1135957, 4543829, 18175317, 72701269, 290805077, 1163220309, 4652881237, 18611524949, 74446099797, 297784399189, 1191137596757, 4764550387029, 19058201548117, 76232806192469, 304931224769877, 1219724899079509

Offset: 0

Vincenzo Librandi, Jan 09 2020

After 4, these numbers are the third column of the rectangular array in A238475.

Similar to A272743.
Cf. A000302, A002450, A052909, A178415, A237930, A238475, A347834.
Together with 1: first bisection of A136326.

Magma
```
[4^(n+1)+(4^n-1)/3: n in [0..30]];
```

Table[(4^(n + 1) + (4^n - 1) / 3), {n, 0, 30}]

G.f.: (4 - 3*x) / ((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
a(n) = 4*a(n-1) + 1 for n > 0.
a(n) = (13*4^n -1)/3, for n >= 0. - Wolfdieter Lang, Sep 16 2021
a(n) = A178415(5, n) = A347834(7, n-1), arrays, for n >= 1. - Wolfdieter Lang, Nov 29 2021

A238055 a(n) = (13*3^n-1)/2.

Original entry on oeis.org

6, 19, 58, 175, 526, 1579, 4738, 14215, 42646, 127939, 383818, 1151455, 3454366, 10363099, 31089298, 93267895, 279803686, 839411059, 2518233178, 7554699535, 22664098606, 67992295819, 203976887458, 611930662375, 1835791987126, 5507375961379, 16522127884138

Offset: 0

Philippe Deléham, Feb 17 2014

			Ternary....................Decimal
20...............................6
201.............................19
2011............................58
20111..........................175
201111.........................526
2011111.......................1579
20111111......................4738
201111111....................14215, etc.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A000244, A003462, A052909, A060816, A237930.

a(n) = 3*a(n-1) + 1, a(0)=6.
a(n) = 4*a(n-1) - 3*a(n-2), a(0)=6, a(1)=19.
a(n) = 2*A237930(n) - A003462(n).
a(n) = A052909(n+1) + A000244(n).
a(n) = A237930(n) + A000244(n+1).
a(n) = 13*A003462(n) + 6.
G.f.: (6-5*x)/((1-x)*(1-3*x)).
E.g.f.: exp(x)*(13*exp(2*x) - 1)/2. - Stefano Spezia, Aug 28 2023

A238206 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) is A007494(k) and T(n,k) = 3*T(n-1,k) + 1 for n>0.

Original entry on oeis.org

0, 2, 1, 3, 7, 4, 5, 10, 22, 13, 6, 16, 31, 67, 40, 8, 19, 49, 94, 202, 121, 9, 25, 58, 148, 283, 607, 364, 11, 28, 76, 175, 445, 850, 1822, 1093, 12, 34, 85, 229, 526, 1336, 2551, 5467, 3280, 14, 37, 103, 256, 688, 1579, 4009, 7654, 16402, 9841, 15, 43, 112, 310

Offset: 0

Philippe Deléham, Feb 20 2014

Permutation of nonnegative integers.

			Square array begins:
0, 2, 3, 5, 6, 8, 9, ...
1, 7, 10, 16, 19, 25, 28, ...
4, 22, 31, 49, 58, 76, 85, ...
13, 67, 94, 148, 175, 229, 256, ...
40, 202, 283, 445, 523, 688, 769, ...
121, 607, 850, 1336, 1579, 2065, 2308, ...
364, 1822, 2551, 4009, 4738, 6196, 6925, ...
1093, 5467, 7654, 12028, 14215, 18589, 20776, ...
3280, 16402, 22963, 36085, 42646, 55768, 62329, ...
9841, 49207, 68890, 108256, 127939, 167305, 186988, ...
...

Cf. A003462, A007494, A052909, A060816, A237930, A238055

T(n,k) = T(0,k)*3^n + T(n,0) where T(0,k) = (6*k + 1 -(-1)^k)/4 = A007494(k) and T(n,0) = (3^n - 1)/2 = A003462(n).

cp's OEIS Frontend

A182950 Joint-rank array of the numbers (3*i+2)*3^j, where i>=0, j>=0, by antidiagonals.

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A198646 a(n) = 11*3^n-1.

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A330246 a(n) = 4^(n+1) + (4^n-1)/3.

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A238055 a(n) = (13*3^n-1)/2.

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A238206 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) is A007494(k) and T(n,k) = 3*T(n-1,k) + 1 for n>0.

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A182950 Joint-rank array of the numbers (3i+2)3^j, where i>=0, j>=0, by antidiagonals.