A105281 -id:A105281 - cp's OEIS frontend

A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 14, 12, 4, 0, 0, 5, 30, 39, 20, 5, 0, 0, 6, 62, 120, 84, 30, 6, 0, 0, 7, 126, 363, 340, 155, 42, 7, 0, 0, 8, 254, 1092, 1364, 780, 258, 56, 8, 0, 0, 9, 510, 3279, 5460, 3905, 1554, 399, 72, 9, 0

Offset: 0

Alois P. Heinz, Aug 19 2013

A(n,k) is the total sum of lengths of longest ending contiguous subsequences with the same value over all s in {1,...,n}^k:
A(4,1) = 4 = 1+1+1+1: [1], [2], [3], [4].
A(1,4) = 4: [1,1,1,1].
A(3,2) = 12 = 2+1+1+1+2+1+1+1+2: [1,1], [1,2], [1,3], [2,1], [2,2], [2,3], [3,1], [3,2], [3,3].
A(2,3) = 14 = 3+1+1+2+2+1+1+3: [1,1,1], [1,1,2], [1,2,1], [1,2,2], [2,1,1], [2,1,2], [2,2,1], [2,2,2].

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,      0, ...
  0, 1,  2,   3,    4,     5,      6,      7, ...
  0, 2,  6,  14,   30,    62,    126,    254, ...
  0, 3, 12,  39,  120,   363,   1092,   3279, ...
  0, 4, 20,  84,  340,  1364,   5460,  21844, ...
  0, 5, 30, 155,  780,  3905,  19530,  97655, ...
  0, 6, 42, 258, 1554,  9330,  55986, 335922, ...
  0, 7, 56, 399, 2800, 19607, 137256, 960799, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000004, A001477, A002378, A027444, A027445, A152031, A228290, A228291, A228292, A228293, A228294.
Rows n=0-11 give: A000004, A001477, A000918(k+1), A029858(k+1), A080674, A104891, A105281, A104896, A052379(k-1), A052386, A105279, A105280.
Main diagonal gives A031972.
Lower diagonal gives A226238.
Cf. A228250.

A:= (n, k)-> `if`(n=1, k, (n/(n-1))*(n^k-1)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[0, 0] = 0; a[1, k_] := k; a[n_, k_] := n*(n^k-1)/(n-1); Table[a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

A(1,k) = k, else A(n,k) = n/(n-1)*(n^k-1).
A(n,k) = Sum_{i=1..k} n^i.
A(n,k) = Sum_{i=1..k+1} binomial(k+1,i)*A(n-i,k)*(-1)^(i+1) for n>k, given values A(0,k), A(1,k),..., A(k,k). - Yosu Yurramendi, Sep 03 2013

A125682 a(n) = 3*(6^n - 1)/5.

Original entry on oeis.org

3, 21, 129, 777, 4665, 27993, 167961, 1007769, 6046617, 36279705, 217678233, 1306069401, 7836416409, 47018498457, 282110990745, 1692665944473, 10155995666841, 60935974001049, 365615844006297, 2193695064037785, 13162170384226713, 78973022305360281, 473838133832161689

Offset: 1

Zerinvary Lajos, Jan 31 2007

The base-6 numbers 3_6, 33_6, 333_6, 3333_6, 33333_6, 333333_6, ... converted to base 10.

Also the total number of holes in a certain triangle fractal (start with 6 triangles, 3 holes) after n iterations. See illustration in Ngaokrajang link. - Jens Ahlström, Aug 29 2023

			Base 6        Base 10
3 ............. 3 = 3*6^0
33 ........... 21 = 3*6^1 + 3*6^0
333 ......... 129 = 3*6^2 + 3*6^1 + 3*6^0
3333 ........ 777 = 3*6^3 + 3*6^2 + 3*6^1 + 3*6^0, etc.

Bruno Berselli, Table of n, a(n) for n = 1..1000
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. A000400, A003464, A005610, A024062, A105281, A125687, A146884.

[(6^n-1)*3/5: n in [1..22]]; // Bruno Berselli, Apr 18 2012

Maple
```
seq((6^n-1)*3/5, n=1..27);
```

a[n_]:=(6^n-1)*3/5; Table[a[n],{n,1,22}] (* Robert P. P. McKone, Aug 29 2023 *)

G.f.: 3*x/((1-x)*(1-6*x)). - Bruno Berselli, Apr 18 2012
a(n) = 7*a(n-1) - 6*a(n-2). - Wesley Ivan Hurt, Dec 25 2021
From Elmo R. Oliveira, Mar 29 2025: (Start)
E.g.f.: 3*exp(x)*(exp(5*x) - 1)/5.
a(n) = 3*A003464(n). (End)

Edited by N. J. A. Sloane, Feb 02 2007

Definition rewritten (with Lajos formula) from Bruno Berselli, Apr 18 2012

A247840 a(n) = Sum_{k=2..n} 6^k.

Original entry on oeis.org

0, 36, 252, 1548, 9324, 55980, 335916, 2015532, 12093228, 72559404, 435356460, 2612138796, 15672832812, 94036996908, 564221981484, 3385331888940, 20311991333676, 121871948002092, 731231688012588, 4387390128075564, 26324340768453420

Offset: 1

Vincenzo Librandi, Sep 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. similar sequences listed in A247817.

Cf. A000400, A003464, A105281.

[0] cat [&+[6^k: k in [2..n]]: n in [2..30]];

Magma
```
[(6^(n+1)-36)/5: n in [1..30]];
```

RecurrenceTable[{a[1] == 0, a[n] == a[n-1] + 6^n}, a, {n, 30}] (* or *) CoefficientList[Series[36 x / ((1 - x) (1 - 6 x)), {x, 0, 30}], x]
Join[{0},Accumulate[6^Range[2,30]]] (* or *) LinearRecurrence[{7,-6},{0,36},30] (* Harvey P. Dale, Jun 11 2016 *)

a(n) = sum(k=2, n, 6^k); \\ Michel Marcus, Sep 25 2014

G.f.: 36*x^2/((1-x)*(1-6*x)).
a(n) = a(n-1) + 6^n = (6^(n+1) - 36)/5 = 7*a(n-1) - 6*a(n-2).
a(n) = A105281(n) - 6. - Michel Marcus, Sep 25 2014
a(n) = 36 * A003464(n-1). - Alois P. Heinz, Jan 14 2025

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A228275 A(n,k) = Sum_{i=1..k} n^i; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A125682 a(n) = 3*(6^n - 1)/5.

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A247840 a(n) = Sum_{k=2..n} 6^k.

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