A121263 -id:A121263 - cp's OEIS frontend

A121864 See Comments lines for definition.

16, 50, 406, 1258008, 25465014649108, 208080288305986199465852412572946560

Offset: 4

N. J. A. Sloane, Aug 31 2006, corrected Sep 05 2006

Let "N_b" denote "N read in base b" and let "N" denote "N written in base 10" (as in normal life). The sequence is given by 16, 32_16, (64_32)_16, ((128_64)_32)_16, etc., or in other words
......16....32.....64....128.......etc.
..............16.....32.....64.........
.......................16.....32.......
................................16.....
where the subscripts are evaluated from the top downwards
More precisely, "N_b" means "Take decimal expansion of N and evaluate it as if it were a base-b expansion".
The next term is too large to include.
A "dungeon" of numbers.

			(64_32)_16 = (6*32 + 4)_16 = 196_16 = 1*256 + 9*16 + 6 = 406.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 393-402.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, arXiv:math/0611293 [math.NT], 2006-2007.
David Applegate, Marc LeBrun, N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.

Cf. A121863, A121263, A121266, A121264, A121265, A121295, A121296, A111050, A121866, A122029, A122030.

A122030 See Comments lines for definition.

Original entry on oeis.org

16, 38, 200, 2324, 1189028, 4996371438596, 54444310605180563002616118404, 1333929961635427338189657222796065386478968915403224556066936061387348857093217691444

Offset: 4

N. J. A. Sloane, Aug 31 2006

Let "N_b" denote "N read in base b" and let "N" denote "N written in base 10" (as in normal life). The sequence is given by 16, 16_32, 16_(32_64), 16_(32_(64_128)), etc., or in other words
......16....16.....16.....16.......etc.
..............32.....32.....32.........
.......................64.....64.......
................................128....
where the subscripts are evaluated from the bottom upwards
More precisely, "N_b" means "Take decimal expansion of N and evaluate it as if it were a base-b expansion".
The next term is too large to include.
A "dungeon" of numbers.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 393-402.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, arXiv:math/0611293 [math.NT], 2006-2007.
David Applegate, Marc LeBrun, N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.

Cf. A121863, A121263, A121266, A121264, A121265, A121295, A121296, A111050, A121866, A121864.

A124075 a(n) = 2^(3^(4^...^n)...).

Original entry on oeis.org

2, 8, 2417851639229258349412352

Offset: 2

David Applegate and N. J. A. Sloane, Nov 08 2006

The next term is too large to include.

The next term, a(5) = 2^(3^(4^5)), has 1.124...*10^488 digits. - Amiram Eldar, Jul 13 2025

			a(4) = 2^(3^4) = 2417851639229258349412352.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 393-402.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, arXiv:math/0611293 [math.NT], 2006-2007.
David Applegate, Marc LeBrun, N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.

Cf. A014221, A049384, A121263, A121265, A121295, A121296.

a[n_] := Fold[#2^#1&, n, Range[2, n-1] // Reverse];
Table[a[n], {n, 2, 4}] (* Jean-François Alcover, Oct 10 2018 *)

A367964 Triangle of 2-parameter triangular numbers, read by rows. T(n, k) = (n(n + 1) + k(k + 1)) / 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 6, 7, 9, 12, 10, 11, 13, 16, 20, 15, 16, 18, 21, 25, 30, 21, 22, 24, 27, 31, 36, 42, 28, 29, 31, 34, 38, 43, 49, 56, 36, 37, 39, 42, 46, 51, 57, 64, 72, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 55, 56, 58, 61, 65, 70, 76, 83, 91, 100, 110

Offset: 0

Peter Luschny, Dec 07 2023

If the rows of the triangle are extended for k > n, the array A144216 is created, which is symmetrical to the main diagonal and therefore contains no new information compared to this triangle.

			Triangle T(n, k) starts:
  0 |  0;
  1 |  1,  2;
  2 |  3,  4,  6;
  3 |  6,  7,  9, 12;
  4 | 10, 11, 13, 16, 20;
  5 | 15, 16, 18, 21, 25, 30;
  6 | 21, 22, 24, 27, 31, 36, 42;
  7 | 28, 29, 31, 34, 38, 43, 49, 56;
  8 | 36, 37, 39, 42, 46, 51, 57, 64, 72;
  9 | 45, 46, 48, 51, 55, 60, 66, 73, 81,  90;
 10 | 55, 56, 58, 61, 65, 70, 76, 83, 91, 100, 110;
.
Start at row 0, column 0 with 0. Go down by adding the column index in step n. At row n, restart the counting and go n steps right by adding the row index in step n, then change direction and go down again by adding the column index. After 3*n steps on this path you are at T(2*n, n) which is 2*triangular(n) + (triangular(2*n) - triangular(n)) = (5*n^2 + 3*n)/2. These are the sliced heptagonal numbers A147875 (see the illustration of Leo Tavares).
.
The equation T(n, k) = (n*(n + 1) + k*(k + 1))/2 can be extended to all n, k in ZZ.
  [n\k] ... -6  -5  -4  -3  -2  -1   0   1   2   3   4   5  ...
  -------------------------------------------------------------
  [-5] ..., 25, 20, 16, 13, 11, 10, 10, 11, 13, 16, 20, 25, ...
  [-4] ..., 21, 16, 12,  9,  7,  6,  6,  7,  9, 12, 16, 21, ...
  [-3] ..., 18, 13,  9,  6,  4,  3,  3,  4,  6,  9, 13, 18, ...
  [-2] ..., 16, 11,  7,  4,  2,  1,  1,  2,  4,  7, 11, 16, ...
  [-1] ..., 15, 10,  6,  3,  1,  0,  0,  1,  3,  6, 10, 15, ...
  [ 0] ..., 15, 10,  6,  3,  1,  0,  0,  1,  3,  6, 10, 15, ...
  [ 1] ..., 16, 11,  7,  4,  2,  1,  1,  2,  4,  7, 11, 16, ...
  [ 2] ..., 18, 13,  9,  6,  4,  3,  3,  4,  6,  9, 13, 18, ...
  [ 3] ..., 21, 16, 12,  9,  7,  6,  6,  7,  9, 12, 16, 21, ...
  [ 4] ..., 25, 20, 16, 13, 11, 10, 10, 11, 13, 16, 20, 25, ...

Columns: A000217, A000124, A152950, A166136, A121263.
Diagonals: A002378, A000290, A002061, A059100, A027689.
Cf. A147875 (T(2*n, n)), A016061 (row sums), A367965 (alternating row sums), A143216 (the multiplicative equivalent), A144216 (extended array).

T := (n, k) -> (n*(n + 1) + k*(k + 1)) / 2:
for n from 0 to 10 do seq(T(n, k), k = 0..n) od;

Module[{n=1},NestList[Append[#+n,n*++n]&,{0},10]] (* or *)
Table[(n(n+1)+k(k+1))/2,{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 07 2023 *)

# A purely additive construction:
from functools import cache
@cache
def a_row(n: int) -> list[int]:
    if n == 0: return [0]
    row = a_row(n - 1) + [0]
    for k in range(n): row[k] += n
    row[n] = row[n - 1] + n
    return row

Recurrence: T(n, n) = n + T(n, n-1) starting with T(0, 0) = 0.
For k <> n: T(n, k) = n + T(n-1, k).
T(n, k) = t(n) + t(k), where t(n) are the triangular numbers A000217.
G.f.: (x + x*(2 - 5*x + x^2)*y + x^4*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Dec 07 2023

A122029 See Comments lines for definition.

Original entry on oeis.org

16, 38, 200, 32768, 12918916616, 1242818253229988572210659846, 1900850177472859316749829932381453683166126327573485314289555274100802310696341510

Offset: 4

N. J. A. Sloane, Aug 31 2006

Let "N_b" denote "N read in base b" and let "N" denote "N written in base 10" (as in normal life). The sequence is given by 16, 16_32, (16_32)_64, ((16_32)_64)_128, etc., or in other words
......16....16.....16.....16.......etc.
..............32.....32.....32.........
.......................64.....64.......
................................128....
where the subscripts are evaluated from the top downwards
More precisely, "N_b" means "Take decimal expansion of N and evaluate it as if it were a base-b expansion".
The next term is too large to include.
A "dungeon" of numbers.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 393-402.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, arXiv:math/0611293 [math.NT], 2006-2007.
David Applegate, Marc LeBrun, N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.

Cf. A121863, A121263, A121266, A121264, A121265, A121295, A121296, A111050, A121866, A121864.

rebase(n,bas)={ local(resul,i) ; resul= n % 10 ; i=1 ; while(n>0, n = n \10 ; resul += (n%10)*bas^i ; i++ ; ) ; return(resul) ; } { a=16 ; for(p=5,10, print(a) ; a=rebase(a,2^p) ; ) ; } \\ R. J. Mathar, Sep 01 2006

Corrected and extended by R. J. Mathar, Sep 01 2006

A121297 For definition see Comments lines.

Original entry on oeis.org

11, 14, 21, 39, 78, 211, 1954, 63163, 17163259, 316235916142, 7475840758734855197, 77068358083998565749275388634420, 56080446471298599543571746837309517827424625680076701163

Offset: 10

N. J. A. Sloane, Aug 25 2006

Using N_b to denote "N read in base b", the sequence is
......11....11.....11.....11.......etc.
..............13.....13.....13.........
.......................17.....17.......
................................19.....
where the subscripts are evaluated from the top downwards.
Analog of A121265 using primes >= 11.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 393-402.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, arXiv:math/0611293 [math.NT], 2006-2007.
David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.

Cf. A121263, A121265, A121295, A121296.

asubb := proc(a,b) local t1; t1:=convert(a,base,10); add(t1[j]*b^(j-1),j=1..nops(t1)): end;
t1:=[10]; for n from 1 to 12 do t2:=f(t1[n],ithprime(n+5)); t1:=[op(t1),t2]; od: t1;

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A121864 See Comments lines for definition.

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A122030 See Comments lines for definition.

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A124075 a(n) = 2^(3^(4^...^n)...).

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A367964 Triangle of 2-parameter triangular numbers, read by rows. T(n, k) = (n*(n + 1) + k*(k + 1)) / 2.

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A122029 See Comments lines for definition.

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A121297 For definition see Comments lines.

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A367964 Triangle of 2-parameter triangular numbers, read by rows. T(n, k) = (n(n + 1) + k(k + 1)) / 2.