A055129 -id:A055129 - cp's OEIS frontend

A368296 Square array T(n,k), n >= 2, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/2).

1, 1, 1, 1, 2, 2, 1, 3, 4, 2, 1, 4, 8, 6, 3, 1, 5, 14, 18, 9, 3, 1, 6, 22, 44, 39, 12, 4, 1, 7, 32, 90, 135, 81, 16, 4, 1, 8, 44, 162, 363, 408, 166, 20, 5, 1, 9, 58, 266, 813, 1455, 1228, 336, 25, 5, 1, 10, 74, 408, 1599, 4068, 5824, 3688, 677, 30, 6

Offset: 2

Seiichi Manyama, Dec 20 2023

			Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  1,  2,   3,    4,    5,     6,     7, ...
  2,  4,   8,   14,   22,    32,    44, ...
  2,  6,  18,   44,   90,   162,   266, ...
  3,  9,  39,  135,  363,   813,  1599, ...
  3, 12,  81,  408, 1455,  4068,  9597, ...
  4, 16, 166, 1228, 5824, 20344, 57586, ...

Seiichi Manyama, Antidiagonals n = 2..141, flattened

Columns k=0..8 give A004526, A002620, A178420, A097137, A097138, A097139, A178719, A178730, A178827.

Cf. A055129, A126885, A368343.

T(n, k) = (-((n+1)\2)+sum(j=1, n, j*k^(n-j)))/(k+1);

T(n,k) = T(n-2,k) + Sum_{j=0..n-2} k^j.
T(n,k) = 1/(k+1) * (-floor((n+1)/2) + Sum_{j=1..n} j*k^(n-j)).
T(n,k) = 1/(k-1) * Sum_{j=0..n} floor(k^j/(k+1)) = Sum_{j=0..n} floor(k^j/(k^2-1)) for k > 1.
T(n,k) = (k+1)*T(n-1,k) - (k-1)*T(n-2,k) - (k+1)*T(n-3,k) + k*T(n-4,k).
G.f. of column k: x^2/((1-x) * (1-k*x) * (1-x^2)).
T(n,k) = 1/(k-1) * (floor(k^(n+1)/(k^2-1)) - floor((n+1)/2)) for k > 1.

A053717 a(n) = n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1.

Original entry on oeis.org

1, 8, 255, 3280, 21845, 97656, 335923, 960800, 2396745, 5380840, 11111111, 21435888, 39089245, 67977560, 113522235, 183063616, 286331153, 435984840, 648232975, 943531280, 1347368421, 1891142968, 2613136835, 3559590240, 4785883225, 6357828776, 8353082583

Offset: 0

Henry Bottomley, Mar 23 2000

a(n) = 11111111 in base n for n>0.

			a(3) = 3280 because 11111111 base 3 = 2187+729+243+81+27+9+3+1 = 3280.

Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

8th row of the array A055129.

Cf. A104878.

Table[FromDigits["11111111",n],{n,1,30}] (* or *) Table[n^7+n^6+n^5+n^4+n^3+n^2+n+1,{n,1,60}] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)

a(n) = (n^8-1)/(n-1) for n != 1.
G.f.: 1 -x*(x^7-8*x^6-57*x^5-1016*x^4-2297*x^3-1464*x^2-191*x-8)/(x-1)^8. - Colin Barker, Oct 29 2012
E.g.f.: exp(x)*(1 + 7*x + 120*x^2 + 423*x^3 + 426*x^4 + 156*x^5 + 22*x^6 + x^7). - Stefano Spezia, Oct 03 2024
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8). - Wesley Ivan Hurt, Jun 19 2025

a(0)=1 prepended by Alois P. Heinz, May 04 2021

A062160 Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by falling antidiagonals.

Original entry on oeis.org

0, 1, 0, -1, 1, 0, 1, 0, 1, 0, -1, 1, 1, 1, 0, 1, 0, 3, 2, 1, 0, -1, 1, 5, 7, 3, 1, 0, 1, 0, 11, 20, 13, 4, 1, 0, -1, 1, 21, 61, 51, 21, 5, 1, 0, 1, 0, 43, 182, 205, 104, 31, 6, 1, 0, -1, 1, 85, 547, 819, 521, 185, 43, 7, 1, 0, 1, 0, 171, 1640, 3277, 2604, 1111, 300, 57, 8, 1, 0, -1, 1, 341, 4921, 13107, 13021, 6665, 2101, 455, 73, 9, 1, 0

Offset: 0

Henry Bottomley, Jun 08 2001

For n >= 1, T(n, k) equals the number of walks of length k between any two distinct vertices of the complete graph K_(n+1). - Peter Bala, May 30 2024

			From _Seiichi Manyama_, Apr 12 2019: (Start)
Square array begins:
   0, 1, -1,  1,  -1,    1,    -1,      1, ...
   0, 1,  0,  1,   0,    1,     0,      1, ...
   0, 1,  1,  3,   5,   11,    21,     43, ...
   0, 1,  2,  7,  20,   61,   182,    547, ...
   0, 1,  3, 13,  51,  205,   819,   3277, ...
   0, 1,  4, 21, 104,  521,  2604,  13021, ...
   0, 1,  5, 31, 185, 1111,  6665,  39991, ...
   0, 1,  6, 43, 300, 2101, 14706, 102943, ... (End)

Seiichi Manyama, Antidiagonals n = 0..139, flattened
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

Rows include A062157, A000035, A001045, A015518, A015521, A015531, A015540, A015552, A015565, A015577, A015585, A015592, A015609.
Columns include A000004, A000012, A023443, A002061, A062158, A060884, A062159, A060888.
Related to repunits in negative bases (cf. A055129 for positive bases).
Main diagonal gives A081216.
Cf. A109502.

seq(print(seq((n^k - (-1)^k)/(n+1), k = 0..10)), n = 0..10); # Peter Bala, May 31 2024

T[n_,k_]:=(n^k - (-1)^k)/(n+1); Join[{0},Table[Reverse[Table[T[n-k,k],{k,0,n}]],{n,12}]]//Flatten (* Stefano Spezia, Feb 20 2024 *)

T(n, k) = n^(k-1) - n^(k-2) + n^(k-3) - ... + (-1)^(k-1) = n^(k-1) - T(n, k-1) = n*T(n, k-1) - (-1)^k = (n - 1)*T(n, k-1) + n*T(n, k-2) = round[n^k/(n+1)] for n > 1.
T(n, k) = (-1)^(k+1) * resultant( n*x + 1, (x^k-1)/(x-1) ). - Max Alekseyev, Sep 28 2021
G.f. of row n: x/((1+x) * (1-n*x)). - Seiichi Manyama, Apr 12 2019
E.g.f. of row n: (exp(n*x) - exp(-x))/(n+1). - Stefano Spezia, Feb 20 2024
From Peter Bala, May 31 2024: (Start)
Binomial transform of the m-th row: Sum_{k = 0..n} binomial(n, k)*T(m, k) = (m + 1)^(n-1) for n >= 1.
Let R(m, x) denote the g.f. of the m-th row of the square array. Then R(m_1, x) o R(m_2, x) = R(m_1 + m_2 + m_1*m_2, x),  where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A109502.
T(m_1 + m_2 + m_1*m_2, k) = Sum_{i = 0..k} Sum_{j = i..k} binomial(k, i)* binomial(k-i, j-i)*T(m_1, j)*T(m_2, k-i).  (End)

A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_jk^j, then T(n,k) = Sum_{j>=1} d_jj*k^(j-1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 1, 5, 0, 0, 0, 0, 0, 1, 2, 5, 0, 0, 0, 0, 0, 1, 1, 2, 12, 0, 0, 0, 0, 0, 0, 1, 1, 2, 12, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 13, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 6, 13

Offset: 0

Pontus von Brömssen, Sep 04 2020

			Array begins:
  n\k|  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16
  ---|---------------------------------------------
   0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   1 |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   2 |  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   3 |  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0
   4 |  4  1  1  0  0  0  0  0  0  0  0  0  0  0  0
   5 |  4  1  1  1  0  0  0  0  0  0  0  0  0  0  0
   6 |  5  2  1  1  1  0  0  0  0  0  0  0  0  0  0
   7 |  5  2  1  1  1  1  0  0  0  0  0  0  0  0  0
   8 | 12  2  2  1  1  1  1  0  0  0  0  0  0  0  0
   9 | 12  6  2  1  1  1  1  1  0  0  0  0  0  0  0
  10 | 13  6  2  2  1  1  1  1  1  0  0  0  0  0  0
  11 | 13  6  2  2  1  1  1  1  1  1  0  0  0  0  0
  12 | 16  7  3  2  2  1  1  1  1  1  1  0  0  0  0
  13 | 16  7  3  2  2  1  1  1  1  1  1  1  0  0  0
  14 | 17  7  3  2  2  2  1  1  1  1  1  1  1  0  0
  15 | 17  8  3  3  2  2  1  1  1  1  1  1  1  1  0
  16 | 32  8  8  3  2  2  2  1  1  1  1  1  1  1  1
64 = 2*3^3 + 1*3^2 + 0*3^1 + 1*3^0, so T(64,3) = 2*3*3^2 + 1*2*3^1 + 0*1*3^0 = 60. Alternatively, using the formula T(n,k) = floor(n/k) + k*T(floor(n/k),k), we get T(64,3) = 21 + 3*T(21,3) = 21 + 3*(7 + 3*T(7,3)) = 42 + 9*(2 + 3*T(2,3)) = 60.

Pontus von Brömssen, Antidiagonals n = 0..99, flattened
M. J. Bannister, Z. Cheng, W. E. Devanny, and D. Eppstein, Superpatterns and universal point sets, Journal of Graph Algorithms and Applications 18(2) (2014), 177-209.

Cf. A136013 (column k=2), A080277 (every second term of column k=2), A080333 (every third term of column k=3).

Cf. A007824, A055129, A286561.

import sympy
def A333979(n,k):
  d=sympy.ntheory.factor_.digits(n,k)
  return sum(j*d[-j-1]*k**(j-1) for j in range(1,len(d)-1))

# Second program (faster)
def A333979(n,k):
  return n//k+k*A333979(n//k,k) if n>=k else 0

T(n,k) = floor(n/k) + k*T(floor(n/k),k). Proof: With n = Sum_{j>=0} d_j*k^j we have floor(n/k) + k*T(floor(n/k),k) = Sum_{j>=1} (d_j*k^(j-1) + k*d_j*(j-1)*k^(j-2)) = Sum_{j>=1} d_j*j*k^(j-1) = T(n,k).
T(n,k) = T(n-1,k) + A055129(A286561(n,k),k). Proof: Let n = Sum_{j>=0} d_j*k^j and pick v so that d_j = 0 for j < v and d_v > 0 (so v = A286561(n,k)). Then n - 1 = sum_{j>=0} e_j*k^j, where e_j = k - 1 for j < v, e_v = d_v - 1, and e_j = d_j for j > v. We get T(n,k) - T(n-1,k) = Sum_{j>=1} j*(d_j-e_j)*k^(j-1) = v*k^(v-1) - (k-1)*Sum_{1<=jA055129(A286561(n,k),k).
For fixed k, T(n,k) ~ n*log(n)/(k*log(k)). (The proof for k = 2 by Bannister et al. (p. 182) can be adapted to general k.)
T(n,k) = Sum_{j>=0} k^j*floor(n/k**(j+1)).

A368343 Square array T(n,k), n >= 3, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/3).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 7, 5, 2, 1, 5, 13, 16, 7, 2, 1, 6, 21, 41, 34, 9, 3, 1, 7, 31, 86, 125, 70, 12, 3, 1, 8, 43, 157, 346, 377, 143, 15, 3, 1, 9, 57, 260, 787, 1386, 1134, 289, 18, 4, 1, 10, 73, 401, 1562, 3937, 5547, 3405, 581, 22, 4

Offset: 3

Seiichi Manyama, Dec 22 2023

			Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  1,  2,   3,    4,    5,     6,     7, ...
  1,  3,   7,   13,   21,    31,    43, ...
  2,  5,  16,   41,   86,   157,   260, ...
  2,  7,  34,  125,  346,   787,  1562, ...
  2,  9,  70,  377, 1386,  3937,  9374, ...
  3, 12, 143, 1134, 5547, 19688, 56247, ...

Seiichi Manyama, Antidiagonals n = 3..142, flattened

Columns k=0..4 give A002264, A130518, A178455, A368344, A368345.

Cf. A055129, A368296.

PARI
```
T(n, k) = sum(j=0, n, k^(n-j)*(j\3));
```

T(n,k) = T(n-3,k) + Sum_{j=0..n-3} k^j.
T(n,k) = 1/(k-1) * Sum_{j=0..n} floor(k^j/(k^2+k+1)) = Sum_{j=0..n} floor(k^j/(k^3-1)) for k > 1.
T(n,k) = (k+1)*T(n-1,k) - k*T(n-2,k) + T(n-3,k) - (k+1)*T(n-4,k) + k*T(n-5,k).
G.f. of column k: x^3/((1-x) * (1-k*x) * (1-x^3)).
T(n,k) = 1/(k-1) * (floor(k^(n+1)/(k^3-1)) - floor((n+1)/3)) for k > 1.

A125598 a(n) = ((n+1)^(n-1) - 1)/n.

Original entry on oeis.org

0, 1, 5, 31, 259, 2801, 37449, 597871, 11111111, 235794769, 5628851293, 149346699503, 4361070182715, 139013933454241, 4803839602528529, 178901440719363487, 7143501829211426575, 304465936543600121441

Offset: 1

Alexander Adamchuk, Nov 26 2006

nonn

Odd prime p divides a(p-2).
a(n) is prime for n = {3,4,6,74, ...}; prime terms are {5, 31, 2801, ...}.
a(n) is the (n-1)-th generalized repunit in base (n+1). For example, a(5) = 259 which is 1111 in base 6. - Mathew Englander, Oct 20 2020

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A000272 (n^(n-2)), A125599.

Cf. other sequences of generalized repunits, such as A125118, A053696, A055129, A060072, A031973, A173468, A023037, A119598, A085104, and A162861.

[((n+1)^(n-1) -1)/n: n in [1..25]]; // G. C. Greubel, Aug 15 2022

Mathematica
```
Table[((n+1)^(n-1)-1)/n, {n,25}]
```

[gaussian_binomial(n,1,n+2) for n in range(0,18)] # Zerinvary Lajos, May 31 2009

a(n) = ((n+1)^(n-1) - 1)/n.
a(n) = (A000272(n+1) - 1)/n.
a(2k-1)/(2k+1) = A125599(k) for k>0.
From Mathew Englander, Dec 17 2020: (Start)
a(n) = (A060072(n+1) - A083069(n-1))/2.
For n > 1, a(n) = Sum_{k=0..n-2} (n+1)^k.
For n > 1, a(n) = Sum_{j=0..n-2} n^j*C(n-1,j+1). (End)

A067763 Square array read by antidiagonals of base n numbers written as 122...222 with k 2's (and a suitable interpretation for n=0, 1 or 2).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 7, 10, 5, 1, 2, 9, 22, 17, 6, 1, 2, 11, 46, 53, 26, 7, 1, 2, 13, 94, 161, 106, 37, 8, 1, 2, 15, 190, 485, 426, 187, 50, 9, 1, 2, 17, 382, 1457, 1706, 937, 302, 65, 10, 1, 2, 19, 766, 4373, 6826, 4687, 1814, 457, 82, 11, 1, 2, 21, 1534, 13121

Offset: 0

Henry Bottomley, Feb 06 2002

Start with a node; step one is to connect that node to n+1 new nodes so that it is of degree n+1; further steps are to connect each existing node of degree 1 to n new nodes so that it is of degree n+1; T(n,k) is the total number of nodes after k steps.

			Rows start: 1,2,2,2,2,2,...; 1,3,5,7,9,11,...; 1,4,10,22,46,94,...; 1,5,17,53,161,485,... T(3,2) =122 base 3 =17.

Rows include A040000, A005408, A033484, A048473, A020989, A057651, A061801 etc. For negative n (not shown) absolute values of rows would effectively include A000012, A014113, A046717.

T(n, k) =((n+1)*n^k-2)/(n-1) [with T(1, k)=2k+1] =n*T(n, k-1)+2 =(n+1)*T(n, k-1)-n*T(n, k-2) =T(n, k-1)+(1+1/n)*n^k =A055129(k, n)+A055129(k-1, n). Coefficient of x^k in expansion of (1+x)/((1-x)(1-nx)).

A122873 Triangle T(n,k) = smallest number whose square has largest digit k in base n, 0<=k

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 0, 1, 3, 7, 0, 1, 6, 4, 2, 0, 1, 7, 3, 2, 14, 0, 1, 3, 11, 2, 6, 12, 0, 1, 4, 5, 2, 13, 7, 11, 0, 1, 10, 91, 2, 7, 28, 4, 20, 0, 1, 11, 111, 2, 5, 4, 24, 9, 3, 0, 1, 12, 5, 2, 4, 26, 9, 24, 3, 19, 0, 1, 5, 6, 2, 8, 19, 32, 10, 3, 11, 40, 0, 1, 14, 4, 2, 24, 9, 35, 28, 3, 6, 12, 5

Offset: 1

Franklin T. Adams-Watters, Oct 23 2006

Repunits in base n are an upper bound, since 11...11^2 = 12...(k)...21 in any sufficiently large base.

			Row 10: 0,1,11,111,2,5,4,24,9,3, has squares 0,1,121,12321,4,25,16,576,81,9, with digits 0,...,9 as the respective largest digits.

Cf. A055129, A000290, A054055.

cp's OEIS Frontend

A368296 Square array T(n,k), n >= 2, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A053717 a(n) = n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A062160 Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by falling antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_j*k^j, then T(n,k) = Sum_{j>=1} d_j*j*k^(j-1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Python

Formula

A368343 Square array T(n,k), n >= 3, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A125598 a(n) = ((n+1)^(n-1) - 1)/n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A067763 Square array read by antidiagonals of base n numbers written as 122...222 with k 2's (and a suitable interpretation for n=0, 1 or 2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A122873 Triangle T(n,k) = smallest number whose square has largest digit k in base n, 0<=k

Views

Author

Keywords

A333979 Array read by antidiagonals, n >= 0, k >= 2: T(n,k) is the "digital derivative" of n in base k; if the base k representation of n is Sum_{j>=0} d_jk^j, then T(n,k) = Sum_{j>=1} d_jj*k^(j-1).