A014830 -id:A014830 - cp's OEIS frontend

A126885 T(n,k) = n*T(n,k-1) + k, with T(n,1) = 1, square array read by ascending antidiagonals (n >= 0, k >= 1).

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 11, 10, 5, 1, 6, 18, 26, 15, 6, 1, 7, 27, 58, 57, 21, 7, 1, 8, 38, 112, 179, 120, 28, 8, 1, 9, 51, 194, 453, 543, 247, 36, 9, 1, 10, 66, 310, 975, 1818, 1636, 502, 45, 10, 1, 11, 83, 466, 1865, 4881, 7279, 4916, 1013, 55, 11

Offset: 0

Gary W. Adamson, Dec 30 2006

			Square array begins:
  n\k | 1   2   3   4    5     6      7       8 ...
  -------------------------------------------------
    0 | 1   2   3   4    5     6      7       8 ... A000027
    1 | 1   3   6  10   15    21     28      36 ... A000217
    2 | 1   4  11  26   57   120    247     502 ... A000295
    3 | 1   5  18  58  179   543   1636    4916 ... A000340
    4 | 1   6  27 112  453  1818   7279   29124 ... A014825
    5 | 1   7  38 194  975  4881  24412  122068 ... A014827
    6 | 1   8  51 310 1865 11196  67183  403106 ... A014829
    7 | 1   9  66 466 3267 22875 160132 1120932 ... A014830
    8 | 1  10  83 668 5349 42798 342391 2739136 ... A014831
    ...

Antidiagonal sums are A134195.
Main diagonal gives A062805.
Cf. A000027, A000217, A000295, A000340, A014825, A014827, A014829, A014830, A014831.

T(n, k) := if k = 1 then 1 else n*T(n, k - 1) + k$
create_list(T(n - k + 1, k), n, 0, 20, k, 1, n + 1);
/* Franck Maminirina Ramaharo, Jan 26 2019 */

T(1,k) = k*(k + 1)/2, and T(n,k) = (k - (k + 1)*n + n^(k + 1))/(n^2 - 2*n + 1) elsewhere.
T(n,k) = third entry in the vector M^k * (1, 0, 0), where M is the following 3 X 3 matrix:
  1, 0, 0
  1, 1, 0
  1, 1, n.

Edited and name clarified by Franck Maminirina Ramaharo, Jan 26 2019

A048439 Take the first n numbers written in base 7, concatenate them, then convert from base 7 to base 10.

Original entry on oeis.org

1, 9, 66, 466, 3267, 22875, 1120882, 54923226, 2691238083, 131870666077, 6461662637784, 316621469251428, 15514451993319985, 760208147672679279, 37250199235961284686, 1825259762562102949630, 89437728365543044531887, 4382448689911609182062481

Offset: 1

Patrick De Geest, May 15 1999

The first two primes in this sequence occur for n = 10 (a(10) = 131870666077) and n = 37 (a(37) = 569432644200356239518976257368822195317881440478377541397) (email from Kurt Foster, Oct 24 2015). What is the next prime? - N. J. A. Sloane, Oct 25 2015

After a(37), there are no more primes through a(4000) = 2.2670...*10^14538. - Jon E. Schoenfield, Jan 19 2018

			a(8): (1)(2)(3)(4)(5)(6)(10)(11) = 1234561011_7 = 54923226.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014830.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: this sequence, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*7^(1+Ilog(7, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

a[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 7], 7]; Array[a, 20] (* Vincenzo Librandi, Dec 30 2012 *)

A353098 a(1) = 6; for n>1, a(n) = 7 * a(n-1) + 7 - n.

Original entry on oeis.org

6, 47, 333, 2334, 16340, 114381, 800667, 5604668, 39232674, 274628715, 1922401001, 13456807002, 94197649008, 659383543049, 4615684801335, 32309793609336, 226168555265342, 1583179886857383, 11082259208001669, 77575814456011670, 543030701192081676

Offset: 1

Seiichi Manyama, Apr 23 2022

Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Cf. A064617, A353094, A353095, A353096, A353097, A353099, A353100.

Cf. A014830.

LinearRecurrence[{9, -15, 7}, {6, 47, 333}, 21] (* Amiram Eldar, Apr 23 2022 *)

my(N=30, x='x+O('x^N)); Vec(x*(6-7*x)/((1-x)^2*(1-7*x)))

PARI
```
a(n) = (5*7^(n+1)+6*n-35)/36;
```

b(n, k) = sum(j=0, n-1, (k-n+j)*k^j);
a(n) = b(n, 7);

G.f.: x * (6 - 7 * x)/((1 - x)^2 * (1 - 7 * x)).
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).
a(n) = 5 * A014830(n) + n.
a(n) = (5*7^(n+1) + 6*n - 35)/36.
a(n) = Sum_{k=0..n-1} (7 - n + k)*7^k.
E.g.f.: exp(x)*(35*(exp(6*x) - 1) + 6*x)/36. - Stefano Spezia, May 29 2023

A014854 Numbers k that divide s(k), where s(1)=1, s(j)=7*s(j-1)+j.

Original entry on oeis.org

1, 3, 7, 9, 21, 27, 57, 63, 81, 171, 189, 203, 243, 333, 399, 513, 567, 609, 729, 999, 1083, 1197, 1539, 1701, 1827, 2187, 2331, 2943, 2997, 3249, 3591, 4617, 5103, 5481, 5887, 6327, 6561, 6993, 7581, 8829, 8991, 9567, 9747, 10773, 11571, 11977

Offset: 1

N. J. A. Sloane, Olivier Gérard

nonn

s(n) = A014830(n).

cp's OEIS Frontend

A126885 T(n,k) = n*T(n,k-1) + k, with T(n,1) = 1, square array read by ascending antidiagonals (n >= 0, k >= 1).

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A048439 Take the first n numbers written in base 7, concatenate them, then convert from base 7 to base 10.

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A353098 a(1) = 6; for n>1, a(n) = 7 * a(n-1) + 7 - n.

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A014854 Numbers k that divide s(k), where s(1)=1, s(j)=7*s(j-1)+j.

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