A253208 -id:A253208 - cp's OEIS frontend

A242329 a(n) = 5^n + 4.

5, 9, 29, 129, 629, 3129, 15629, 78129, 390629, 1953129, 9765629, 48828129, 244140629, 1220703129, 6103515629, 30517578129, 152587890629, 762939453129, 3814697265629, 19073486328129, 95367431640629, 476837158203129, 2384185791015629, 11920928955078129

Offset: 0

Vincenzo Librandi, May 13 2014

Subsequence of A226810. - Bruno Berselli, May 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A003463, A034474, A132079, A178676, A226810, A242328, A253208 (similar sequence).

Magma
```
[5^n+4: n in [0..30]];
```

Table[5^n + 4, {n, 0, 30}]
LinearRecurrence[{6,-5},{5,9},30] (* Harvey P. Dale, Mar 15 2025 *)

G.f.: (5-21*x)/((1-x)*(1-5*x)).
a(n) = 6*a(n-1) - 5*a(n-2) for n > 1.
From Elmo R. Oliveira, Dec 06 2023: (Start)
a(n) = A000351(n)+4 = A034474(n)+3 = A242328(n)+2.
a(n) = 5*a(n-1) - 16 with a(0) = 5.
E.g.f.: exp(5*x) + 4*exp(x). (End)

A327840 Numbers m that divide 4^m + 3.

Original entry on oeis.org

1, 7, 16387, 4509253, 24265177, 42673920001, 103949349763, 12939780075073

Offset: 1

Juri-Stepan Gerasimov, Sep 27 2019

Number of solutions < 10^9 to k^n == k-1 (mod n): 1 (if k = 1), 188 (if k = 2, see A006521), 5 (if k = 3, see A015973), 5 (if k = 4, see this sequence), 5 (if k = 5), 10 (if k = 6), 10 (if k = 7), 7 (if k = 8), 5 (if k = 9), 8 (if k = 10), 11 (if k = 11), 8 (if k = 12), 9 (if k = 13), 4 (if k = 14), 3 (if k = 15), 6 (if k = 16), 7 (if k = 17), 7 (if k = 18), ...

a(9) > 10^15. - Max Alekseyev, Nov 10 2022

Solutions to k^n == 1-k (mod n): A006521 (k = 2), A015973 (k = 3), this sequence (k = 4), A123047 (k = 5), A327943 (k = 6).
Solutions to 4^n == k (mod n): A000079 (k = 0), A015950 (k = -1), A014945 (k = 1), A130421 (k = 2), this sequence (k = -3), A130422 (k = 3).
Cf. A015940, A253208.

[1] cat [n: n in [1..10^8] | Modexp(4,n,n) + 3 eq n];

Select[Range[10^7], IntegerQ[(PowerMod[4, #, # ]+3)/# ]&] (* Metin Sariyar, Sep 28 2019 *)

is(n)=Mod(4,n)^n==-3 \\ Charles R Greathouse IV, Sep 29 2019

a(6)-a(7) from Giovanni Resta, Sep 29 2019

a(8) from Max Alekseyev, Nov 10 2022

A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 8, 3, 4, 3, 0, 1, 16, 3, 16, 5, 3, 0, 1, 32, 3, 64, 9, 6, 7, 0, 1, 64, 3, 256, 17, 12, 7, 1, 0, 1, 128, 3, 1024, 33, 24, 7, 8, 3, 0, 1, 256, 3, 4096, 65, 48, 7, 64, 9, 3, 0, 1, 512, 3, 16384, 129, 96, 7, 512, 33, 10, 7

Offset: 0

Alois P. Heinz, Jan 15 2021

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0,        0, ...
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 2,  4,   8,   16,    32,     64,     128,      256, ...
  3, 3,  3,   3,    3,     3,      3,       3,        3, ...
  1, 4, 16,  64,  256,  1024,   4096,   16384,    65536, ...
  3, 5,  9,  17,   33,    65,    129,     257,      513, ...
  3, 6, 12,  24,   48,    96,    192,     384,      768, ...
  7, 7,  7,   7,    7,     7,      7,       7,        7, ...
  1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0-2, 4 give: A038573, A001477, A084471, A084473.
Rows n=0..17, 19 give: A000004, A000012, A000079, A010701, A000302, A000051(k+1), A007283, A010727, A001018, A087289, A007582(k+1), A062709(k+2), A164346, A181565(k+1), A005009, A181404(k+3), A001025, A199493, A253208(k+1).
Main diagonal gives A340667.
Cf. A000120, A023416.

A:= (n, k)-> Bits[Join](subs(0=[0$k][], Bits[Split](n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n<2, n,
     `if`(irem(n, 2, 'r')=1, A(r, k)*2+1, A(r, k)*2^k))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence @@ Table[0, {k}], 2];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 02 2021 *)

A000120(A(n,k)) = A000120(n) = log_2(A(n,0)+1).

A023416(A(n,k)) = k * A023416(n) for n >= 1.

A253211 a(n) = 8^n + 7.

Original entry on oeis.org

8, 15, 71, 519, 4103, 32775, 262151, 2097159, 16777223, 134217735, 1073741831, 8589934599, 68719476743, 549755813895, 4398046511111, 35184372088839, 281474976710663, 2251799813685255, 18014398509481991, 144115188075855879, 1152921504606846983

Offset: 0

Vincenzo Librandi, Dec 30 2014

Subsequence of A226825.

Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. similar sequences listed in A253208.

Cf. A164786, A226825.

Magma
```
[8^n+7: n in [0..30]];
```

Table[8^n + 7, {n, 0, 40}]
8^Range[0,20]+7 (* or *) LinearRecurrence[{9,-8},{8,15},30] (* Harvey P. Dale, Feb 25 2024 *)

G.f.: (8 - 57*x)/((1 - x)*(1 - 8*x)).

a(n) = 9*a(n-1) - 8*a(n-2) for n>1.

A253213 a(n) = 10^n + 9.

Original entry on oeis.org

10, 19, 109, 1009, 10009, 100009, 1000009, 10000009, 100000009, 1000000009, 10000000009, 100000000009, 1000000000009, 10000000000009, 100000000000009, 1000000000000009, 10000000000000009, 100000000000000009, 1000000000000000009, 10000000000000000009, 100000000000000000009

Offset: 0

Vincenzo Librandi, Dec 30 2014

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. similar sequences listed in A253208.

Cf. A170955.

Magma
```
[10^n+9: n in [0..30]];
```

Table[10^n + 9, {n, 0, 40}]
LinearRecurrence[{11,-10},{10,19},40] (* Harvey P. Dale, Jun 29 2018 *)

a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: (10 - 91*x)/((1 - x)*(1 - 10*x)).
E.g.f.: exp(x)*(exp(9*x) + 9). - Elmo R. Oliveira, Sep 15 2024

a(20) from Elmo R. Oliveira, Sep 15 2024

a(20) corrected by Sean A. Irvine, Sep 22 2024

A253209 a(n) = 6^n + 5.

Original entry on oeis.org

6, 11, 41, 221, 1301, 7781, 46661, 279941, 1679621, 10077701, 60466181, 362797061, 2176782341, 13060694021, 78364164101, 470184984581, 2821109907461, 16926659444741, 101559956668421, 609359740010501, 3656158440062981, 21936950640377861, 131621703842267141

Offset: 0

Vincenzo Librandi, Dec 29 2014

Subsequence of A226814.

Index entries for linear recurrences with constant coefficients, signature (7,-6).

Cf. similar sequences listed in A253208.

Cf. A164784, A226814.

Magma
```
[6^n+5: n in [0..30]];
```
Mathematica
```
Table[6^n + 5, {n, 0, 30}]
```

G.f.: (6 - 31*x) / ((1 - x)*(1 - 6*x)).

a(n) = 7*a(n-1) - 6*a(n-2) for n>1.

A253210 a(n) = 7^n + 6.

Original entry on oeis.org

7, 13, 55, 349, 2407, 16813, 117655, 823549, 5764807, 40353613, 282475255, 1977326749, 13841287207, 96889010413, 678223072855, 4747561509949, 33232930569607, 232630513987213, 1628413597910455, 11398895185373149, 79792266297612007, 558545864083284013

Offset: 0

Vincenzo Librandi, Dec 29 2014

Subsequence of A226819.

Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. similar sequences listed in A253208.

Cf. A164783, A226819.

Magma
```
[7^n+6: n in [0..30]];
```
Mathematica
```
Table[7^n + 6, {n, 0, 30}]
```

G.f.: (7 - 43*x) / ((1 - x)*(1 - 7*x)).

a(n) = 8*a(n-1) - 7*a(n-2) for n>1.

A253212 a(n) = 9^n + 8.

Original entry on oeis.org

9, 17, 89, 737, 6569, 59057, 531449, 4782977, 43046729, 387420497, 3486784409, 31381059617, 282429536489, 2541865828337, 22876792454969, 205891132094657, 1853020188851849, 16677181699666577, 150094635296999129, 1350851717672992097, 12157665459056928809

Offset: 0

Vincenzo Librandi, Dec 30 2014

Subsequence of A226832.

Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. similar sequences listed in A253208.

Cf. A177095, A226832.

Magma
```
[9^n+8: n in [0..30]];
```

Table[9^n + 8, {n, 0, 40}]
LinearRecurrence[{10,-9},{9,17},30] (* Harvey P. Dale, Jul 02 2021 *)

G.f.: (9 - 73*x)/((1 - x)*(1 - 9*x)).

a(n) = 10*a(n-1) - 9*a(n-2) for n>1.

cp's OEIS Frontend

A242329 a(n) = 5^n + 4.

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A327840 Numbers m that divide 4^m + 3.

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A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A253211 a(n) = 8^n + 7.

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A253213 a(n) = 10^n + 9.

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A253209 a(n) = 6^n + 5.

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A253210 a(n) = 7^n + 6.

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A253212 a(n) = 9^n + 8.

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