A080674 -id:A080674 - cp's OEIS frontend

A104891 a(0) = 0; a(n) = 5*a(n-1) + 5.

0, 5, 30, 155, 780, 3905, 19530, 97655, 488280, 2441405, 12207030, 61035155, 305175780, 1525878905, 7629394530, 38146972655, 190734863280, 953674316405, 4768371582030, 23841857910155, 119209289550780, 596046447753905, 2980232238769530, 14901161193847655

Offset: 0

Alexandre Wajnberg, Apr 24 2005

Number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3 and 4 as a digit.

Number of monic irreducible polynomials of degree 1 in GF(5)[x1,...,xn]. - Max Alekseyev, Jan 23 2006

			a(3) = 5*a(2) + 5 = 5*30 + 5 = 155.

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

Cf. A000225, A000918, A029858, A052379, A052386, A080674.

Row n=5 of A228275.

[5*(5^n -1)/4: n in [0..30]]; // G. C. Greubel, Jun 15 2021

a:=n->add(5^j,j=1..n): seq(a(n),n=0..30); # Zerinvary Lajos, Jun 27 2007

RecurrenceTable[{a[n]==5*a[n-1]+5, a[0]==0}, a, {n, 0, 30}] (* Vaclav Kotesovec, Jul 25 2014 *)
NestList[5#+5&,0,30] (* Harvey P. Dale, Oct 04 2019 *)

concat(0, Vec(5*x/((x-1)*(5*x-1)) + O(x^30))) \\ Colin Barker, Jul 25 2014

[5*(5^n -1)/4 for n in (0..30)] # G. C. Greubel, Jun 15 2021

a(n) = 5*(5^n - 1)/4. - Max Alekseyev, Jan 23 2006
a(n) = a(n-1) + 5^n with a(0)=0. - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Jul 25 2014: (Start)
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: 5*x / ((1-x)*(1-5*x)). (End)
E.g.f.: (5/4)*(exp(5*x) - exp(x)). - G. C. Greubel, Jun 15 2021

A104896 a(0) = 0; a(n) = 7*a(n-1) + 7.

Original entry on oeis.org

0, 7, 56, 399, 2800, 19607, 137256, 960799, 6725600, 47079207, 329554456, 2306881199, 16148168400, 113037178807, 791260251656, 5538821761599, 38771752331200, 271402266318407, 1899815864228856, 13298711049601999, 93090977347214000, 651636841430498007

Offset: 0

Alexandre Wajnberg, Apr 24 2005

Conjecture: this is also the number of integers from 0 to 10^n - 1 that lack 0, 1 and 2 as a digit.

Number of monic irreducible polynomials of degree 1 in GF(7)[x1,...,xn]. - Max Alekseyev, Jan 23 2006

Alois P. Heinz, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A000918, A029858, A052379, A052386, A080674.

Row n=7 of A228275.

[(7/6)*(7^n -1): n in [0..30]]; // G. C. Greubel, Jun 09 2021

a:=n->sum (7^j,j=1..n): seq(a(n), n=0..30); # Zerinvary Lajos, Oct 03 2007

RecurrenceTable[{a[n]==7*a[n-1]+7,a[0]==0},a,{n,0,30}] (* Vaclav Kotesovec, Jul 25 2014 *)

concat(0, Vec(7*x/((x-1)*(7*x-1)) + O(x^30))) \\ Colin Barker, Jul 25 2014

[(7/6)*(7^n -1) for n in (0..30)] # G. C. Greubel, Jun 09 2021

a(n) = (7^(n+1) - 7) / 6. - Max Alekseyev, Jan 23 2006
a(n) = a(n-1) + 7^n with a(0)=0. - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Jul 25 2014: (Start)
a(n) = 8*a(n-1) - 7*a(n-2).
G.f.: 7*x / ((x-1)*(7*x-1)). (End)
E.g.f.: (7/6)*(exp(7*x) - exp(x)). - G. C. Greubel, Jun 09 2021

A105281 a(0)=0; a(n) = 6*a(n-1) + 6.

Original entry on oeis.org

0, 6, 42, 258, 1554, 9330, 55986, 335922, 2015538, 12093234, 72559410, 435356466, 2612138802, 15672832818, 94036996914, 564221981490, 3385331888946, 20311991333682, 121871948002098, 731231688012594, 4387390128075570, 26324340768453426, 157946044610720562

Offset: 0

Alexandre Wajnberg, Apr 25 2005

Number of integers from 0 to (10^n) - 1 that lack 0, 1, 2 and 3 as a digit.

a(n) is the expected number of tosses of a single die needed to obtain for the first time a string of n consecutive 6's. - Jean M. Morales, Aug 04 2012

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

Cf. A000918, A003464, A029858, A052379, A052386, A080674.

Row n=6 of A228275.

a:=n->add(6^j,j=1..n): seq(a(n),n=0..30); # Zerinvary Lajos, Oct 03 2007

NestList[6#+6&,0,30] (* Harvey P. Dale, Jul 24 2012 *)

PARI
```
a(n)=if(n<0,0, (6^n-1)*6/5)
```

a(n) = 6^n + a(n-1) (with a(0)=0). - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Jan 28 2013: (Start)
a(n) = 7*a(n-1) - 6*a(n-2).
G.f.: 6*x/((x-1)*(6*x-1)). (End)
From Elmo R. Oliveira, Mar 16 2025: (Start)
E.g.f.: 6*exp(x)*(exp(5*x) - 1)/5.
a(n) = 6*(6^n - 1)/5.
a(n) = 6*A003464(n). (End)

More terms from Harvey P. Dale, Jul 24 2012

A263133 Numbers m such that binomial(4*m + 3, m) is odd.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 10, 11, 15, 21, 23, 31, 42, 43, 47, 63, 85, 87, 95, 127, 170, 171, 175, 191, 255, 341, 343, 351, 383, 511, 682, 683, 687, 703, 767, 1023, 1365, 1367, 1375, 1407, 1535, 2047, 2730, 2731, 2735, 2751, 2815, 3071, 4095, 5461, 5463, 5471, 5503

Offset: 1

Peter Bala, Oct 11 2015

The even terms in the sequence are A020988. If m is a term in the sequence then 2*m + 1 is also a term in the sequence. Repeatedly applying the transformation m -> 2*m + 1 to the terms of A020988 produces all the terms of this sequence. See the example below.

2*a(n) gives the values of m such that binomial(4*m + 6, m) is odd.

			1) This sequence can be read from Table 1 below in a sequence of 'knight moves' (2 down and 1 to the left) starting from the first two rows. For example, starting at 42 in the first row we jump 42 -> 43 -> 47 -> 63, then return to the second row at 85 and jump 85 -> 87 -> 95 -> 127, followed by 170 -> 171 -> 175 -> 191 -> 255, and so on.
...........................................................
. Table 1. 2^n*ceiling((2^(2*k + 1) - 1)/3) - 1, n,k >= 0 .
...........................................................
  n\k|   0    1    2    3    4    5
  ---+---------------------------------
   0 |   0    2   10   42  170  682 ...
   1 |   1    5   21   85  341  ...
   2 |   3   11   43  171  683  ...
   3 |   7   23   87  343  ...
   4 |  15   47  175  687  ...
   5 |  31   95  351  ...
   6 |  63  191  703  ...
   7 | 127  383  ...
   8 | 255  767  ...
   9 | 511  ...
   ...
The first row of the table is A020988. The columns of the table are obtained by repeatedly applying the transformation m -> 2*m + 1 to the entries in the first row.
2) Alternatively, this sequence can be read from Table 2 below by starting with a number on the top row and moving in a series of 'knight moves' (1 down and 2 to the left) through the table as far as you can, before returning to the next number in the top row and repeating the process. For example, starting at 10 in the first row we move 10 -> 11 -> 15, then return to the top row at 21 and move 21 -> 23 -> 31, before returning to the top row at 42 and so on.
........................................................
. Table 2. (4^n)*ceiling(2^k/3) - 1 for n >= 0, k >= 1 .
........................................................
n\k|    1    2    3    4     5     6     7     8    9   10
---+---------------------------------------------------------
  0|    0    1    2    5    10    21    42    85  170  682...
  1|    3    7   11   23    43    87   171   343  683  ...
  2|   15   31   47   95   175   351   687  1375  ...
  3|   63  127  191  383   703  1407  2751  5503  ...
  4|  255  511  767 1535  2815  5631 11007 22015  ...
  5| 1023 2047 3071 6143 11263 22527 44031 88063  ...
  6| 4095 ...
  ...
The first row of the table is A000975. The columns of the table are obtained by repeatedly applying the transformation m -> 4*m + 3 to the entries in the first row.

Cf. A000975.

Other odd binomials: A263132 (4*m-1,m), A002450 (4*m+1,m), A020988 (4*m+2,m), A080674 (4*m+4,m), A118113 (3*m-2,m), A003714 (3*m,m).

[n: n in [0..6000] | Binomial(4*n+3, n) mod 2 eq 1]; // Vincenzo Librandi, Oct 12 2015

for n from 1 to 4096 do if mod(binomial(4*n+3, n), 2) = 1 then print(n) end if end do;

Select[Range[0,5600],OddQ[Binomial[4#+3,#]]&] (* Harvey P. Dale, Apr 15 2019 *)

for(n=0, 1e4, if (binomial(4*n+3, n) % 2 == 1, print1(n", "))) \\ Altug Alkan, Oct 11 2015

a(n) = my(r,s=sqrtint(4*n-3,&r)); (1<Kevin Ryde, Jul 06 2025

Python

A263133_list = [m for m in range(10**6) if not ~(4*m+3) & m] # Chai Wah Wu, Feb 07 2016

Formula

a(n) = A263132(n) - 1.

m is a term if and only if m AND NOT (4*m+3) = 0 where AND and NOT are bitwise operators. - Chai Wah Wu, Feb 07 2016

a(n) = (2^A000267(n) + 2^A384688(n))/3 - 1, for n >= 1. - Kevin Ryde, Jul 06 2025

Extensions

More terms from Vincenzo Librandi, Oct 12 2015

cp's OEIS Frontend

A104891 a(0) = 0; a(n) = 5*a(n-1) + 5.

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A104896 a(0) = 0; a(n) = 7*a(n-1) + 7.

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A105281 a(0)=0; a(n) = 6*a(n-1) + 6.

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A263133 Numbers m such that binomial(4*m + 3, m) is odd.

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A105280 a(0)=0; a(n) = 11*a(n-1) + 11.

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A155721 Positions of parity change in A033035.

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