A024049 -id:A024049 - cp's OEIS frontend

A103455 a(n) = 0^n + 5^n - 1.

1, 4, 24, 124, 624, 3124, 15624, 78124, 390624, 1953124, 9765624, 48828124, 244140624, 1220703124, 6103515624, 30517578124, 152587890624, 762939453124, 3814697265624, 19073486328124, 95367431640624, 476837158203124

Offset: 0

Paul Barry, Feb 06 2005

A transform of 5^n under the matrix A103452.

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A024049, A103452.

[0^n+5^n-1: n in [0..30]]; // Vincenzo Librandi, Jun 06 2011

Join[{1},5^Range[20]-1] (* Harvey P. Dale, Nov 15 2011 *)

[1]+[5^n -1 for n in [1..40]] # G. C. Greubel, Jun 21 2021

G.f.: (1 - 2*x + 5*x^2)/((1-x)*(1-5*x)).
a(n) = Sum_{k=0..n} A103452(n, k)*5^k.
a(n) = Sum_{k=0..n} (2*0^(n-k) - 1)*0^(k*(n-k))*5^k.
a(n) = A024049(n), n > 0. - R. J. Mathar, Aug 30 2008
E.g.f.: 1 - exp(x) + exp(5*x). - G. C. Greubel, Jun 21 2021

A137410 a(n) = (5^n - 3)/2.

Original entry on oeis.org

-1, 1, 11, 61, 311, 1561, 7811, 39061, 195311, 976561, 4882811, 24414061, 122070311, 610351561, 3051757811, 15258789061, 76293945311, 381469726561, 1907348632811, 9536743164061, 47683715820311, 238418579101561, 1192092895507811, 5960464477539061, 29802322387695311, 149011611938476561

Offset: 0

Ctibor O. Zizka, Apr 15 2008

Sequence is a(n) = a(n;5,3,1) where a(n;A,B,r) = (A^n - B^r)/(A - B) for arbitrary integers A, B, r with A != B.
Primes of this form are sometimes of interest, examples:
A=2, B=1, r=1 gives A000225 and subsequence of primes: A001348,
A=3, B=1, r=1 gives A003462 and subsequence of primes: A028491,
A=3, B=2, r=1 gives A058481 and subsequence of primes: A014224,
A=4, B=1, r=1 gives A002450,
A=4, B=2, r=1 gives A083420,
A=4, B=2, r=2 gives A002446,
A=5, B=1, r=1 gives A003463 and subsequence of primes: A004061,
A=5, B=2, r=1 gives A037577.
Sum of n-th row of triangle of powers of 5: 1; 5 1 5; 25 5 1 5 25; 125 25 5 1 5 25 125; ... (cf. Examples). - Philippe Deléham, Feb 24 2014
Integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n (see Campbell and Zujev). - Michel Marcus, Mar 02 2016

			From _Philippe Deléham_, Feb 24 2014: (Start)
a(1) = 1;
a(2) = 5 + 1 + 5 = 11;
a(3) = 25 + 5 + 1 + 5 + 25 = 61;
a(4) = 125 + 25 + 5 + 1 + 5 + 25 + 125 = 311;
etc. (End)

M. F. Hasler, Table of n, a(n) for n = 0..100
Geoffrey B Campbell and Aleksander Zujev, On integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, arXiv:1603.00080 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000040, A000225, A001348, A002446, A002450, A003462, A003463, A004061, A014224, A024049, A028491, A037577, A058481, A083420, A125831.

[(5^n-3)/2: n in [0..25]]; // Vincenzo Librandi, Mar 02 2016

LinearRecurrence[{6, -5}, {1, 11}, 25] (* Vincenzo Librandi, Mar 02 2016 *)
(5^Range[30]-3)/2 (* Harvey P. Dale, Feb 24 2017 *)

a(n) = (5^n - 3) / 2; \\ Michel Marcus, Mar 02 2016

a(n) = (5^n - 3)/2.
From Colin Barker, May 01 2012: (Start)
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: (-1+7*x)/((1-x)*(1-5*x)). (End)
a(n) = 5*a(n-1) + 6, a(1) = 1. - Philippe Deléham, Feb 24 2014
From Elmo R. Oliveira, Dec 11 2023: (Start)
a(n) = A024049(n)/2 - 1 = A125831(n) - 1.
E.g.f.: (1/2)*(exp(5*x) - 3*exp(x)). (End)

More terms from Michel Marcus, Mar 02 2016

Edited and missing term a(0) inserted by M. F. Hasler, Jul 10 2018

A249435 a(1) = 0, after which one less than prime powers p^m with exponent m >= 2.

Original entry on oeis.org

0, 3, 7, 8, 15, 24, 26, 31, 48, 63, 80, 120, 124, 127, 168, 242, 255, 288, 342, 360, 511, 528, 624, 728, 840, 960, 1023, 1330, 1368, 1680, 1848, 2047, 2186, 2196, 2208, 2400, 2808, 3124, 3480, 3720, 4095, 4488, 4912, 5040, 5328, 6240, 6560, 6858, 6888, 7920, 8191, 9408, 10200, 10608, 11448

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

One less than A025475.
Subsequence of A181062 and also a subsequence of A249433 (after the initial zero).
Union of sequences A000225, A024023, A024049, A024075, A024127, etc. without their term a(1).
Apart from the first term, subsequence of A045542.

list(lim)=my(v=List([0])); lim=lim\1+1; for(m=2,logint(lim,2), forprime(p=2,sqrtnint(lim,m), listput(v, p^m-1))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015

Scheme
```
(define (A249435 n) (- (A025475 n) 1))
```

a(n) = A025475(n) - 1.

A114808 The numbers 5^n-1 written in groups of three digits, with leading zeros omitted.

Original entry on oeis.org

424, 124, 624, 312, 415, 624, 781, 243, 906, 241, 953, 124, 976, 562, 448, 828, 124, 244, 140, 624, 122, 70, 312, 461, 35, 156, 243, 51, 757, 812, 415, 258, 789, 62, 476, 293, 945, 312, 438, 146, 972, 656, 241, 907, 348, 632, 812, 495, 367, 431, 640, 624, 476, 837, 158, 203, 124, 238, 418, 579

Offset: 1

Jonathan Vos Post, Feb 19 2006

			4, 24, 124, 624, 3124, 15624, ...

Cf. A024049, A103455, A114645.

L := [] ;
for n from 1 to 30 do
    dggs := ListTools[Reverse](convert(5^n-1,base,10) );
    L := [op(L),op(dggs)] ;
end do:
for k from 1 to nops(L)-3 by 3 do
    op(k,L)*100+op(k+1,L)*10+op(k+2,L) ;
    printf("%d,",%) ;
end do: # R. J. Mathar, Jun 23 2014

FromDigits[#] & /@ Partition[ Flatten@ IntegerDigits@ Table[5^n - 1, {n, 22}], 3] (* Robert G. Wilson v, Jun 23 2014 *)

Definition and terms realigned with A114645 by Robert G. Wilson v, Jun 23 2014

A198762 a(n) = 35^n - 1 = 2A057651(n).

Original entry on oeis.org

2, 14, 74, 374, 1874, 9374, 46874, 234374, 1171874, 5859374, 29296874, 146484374, 732421874, 3662109374, 18310546874, 91552734374, 457763671874, 2288818359374, 11444091796874, 57220458984374, 286102294921874, 1430511474609374, 7152557373046874, 35762786865234374

Offset: 0

Vincenzo Librandi, Oct 30 2011

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

Cf. A024049, A057651, A081655.

Magma
```
[(3*5^n-1): n in [0..30]];
```

CoefficientList[Series[2*(1 + x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
LinearRecurrence[{6,-5},{2,14},30] (* Harvey P. Dale, Mar 19 2013 *)

a(n) = 5*a(n-1) + 4.
a(n) = 6*a(n-1) - 5*a(n-2), n > 1.
G.f.: 2*(1 + x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(3*exp(4*x) - 1). - Elmo R. Oliveira, Mar 29 2025

A198764 6*5^n-1.

Original entry on oeis.org

5, 29, 149, 749, 3749, 18749, 93749, 468749, 2343749, 11718749, 58593749, 292968749, 1464843749, 7324218749, 36621093749, 183105468749, 915527343749, 4577636718749, 22888183593749, 114440917968749, 572204589843749, 2861022949218749

Offset: 0

Vincenzo Librandi, Oct 30 2011

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

Cf. A024049, A057651, A081655.

Magma
```
[6*5^n-1: n in [0..30]]
```

CoefficientList[Series[(5 - x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
6*5^Range[0,30]-1 (* or *) LinearRecurrence[{6,-5},{5,29},30] (* Harvey P. Dale, Dec 21 2014 *)

a(n) = 5*a(n-1)+4.
a(n) = 6*a(n-1)-5*a(n-2), n>1.
G.f.: (5 - x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013

A198763 a(n) = 4*5^n-1.

Original entry on oeis.org

3, 19, 99, 499, 2499, 12499, 62499, 312499, 1562499, 7812499, 39062499, 195312499, 976562499, 4882812499, 24414062499, 122070312499, 610351562499, 3051757812499, 15258789062499, 76293945312499, 381469726562499, 1907348632812499

Offset: 0

Vincenzo Librandi, Oct 30 2011

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

Cf. A024049, A057651, A081655.

Magma
```
[4*5^n-1: n in [0..30]]
```

CoefficientList[Series[(3 + x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
NestList[5#+4&,3,30] (* or *) LinearRecurrence[{6,-5},{3,19},30] (* Harvey P. Dale, Jul 03 2021 *)

a(n) = 5*a(n-1)+4.
a(n) = 6*a(n-1)-5*a(n-2), n>1.
G.f.: (3 + x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013

A198765 7*5^n-1.

Original entry on oeis.org

6, 34, 174, 874, 4374, 21874, 109374, 546874, 2734374, 13671874, 68359374, 341796874, 1708984374, 8544921874, 42724609374, 213623046874, 1068115234374, 5340576171874, 26702880859374, 133514404296874

Offset: 0

Vincenzo Librandi, Oct 30 2011

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

Cf. A024049, A057651, A081655.

Magma
```
[7*5^n-1: n in [0..30]];
```

CoefficientList[Series[2*(3 - x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
7 * 5^Range[0, 19] - 1 (* Alonso del Arte, Dec 05 2013 *)

a(n) = 5*a(n-1) + 4.
a(n) = 6*a(n-1) - 5*a(n-2), n > 1.
G.f.: 2*(3 - x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013

A198766 a(n) = (7*5^n - 1)/2.

Original entry on oeis.org

3, 17, 87, 437, 2187, 10937, 54687, 273437, 1367187, 6835937, 34179687, 170898437, 854492187, 4272460937, 21362304687, 106811523437, 534057617187, 2670288085937, 13351440429687, 66757202148437, 333786010742187, 1668930053710937

Offset: 0

Vincenzo Librandi, Oct 30 2011

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

Cf. A024049, A057651, A081655.

Magma
```
[(7*5^n-1)/2: n in [0..30]];
```

CoefficientList[Series[(3 - x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)
LinearRecurrence[{6,-5},{3,17},30] (* Harvey P. Dale, Jan 23 2015 *)

a(n) = 5*a(n-1)+2.
a(n) = 6*a(n-1)-5*a(n-2), n>1.
G.f.: (3 - x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013
E.g.f.: exp(x)*(7*exp(4*x) - 1)/2. - Stefano Spezia, Mar 08 2025

A198767 8*5^n-1.

Original entry on oeis.org

7, 39, 199, 999, 4999, 24999, 124999, 624999, 3124999, 15624999, 78124999, 390624999, 1953124999, 9765624999, 48828124999, 244140624999, 1220703124999, 6103515624999, 30517578124999, 152587890624999, 762939453124999

Offset: 0

Vincenzo Librandi, Oct 30 2011

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

Cf. A024049, A057651, A081655.

Magma
```
[8*5^n-1: n in [0..30]];
```

CoefficientList[Series[(7 - 3*x)/(1 - 6*x + 5*x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 04 2013 *)

a(n) = 5*a(n-1)+4.
a(n) = 6*a(n-1)-5*a(n-2), n>1.
G.f.: (7 - 3*x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Jan 04 2013

cp's OEIS Frontend

A103455 a(n) = 0^n + 5^n - 1.

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A137410 a(n) = (5^n - 3)/2.

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A249435 a(1) = 0, after which one less than prime powers p^m with exponent m >= 2.

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A114808 The numbers 5^n-1 written in groups of three digits, with leading zeros omitted.

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A198762 a(n) = 3*5^n - 1 = 2*A057651(n).

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A198764 6*5^n-1.

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Magma

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A198763 a(n) = 4*5^n-1.

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A198765 7*5^n-1.

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A198762 a(n) = 35^n - 1 = 2A057651(n).