A125831 -id:A125831 - cp's OEIS frontend

A024049 a(n) = 5^n - 1.

0, 4, 24, 124, 624, 3124, 15624, 78124, 390624, 1953124, 9765624, 48828124, 244140624, 1220703124, 6103515624, 30517578124, 152587890624, 762939453124, 3814697265624, 19073486328124, 95367431640624

Offset: 0

N. J. A. Sloane

Numbers whose base 5 representation is 44444.......4. - Zerinvary Lajos, Feb 03 2007

For n > 0, a(n) is the sum of divisors of 3*5^(n-1). - Patrick J. McNab, May 27 2017

			For n = 5, a(5) = 4*5 + 16*10 + 64*10 + 256*5 + 1024*1 = 3124. - _Bruno Berselli_, Nov 11 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences (2019) Vol. 8, No. 10.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000351, A003463, A125831, A248722.

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

5^Range[0,50]-1 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
LinearRecurrence[{6,-5},{0,4},30] (* Harvey P. Dale, Apr 06 2019 *)

a(n)=5^n-1 \\ Charles R Greathouse IV, Apr 17 2012

G.f.: 1/(1-5*x) - 1/(1-x) = 4*x/((1-5*x)*(1-x)). - Mohammad K. Azarian, Jan 14 2009
E.g.f.: exp(5*x) - exp(x). - Mohammad K. Azarian, Jan 14 2009
a(n+1) = 5*a(n) + 4. - Reinhard Zumkeller, Nov 22 2009
a(n) = Sum_{i=1..n} 4^i*binomial(n,n-i) for n>0, a(0)=0. - Bruno Berselli, Nov 11 2015
a(n) = A000351(n) - 1. - Sean A. Irvine, Jun 19 2019
Sum_{n>=1} 1/a(n) = A248722. - Amiram Eldar, Nov 13 2020
a(n) = 2*A125831(n) = 4*A003463(n). - Elmo R. Oliveira, Dec 10 2023

A350991 Triangular numbers that are palindromes in base 5.

Original entry on oeis.org

0, 1, 3, 6, 36, 78, 378, 1953, 20706, 23436, 48828, 147696, 239778, 426426, 449826, 1220703, 2155926, 6011778, 14625936, 30517578, 74218836, 74316336, 149083278, 314290056, 351562386, 762939453, 7897542681, 9141750936, 10201418541, 19073486328, 35952613476, 38218245156

Offset: 1

Amiram Eldar, Jan 28 2022

This sequence is infinite since A000217((5^k-1)/2) is a term for all k >= 0 (Trigg, 1972).

			6 is a term since 6 = A000217(3) is a triangular number and also a palindromic number in base 5: 6 = 11_5.
36 is a term since 36 = A000217(8) is a triangular number and also a palindromic number in base 5: 36 = 121_5.

Amiram Eldar, Table of n, a(n) for n = 1..68
Charles W. Trigg, Problem 840, Problems and Solutions, Mathematics Magazine, Vol. 45, No. 4 (1972), p. 228; Triangular Palindromes, Solution to Problem 840 by E. P. Starke, ibid., Vol. 46, No. 3 (1973), p. 170.
Charles W. Trigg, Infinite sequences of palindromic triangular numbers, The Fibonacci Quarterly, Vol. 12, No. 2 (1974), pp. 209-212.
Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv:0811.2477 [math.NT], 2008.

Intersection of A000217 and A029952.
The quinary version of A003098.
Cf. A003098, A125831, A350987, A350990, A350992, A350993.

t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 5]] &]

A121177 Catapolyoctagons (see Cyvin et al. for precise definition).

Original entry on oeis.org

0, 2, 12, 62, 312, 1562, 7812, 39062, 195312, 976562, 4882812, 24414062, 122070312, 610351562, 3051757812, 15258789062, 76293945312, 381469726562, 1907348632812, 9536743164062, 47683715820312, 238418579101562, 1192092895507812, 5960464477539062, 29802322387695312

Offset: 1

N. J. A. Sloane, Aug 15 2006

From Petros Hadjicostas, Jul 30 2019: (Start)
The conjecture by Philipp Emanuel Weidmann (see link below) is correct. In Cyvin et al. (1997), this sequence has a double meaning. See Eqs. (6) and (7) and Table I on p. 58 in that paper. The terms of the sequence are related to the enumeration of unbranched catapolyoctagons.
The number of unbranched catapolyoctagons of the symmetry C_{2h} is given by c_r = (1/2) *(5^(floor(r/2)-1) - 1) + (2/5) * binomial(1, r), where r is the number of octagons in the unbranched catapolyoctagon. We get the sequence 0, 0, 0, 2, 2, 12, 12, 62, 62, 312, 312, ... whose bijection (apart for the case r = 1) is the current sequence.
In addition, the number of unbranched catapolyoctagons of the symmetry C_{2v} is given by m_r = (1/2) * (3 - 2*(-1)^r) * 5^(floor(r/2) - 1) - (1/2), where again r is the number of octagons. We get the sequence 0, 0, 2, 2, 12, 12, 62, 62, 312, 312, 1562, 1562, ... whose bijection is the current sequence.
The total number of unbranched catapolygons (with respect to all the symmetry point groups D_{8h}, D_{2h}, C_{2h}, and C_{2v}) is given by i_r = A121101(r).
(End)

S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), 55-70, eq. (6).

P. E. Weidmann, The OEIS Sequencer survey, Apr 11 2015.
Wikipedia, Point groups in three dimensions.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A121101, A125831.

a(n) = (5^n-5)/10 = 2*A003463(n-1) for n >= 1. - Philipp Emanuel Weidmann, cf. link.

G.f.: 2*x^2 / ( (5*x-1)*(x-1) ). - R. J. Mathar, Jul 31 2019

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

A125725 Numbers whose base-7 representation is 222....2.

Original entry on oeis.org

0, 2, 16, 114, 800, 5602, 39216, 274514, 1921600, 13451202, 94158416, 659108914, 4613762400, 32296336802, 226074357616, 1582520503314, 11077643523200, 77543504662402, 542804532636816, 3799631728457714, 26597422099204000

Offset: 1

Zerinvary Lajos, Feb 02 2007

			base 7.......decimal
0..................0
2..................2
22................16
222..............114
2222.............800
22222...........5602
222222.........39216
2222222.......274514
22222222.....1921600
222222222...13451202
etc...........etc.

Davis Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A007310, A023000, A047522, A165759.

Cf. also A002276, A005610, A020988, A024023, A125831, A125835, A125857 for related or similarly constructed sequences.

List([1..25], n-> (7^(n-1) -1)/3); # G. C. Greubel, May 23 2019

[0] cat [n:n in [1..15000000]| Set(Intseq(n,7)) subset [2]]; // Marius A. Burtea, May 06 2019

[(7^(n-1)-1)/3: n in [1..25]]; // Marius A. Burtea, May 06 2019

Maple
```
seq(2*(7^n-1)/6, n=0..25);
```

FromDigits[#,7]&/@Table[PadLeft[{2},n,2],{n,0,25}]  (* Harvey P. Dale, Apr 13 2011 *)
(7^(Range[25]-1) - 1)/3 (* G. C. Greubel, May 23 2019 *)

vector(25, n, (7^(n-1)-1)/3) \\ Davis Smith, Apr 04 2019

[(7^(n-1) -1)/3 for n in (1..25)] # G. C. Greubel, May 23 2019

a(n) = (7^(n-1) - 1)/3 = 2*A023000(n-1).
a(n) = 7*a(n-1) + 2, with a(1)=0. - Vincenzo Librandi, Sep 30 2010
G.f.: 2*x^2 / ( (1-x)*(1-7*x) ). - R. J. Mathar, Sep 30 2013
From Davis Smith, Apr 04 2019: (Start)
A007310(a(n) + 1) = 7^(n - 1).
A047522(a(n + 1)) = -1*A165759(n). (End)
E.g.f.: (exp(7*x) - 7*exp(x) + 6)/21. - Stefano Spezia, Jan 12 2025

Offset corrected by N. J. A. Sloane, Oct 02 2010

A238366 a(n) = 5*a(n-2) + 2, a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 7, 12, 37, 62, 187, 312, 937, 1562, 4687, 7812, 23437, 39062, 117187, 195312, 585937, 976562, 2929687, 4882812, 14648437, 24414062, 73242187, 122070312, 366210937, 610351562, 1831054687, 3051757812, 9155273437, 15258789062, 45776367187, 76293945312

Offset: 0

Philippe Deléham, Feb 25 2014

Row sums of triangle in A152717.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,5,-5).

Cf. A057651, A125831, A152717, A198306.

LinearRecurrence[{1,5,-5},{1,2,7},40] (* Harvey P. Dale, Jul 18 2024 *)

G.f.: (1+x)/((1-x)*(1-5*x^2)).
a(n) = Sum_{k=0..n} A152717(n,k).
a(2*n) = A057651(n).
a(2*n+1) = A125831(n+1) = 2*A003463(n+1).
a(n) = a(n-1) + 5*a(n-2) - 5*a(n-3), a(0) = 1, a(1) = 2, a(2) = 7.
a(n) = A198306(n+1) for n > 1. - Georg Fischer, Oct 23 2018

A275766 a(n) = (5^(2*(n + 1)) - 1)/4.

Original entry on oeis.org

156, 3906, 97656, 2441406, 61035156, 1525878906, 38146972656, 953674316406, 23841857910156, 596046447753906, 14901161193847656, 372529029846191406, 9313225746154785156, 232830643653869628906, 5820766091346740722656, 145519152283668518066406

Offset: 1

Gionata Neri, Aug 07 2016

It seems that these terms are the only numbers n such that n and n + 1 are in A053696.

			3906 written in base 5 is 111111 and 3907 written in base 62 is 111.

Colin Barker, Table of n, a(n) for n = 1..700
Index entries for linear recurrences with constant coefficients, signature (26,-25).

Cf. A003463, A053696, A125831.

Table[(5^(2 (n + 1)) - 1)/4, {n, 16}] (* or *)
Rest@ CoefficientList[Series[6 x (26 - 25 x)/((1 - x) (1 - 25 x)), {x, 0, 16}], x] (* Michael De Vlieger, Aug 28 2016 *)

Vec(6*x*(26-25*x)/((1-x)*(1-25*x)) + O(x^20)) \\ Colin Barker, Aug 24 2016

a(n) = 5^(2*n+2)\4 \\ Charles R Greathouse IV, Aug 28 2016

a(n) = ((A125831(n+1))^3 - 1)/(A125831(n+1) - 1) - 1.
a(n) = A003463(2*(n+1)).
a(n) = 26*a(n-1) - 25*a(n-2), a(1) = 156, a(2) = 3906.
G.f.: 6*x*(26-25*x) / ((1-x)*(1-25*x)). - Colin Barker, Aug 24 2016

cp's OEIS Frontend

A024049 a(n) = 5^n - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A350991 Triangular numbers that are palindromes in base 5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A121177 Catapolyoctagons (see Cyvin et al. for precise definition).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A125725 Numbers whose base-7 representation is 222....2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A238366 a(n) = 5*a(n-2) + 2, a(0) = 1, a(1) = 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A275766 a(n) = (5^(2*(n + 1)) - 1)/4.

Views

Author

Keywords

Comments

Examples