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

-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

Author

Ctibor O. Zizka, Apr 15 2008

Keywords

• sign
• easy

Comments

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

Examples

			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)
		

Links

• 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).

Crossrefs

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

Programs

Magma

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

Mathematica

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

PARI

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

Formula

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)

Extensions

More terms from Michel Marcus, Mar 02 2016

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