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

0,3

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

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).

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)

Example

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)

Mathematica

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

Prog

(PARI) a(n) = (5^n - 3) / 2; \\ Michel Marcus, Mar 02 2016
(Magma) [(5^n-3)/2: n in [0..25]]; // Vincenzo Librandi, Mar 02 2016

Crossrefs

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

Sequence in context: A320145 A106992 A115565 * A268863 A199414 A156095

Adjacent sequences: A137407 A137408 A137409 * A137411 A137412 A137413

Keyword

sign,easy

Author

Ctibor O. Zizka, Apr 15 2008

Extensions

More terms from Michel Marcus, Mar 02 2016

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

Status

approved