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