A137410 - OEIS

%I #41 Dec 15 2023 09:10:33

%S -1,1,11,61,311,1561,7811,39061,195311,976561,4882811,24414061,

%T 122070311,610351561,3051757811,15258789061,76293945311,381469726561,

%U 1907348632811,9536743164061,47683715820311,238418579101561,1192092895507811,5960464477539061,29802322387695311,149011611938476561

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

%C 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.

%C Primes of this form are sometimes of interest, examples:

%C A=2, B=1, r=1 gives A000225 and subsequence of primes: A001348,

%C A=3, B=1, r=1 gives A003462 and subsequence of primes: A028491,

%C A=3, B=2, r=1 gives A058481 and subsequence of primes: A014224,

%C A=4, B=1, r=1 gives A002450,

%C A=4, B=2, r=1 gives A083420,

%C A=4, B=2, r=2 gives A002446,

%C A=5, B=1, r=1 gives A003463 and subsequence of primes: A004061,

%C A=5, B=2, r=1 gives A037577.

%C 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

%C 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

%H M. F. Hasler, <a href="/A137410/b137410.txt">Table of n, a(n) for n = 0..100</a>

%H Geoffrey B Campbell and Aleksander Zujev, <a href="http://arxiv.org/abs/1603.00080">On integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n</a>, arXiv:1603.00080 [math.NT], 2016.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-5).

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

%F From _Colin Barker_, May 01 2012: (Start)

%F a(n) = 6*a(n-1) - 5*a(n-2).

%F G.f.: (-1+7*x)/((1-x)*(1-5*x)). (End)

%F a(n) = 5*a(n-1) + 6, a(1) = 1. - _Philippe Deléham_, Feb 24 2014

%F From _Elmo R. Oliveira_, Dec 11 2023: (Start)

%F a(n) = A024049(n)/2 - 1 = A125831(n) - 1.

%F E.g.f.: (1/2)*(exp(5*x) - 3*exp(x)). (End)

%e From _Philippe Deléham_, Feb 24 2014: (Start)

%e a(1) = 1;

%e a(2) = 5 + 1 + 5 = 11;

%e a(3) = 25 + 5 + 1 + 5 + 25 = 61;

%e a(4) = 125 + 25 + 5 + 1 + 5 + 25 + 125 = 311;

%e etc. (End)

%t LinearRecurrence[{6, -5}, {1, 11}, 25] (* _Vincenzo Librandi_, Mar 02 2016 *)

%t (5^Range[30]-3)/2 (* _Harvey P. Dale_, Feb 24 2017 *)

%o (PARI) a(n) = (5^n - 3) / 2; \\ _Michel Marcus_, Mar 02 2016

%o (Magma) [(5^n-3)/2: n in [0..25]]; // _Vincenzo Librandi_, Mar 02 2016

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

%K sign,easy

%O 0,3

%A _Ctibor O. Zizka_, Apr 15 2008

%E More terms from _Michel Marcus_, Mar 02 2016

%E Edited and missing term a(0) inserted by _M. F. Hasler_, Jul 10 2018