A004119 - OEIS

%I M3308 #87 Jul 13 2023 09:19:50

%S 1,4,7,13,25,49,97,193,385,769,1537,3073,6145,12289,24577,49153,98305,

%T 196609,393217,786433,1572865,3145729,6291457,12582913,25165825,

%U 50331649,100663297,201326593,402653185,805306369,1610612737,3221225473,6442450945,12884901889

%N a(0) = 1; thereafter a(n) = 3*2^(n-1) + 1.

%C Also Pisot sequence L(4,7) (cf. A008776).

%C Alternatively, define the sequence S(a(1),a(2)) by: a(n+2) is the least integer such that a(n+2)/a(n+1) > a(n+1)/a(n) for n > 0. This is S(4,7).

%C a(n) = number of terms of the arithmetic progression with first term 2^(2n-1) and last term 2^(2n+1). So common difference is 2^n. E.g., a(2)=7 corresponds to (8,12,16,20,24,28,32). - _Augustine O. Munagi_, Feb 21 2007

%C Equals binomial transform of [1, 3, 0, 3, 0, 3, 0, 3, ...]. - _Gary W. Adamson_, Aug 27 2010

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A004119/b004119.txt">Table of n, a(n) for n = 0..1000</a>

%H D. W. Boyd, <a href="https://www.researchgate.net/profile/David_Boyd7/publication/262181133_Linear_recurrence_relations_for_some_generalized_Pisot_sequences_-_annotated_with_corrections_and_additions/links/00b7d536d49781037f000000.pdf">Linear recurrence relations for some generalized Pisot sequences</a>, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993

%H S. W. Golomb, <a href="http://www.jstor.org/stable/2005337">Properties of the sequence 3.2^n+1</a>, Math. Comp., 30 (1976), 657-663.

%H S. W. Golomb, <a href="/A004119/a004119.pdf">Properties of the sequence 3.2^n+1</a>, Math. Comp., 30 (1976), 657-663. [Annotated scanned copy]

%H A. O. Munagi and T. Shonhiwa, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Shonhiwa/shonhiwa13.html"> On the partitions of a number into arithmetic progressions</a>, J. Integer Sequences, 11 (2008), #08.5.4.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3, -2).

%H <a href="/index/Ph#Pisot">Index entries for Pisot sequences</a>

%F a(n) = 3a(n-1) - 2a(n-2).

%F For n>3, a(3)=13, a(n)= a(n-1)+2*floor(a(n-1)/2). - _Benoit Cloitre_, Aug 14 2002

%F For n>=1, a(n) = A049775(n+1)/2^(n-2). For n>=2, a(n) = 2a(n-1)-1; see also A000051. - _Philippe Deléham_, Feb 20 2004

%F O.g.f.: -(-1-x+3*x^2)/((2*x-1)*(x-1)). - _R. J. Mathar_, Nov 23 2007

%F For n>0, a(n) = 2*a(n-1)-1. - _Vincenzo Librandi_, Dec 16 2015

%F E.g.f.: exp(x)*(1 + 3*sinh(x)). - _Stefano Spezia_, May 06 2023

%p A004119:=-(-1-z+3*z**2)/(2*z-1)/(z-1); # _Simon Plouffe_ in his 1992 dissertation

%t s=4;lst={1,s};Do[s=s+(s-1);AppendTo[lst,s],{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 30 2009 *)

%t Prepend[Table[3*2^n + 1, {n, 0, 32}], 1] (* or *)

%t {1}~Join~LinearRecurrence[{3, -2}, {4, 7}, 33] (* _Michael De Vlieger_, Dec 16 2015 *)

%o (PARI) a(n)=3<<n+1 \\ _Charles R Greathouse IV_, Sep 28 2015

%o (Magma) [1] cat [n le 1 select 4 else 2*Self(n-1)-1: n in [1..40]]; // _Vincenzo Librandi_, Dec 16 2015

%Y Cf. A049988, A049775, A000051, A008776.

%Y A181565 is an essentially identical sequence.

%Y For primes see A002253 and A039687.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, _R. K. Guy_

%E Edited by _N. J. A. Sloane_, Dec 16 2015 at the suggestion of _Bruno Berselli_