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%I #62 Sep 08 2022 08:45:10
%S 1,3,5,9,17,33,65,129,257,513,1025,2049,4097,8193,16385,32769,65537,
%T 131073,262145,524289,1048577,2097153,4194305,8388609,16777217,
%U 33554433,67108865,134217729,268435457,536870913,1073741825,2147483649
%N a(0) = 1; for n>0, a(n) = 2^n + 1.
%C Inverse binomial transform of A005056.
%C Also, A000533 interpreted as binary numbers, written in base 10. Numbers whose representation in base 2 is has n+1 digits and the digit "1" is the initial and final digit and if n>1 then the internal digits are "0" (see example). - _Omar E. Pol_, Feb 24 2008
%C a(n) equals the number of ternary sequences of length n such that no two consecutive terms differ by 1. - _David Nacin_, May 31 2017
%H Vincenzo Librandi, <a href="/A083318/b083318.txt">Table of n, a(n) for n = 0..1000</a>
%H Ramon Carbó-Dorca, <a href="https://doi.org/10.13140/RG.2.2.25069.13280">Boolean Hypercubes: The Origin of a Tagged Recursive Logic and the Limits of Artificial Intelligence</a>, Universitat de Girona (Spain, 2020).
%H Quynh Nguyen, Jean Pedersen, and Hien T. Vu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Pedersen/pedersen2.html">New Integer Sequences Arising From 3-Period Folding Numbers</a>, Vol. 19 (2016), Article 16.3.1. Cites this sequence.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2).
%F a(n) = 2^n + 1^n - 0^n.
%F G.f.: (1-2*x^2)/((1-x)*(1-2x)).
%F E.g.f.: exp(2*x) + exp(x) - exp(0).
%F a(n) = Sum_{k=0..n} 0^(k*(n-k))*2^(n-k). - _Paul Barry_, Feb 09 2005
%F a(n) = Min{m: A008687(m) = n+1}. - _Reinhard Zumkeller_, Jul 25 2006
%F Row sums of triangle A132749; = binomial transform of [1, 2, 0, 2, 0, 2, 0, 2, ...]. - _Gary W. Adamson_, Aug 28 2007
%F A020650(a(n)) = 1. - _Yosu Yurramendi_, Jun 01 2016
%e From _Omar E. Pol_, Feb 24 2008: (Start)
%e ------------------------------
%e n .... a(n) .. a(n) in base 2
%e ------------------------------
%e 0 ..... 1 ..... 1
%e 1 ..... 3 ..... 11
%e 2 ..... 5 ..... 101
%e 3 ..... 9 ..... 1001
%e 4 .... 17 ..... 10001
%e 5 .... 33 ..... 100001
%e 6 .... 65 ..... 1000001
%e 7 ... 129 ..... 10000001
%e 8 ... 257 ..... 100000001
%e 9 ... 513 ..... 1000000001
%e (End)
%e G.f. = 1 + 3*x + 5*x^2 + 9*x^3 + 17*x^4 + 33*x^5 + 65*x^6 + 129*x^7 + ... - _Michael Somos_, Jun 04 2016
%p seq(`if`(n=0, 1, 2^n + 1), n=0..40); # _G. C. Greubel_, Nov 20 2019
%t Join[{1},2^Range[40]+1] (* _Harvey P. Dale_, May 17 2013 *)
%o (Magma) [2^n+1-0^n : n in [0..40]]; // _Vincenzo Librandi_, Sep 01 2011
%o (PARI) {a(n) = if( n<1, n==0, 2^n + 1)}; /* _Michael Somos_, Jun 04 2016 */
%o (Sage) [1]+[2^n +1 for n in (1..40)] # _G. C. Greubel_, Nov 20 2019
%o (GAP) Concatenation([1], List([1..40], n-> 2^n +1)); # _G. C. Greubel_, Nov 20 2019
%Y Except for the leading term, the same as A000051.
%Y Cf. A000533, A083319, A132749.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Apr 25 2003
%E Edited by _N. J. A. Sloane_, Sep 28 2007
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