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%I #64 Feb 11 2024 00:09:09
%S 2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N a(0)=2, a(n)=1 for n >= 1.
%C Arises in Gilbreath-Proth conjecture; see A036262.
%C a(n) is also the continued fraction for (3+sqrt(5))/2. - _Enrique Pérez Herrero_, May 16 2010
%C a(n) is also the denominator for odd Bernoulli Numbers. - _Enrique Pérez Herrero_, Jul 17 2010
%C a(n) = 3 - A040000(n); a(n) = A182579(n+1,1). - _Reinhard Zumkeller_, May 07 2012
%C From _Paul Curtz_, Feb 04 2014: (Start)
%C Difference table of a(n):
%C 2, 1, 1, 1, 1, 1, 1, ...
%C -1, 0, 0, 0, 0, 0, 0, ...
%C 1, 0, 0, 0, 0, 0, 0, ...
%C -1, 0, 0, 0, 0, 0, 0, ...
%C 1, 0, 0, 0, 0, 0, 0, ...
%C -1, 0, 0, 0, 0, 0, 0, ... .
%C a(n) is an autosequence of second kind. Its inverse binomial transform is the signed sequence with the main diagonal (here A000038) double of the following diagonal (here A000007). Here the other diagonals are also A000007.
%C b(n) = A000032(n) - a(n) = 0, 0, 2, 3, 6, 10, 17, 28, ... = 0, followed by A001610(n) is the autosequence of second kind preceding A000032(n).
%C The corresponding autosequence of first kind, 0 followed by 1's, is A057427(n).
%C The Akiyama-Tanigawa transform applied to a(n) yields a(n).
%C (End)
%C Harmonic or factorial (base) expansion of e, cf. MathWorld link. - _M. F. Hasler_, Nov 25 2018
%H Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Gewurz/gewurz5.html">Sequences realized as Parker vectors of oligomorphic permutation groups</a>, J. Integer Seqs., Vol. 6, 2003.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicExpansion.html">Harmonic Expansion</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F a(n) = A027642(2n+1). - _Enrique Pérez Herrero_, Jul 17 2010
%F G.f.: (2-x)/(1-x). - _Wolfdieter Lang_, Oct 05 2014
%F Sum_{k>=1} a(n)/n! = exp(1). - _G. C. Greubel_, Nov 26 2018
%t A054977[1]:=2;
%t A054977[n_]:=1; (* _Enrique Pérez Herrero_, May 16 2010 *)
%t PadRight[{2},120,{1}] (* _Harvey P. Dale_, Mar 30 2018 *)
%o (Haskell)
%o a054977 0 = 2; a054977 n = 1
%o a054977_list = 2 : repeat 1 -- _Reinhard Zumkeller_, May 07 2012
%o (PARI) a(n)=if(n,1,2) \\ _Charles R Greathouse IV_, Mar 23 2016
%o (PARI) contfrac((sqrt(5)+3)/2)[^-1] \\ or A068446_vec(30,exp(1)) illustrate that this is the c.f. resp. factoriadic expansion of these two constants. - _M. F. Hasler_, Nov 28 2018
%o (Magma) ContinuedFraction((1+Sqrt(5))^2/4); // _G. C. Greubel_, Nov 26 2018
%o (Sage) continued_fraction(golden_ratio^2) # _G. C. Greubel_, Nov 26 2018
%o (Python)
%o def A054977(n):
%o return 1 if n else 2 # _Chai Wah Wu_, Dec 20 2018
%Y Cf. A036262, A054978, A027642, A002445, A174419.
%K nonn,easy,mult
%O 0,1
%A _Henry Gould_, May 29 2000
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