A054977 a(0)=2, a(n)=1 for n >= 1.

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, 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, 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

Offset: 0

Author

Henry Gould, May 29 2000

Keywords

• nonn
• easy
• mult

Comments

Arises in Gilbreath-Proth conjecture; see A036262.

a(n) is also the continued fraction for (3+sqrt(5))/2. - Enrique Pérez Herrero, May 16 2010

a(n) is also the denominator for odd Bernoulli Numbers. - Enrique Pérez Herrero, Jul 17 2010

a(n) = 3 - A040000(n); a(n) = A182579(n+1,1). - Reinhard Zumkeller, May 07 2012

From Paul Curtz, Feb 04 2014: (Start)

Difference table of a(n):

2, 1, 1, 1, 1, 1, 1, ...

-1, 0, 0, 0, 0, 0, 0, ...

1, 0, 0, 0, 0, 0, 0, ...

-1, 0, 0, 0, 0, 0, 0, ...

1, 0, 0, 0, 0, 0, 0, ...

-1, 0, 0, 0, 0, 0, 0, ... .

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.

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).

The corresponding autosequence of first kind, 0 followed by 1's, is A057427(n).

The Akiyama-Tanigawa transform applied to a(n) yields a(n).

(End)

Harmonic or factorial (base) expansion of e, cf. MathWorld link. - M. F. Hasler, Nov 25 2018

Decimal expansion of 19/90. - Elmo R. Oliveira, Aug 09 2024

Links

• Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
• Eric Weisstein's World of Mathematics, Harmonic Expansion
• Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A036262, A054978, A027642, A002445, A174419.

Programs

Haskell

a054977 0 = 2; a054977 n = 1
a054977_list = 2 : repeat 1  -- Reinhard Zumkeller, May 07 2012

Magma

ContinuedFraction((1+Sqrt(5))^2/4); // G. C. Greubel, Nov 26 2018

Mathematica

A054977[1]:=2;
A054977[n_]:=1; (* Enrique Pérez Herrero, May 16 2010 *)
PadRight[{2},120,{1}] (* Harvey P. Dale, Mar 30 2018 *)

PARI

a(n)=if(n,1,2) \\ Charles R Greathouse IV, Mar 23 2016

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

Python

def A054977(n):
    return 1 if n else 2 # Chai Wah Wu, Dec 20 2018

Sage

continued_fraction(golden_ratio^2) # G. C. Greubel, Nov 26 2018

Formula

a(n) = A027642(2*n+1). - Enrique Pérez Herrero, Jul 17 2010

G.f.: (2-x)/(1-x). - Wolfdieter Lang, Oct 05 2014

Sum_{k>=1} a(n)/n! = exp(1). - G. C. Greubel, Nov 26 2018