A007395 Constant sequence: the all 2's sequence.

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Continued fraction for 1 + sqrt(2). - Philippe Deléham, Nov 14 2006

a(n) = A213999(n,1). - Reinhard Zumkeller, Jul 03 2012

The least witness function W(k) is defined for odd composite numbers k. The sequence W(k) does not have its own entry in the OEIS because W(k) = 2 for all k with 9 <= k < 2047; then W(2047)=3. Cf. A089105. - N. J. A. Sloane, Sep 17 2014

a(n) = A254858(n-1,1). - Reinhard Zumkeller, Feb 09 2015

a(n) = number of permutations of length n+2 having exactly one ascent such that the first element the permutation is 2. - Ran Pan, Apr 20 2015

With alternating signs, this is the sequence of determinants of the 3 X 3 matrices m with m(i,j) = Fibonacci(n+i+j-2)^2. - Michel Marcus, Dec 23 2015

For p = prime(n+2), a(n) = ord_p(H_(p-1)), where ord_p denotes the p-adic valuation and H_i = 1 + 1/2 + ... + 1/i is a harmonic sum, except for n = 1944 and n = 157504, where ord_p(H_(p-1)) = 3, and any other term of A088164 that may exist (see Conrad link). The sequence a(n) = ord_p(H_(p-1)) does not have its own entry in the OEIS. - Felix Fröhlich, Mar 16 2016

This sequence is the only infinite bounded sequence of positive integers such that a(n) = (a(n-1) + a(n-2)) / gcd(a(n-1), a(n-2)) for all n >= 2. - Bernard Schott, Dec 28 2018

References

• Titu Andreescu and Dorin Andrica, Number Theory, Birkhäuser, 2009, from 1999 Russian Mathematical Olympiad, p. 347.
• Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, NY, 2000, p. 6.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• 25th All-Russian Mathematical Olympiad, Grade 10, Problem 2, p. 2, 1999.
• Tobias Boege and Thomas Kahle, Construction Methods for Gaussoids, arXiv:1902.11260 [math.CO], 2019.
• Keith Conrad, The p-adic growth of harmonic sums
• Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
• R. H. Hardin, Binary arrays with both rows and cols sorted, symmetries
• Tanya Khovanova, Recursive Sequences
• Mihai Prunescu, On other two representations of the C-recursive integer sequences by terms in modular arithmetic, arXiv:2406.06436 [math.NT], 2024. See p. 18.
• Mihai Prunescu and Lorenzo Sauras-Altuzarra, On the representation of C-recursive integer sequences by arithmetic terms, arXiv:2405.04083 [math.LO], 2024. See p. 16.
• Mihai Prunescu and Joseph M. Shunia, On modular representations of C-recursive integer sequences, arXiv:2502.16928 [math.NT], 2025. See p. 5.
• Aram Tangboonduangjit and Thotsaporn Thanatipanonda, Determinants Containing Powers of Generalized Fibonacci Numbers, arXiv:1512.07025 [math.CO], 2015.
• Eric Weisstein's World of Mathematics, Hamiltonian Cycle
• 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 to divisibility sequences
• Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A000004, A000012, A002061, A010701, A089105, A165900.

Cf. A213999, A254858.

Programs

Haskell

a007395 = const 2
a007395_list = repeat 2  -- Reinhard Zumkeller, May 07 2012

Mathematica

Table[2, {105}]

Maxima

makelist(2,n,0,30); /* Martin Ettl, Nov 09 2012 */

PARI

a(n) = 2 \\ Charles R Greathouse IV, Apr 07 2012

Python

def A007395(n): return 2 # Chai Wah Wu, Nov 10 2022

Formula

G.f.: 2/(1-x), and e.g.f.: 2*e^x. - Mohammad K. Azarian, Dec 22 2008

a(n) = A000005(A000040(n)). - Omar E. Pol, Feb 28 2018

a(n) = A002061(n) - A165900(n). - Torlach Rush, Feb 21 2019