A025192 -id:A025192 - cp's OEIS frontend

A008776 Pisot sequences E(2,6), L(2,6), P(2,6), T(2,6).

2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886, 5083731656658, 15251194969974

Offset: 0

N. J. A. Sloane, David W. Wilson

Definitions of Pisot and related sequences:
Pisot sequence E(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2) + 1/2) = nearest integer to a(n-1)^2/a(n-2), with 0 < x < y.
Pisot sequence L(x, y): a(0) = x, a(1) = y, a(n) = ceiling(a(n-1)^2/a(n-2)).
Pisot sequence P(x, y): a(0) = x, a(1) = y, a(n) = ceiling(a(n-1)^2/a(n-2) - 1/2).
Pisot sequence T(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2)).
Pisot/Shallit sequence S(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2)+1).
A025192 is the main entry for the sequence of numbers 2*3^n.
Number of tilings of a 4 X (4n+4) rectangle into T tetrominoes.
Numbers n such that 3^n = n/2 mod n. Cf. A066601 3^n mod n. - Zak Seidov, Aug 26 2006, Nov 20 2008
For n >= 1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2,3} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007
a(n+1) is the number of compositions of n when there are 2 types of each natural number. - Milan Janjic, Aug 13 2010
2*Sum_{n>=2} 1/A083667(n) = 2*Sum_{n>=2} 2^(-n)*3^(-((n*(n-1))/2)) = Sum_{n>=1} 1/Product_{k=1..n} A008776(k) = Sum_{n>=1} 1/Product_{k=1..n} 2*3^k = 0.17609845431233461692099660022134... . - Alexander R. Povolotsky, Aug 08 2011
Number of monic squarefree polynomials over F_3 of degree n+1. - Charles R Greathouse IV, Feb 07 2012
a(n) is the sum of the elements of the n-th power of the matrix {{1, 2}, {2, 1}}. - Griffin N. Macris, Mar 25 2016
Let D(m) denote the set of divisors of a number m, and consider s1(m) and s2(m) the sums of those divisors that are congruent to 1 and 2 (mod 3) respectively. This sequence lists the numbers m such that s1(m) = 1 and s2(m) = 2. - Michel Lagneau, Feb 09 2017
a(n) is the multiplicative order of k modulo 3^(n+1), where k is any number congruent to 2 or 5 modulo 9. Note that for n > 0, k is a primitive root modulo 3^(n+1) if and only if k == 2, 5 (mod 9). - Jianing Song, Apr 20 2021

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 203).

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..200
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 26.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 170
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Craig Knecht, Sphinx tiling of a repetitive shape.
C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.
C. Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scu. Norm. Sup. Pisa 2 ser, vol 7. no 3-4 (1938) p 205-248.
Index entries for linear recurrences with constant coefficients, signature (3).

Apart from initial term, same as A025192.
Cf. A080643.
Cf. A000244.

List([0..30], n-> 2*3^n); # G. C. Greubel, Sep 11 2019

a008776 = (* 2) . (3 ^)
a008776_list = iterate (* 3) 2  -- Reinhard Zumkeller, Oct 19 2015

[2*3^n: n in [0..30]]; // G. C. Greubel, Sep 11 2019

# E(x,y) is f(n,x,y,1/2), T(x,y) is f(n,x,y,0), and S(x,y) is f(n,x,y,1).
f:=proc(n,x,y,r) option remember;
if n=0 then x
elif n=1 then y
else floor(f(n-1,x,y,r)^2/f(n-2,x,y,r) + r); fi; end;
[seq(f(n,2,6,1/2),n=0..30)];
# N. J. A. Sloane, Jul 30 2016

Table[EulerPhi[3^n], {n, 0, 100}] (* Artur Jasinski, Nov 19 2008 *)
Table[MatrixPower[{{1,2},{1,2}},n][[1]][[2]],{n,0,44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
NestList[3#&,2,50] (* Harvey P. Dale, Nov 28 2022 *)

a(n)=3^n<<1 \\ corrected by Michel Marcus, Aug 03 2015

def A008776(n): return 3**n<<1 # Chai Wah Wu, Apr 02 2025

[2*3^n for n in (0..30)] # G. C. Greubel, Sep 11 2019

a(n) = 2*3^n.
a(n) = 3*a(n-1).
G.f.: 2/(1-3*x). - Philippe Deléham, Oct 08 2007
a(n-1) = phi(3^n). - Artur Jasinski, Nov 19 2008
E.g.f.: 2*exp(3*x). - Mohammad K. Azarian, Jan 15 2009
From Paul Curtz, Jan 20 2009: (Start)
a(n) = A048473(n) + 1.
a(n) = A052919(n+1)-1.
a(n) = A115099(n) - 2.
a(n) = A100774(n) + 2. (End)
If p[i]=2, (i >= 1), and if A is Hessenberg matrix of order n defined by: A[i,j] = p[j-i+1], (i <= j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1)=det A. - Milan Janjic, Apr 29 2010
G.f.: ((1/2)/G(0)-1)/x^2 where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
G.f.: -G(0)/x where G(k) = 1 - 1/(1-2*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 25 2013
G.f.: (1 - 1/Q(0))/x where Q(k) = 1 - x*(2*k-2)/(1 - x*(2*k+5)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: W(0), where W(k) = 1 + 1/(1 - x*(2*k+3)/(x*(2*k+4) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013

Jasinski formula corrected by Charles R Greathouse IV, Feb 18 2011

A002326 Multiplicative order of 2 mod 2n+1.

Original entry on oeis.org

1, 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, 11, 20, 18, 28, 5, 10, 12, 36, 12, 20, 14, 12, 23, 21, 8, 52, 20, 18, 58, 60, 6, 12, 66, 22, 35, 9, 20, 30, 39, 54, 82, 8, 28, 11, 12, 10, 36, 48, 30, 100, 51, 12, 106, 36, 36, 28, 44, 12, 24, 110, 20, 100, 7, 14, 130, 18, 36, 68, 138, 46, 60, 28

Offset: 0

N. J. A. Sloane

In other words, least m > 0 such that 2n+1 divides 2^m-1.
Number of riffle shuffles of 2n+2 cards required to return a deck to initial state. A riffle shuffle replaces a list s(1), s(2), ..., s(m) with s(1), s((i/2)+1), s(2), s((i/2)+2), ... a(1) = 2 because a riffle shuffle of [1, 2, 3, 4] requires 2 iterations [1, 2, 3, 4] -> [1, 3, 2, 4] -> [1, 2, 3, 4] to restore the original order.
Concerning the complexity of computing this sequence, see for example Bach and Shallit, p. 115, exercise 8.
It is not difficult to prove that if 2n+1 is a prime then 2n is a multiple of a(n). But the converse is not true. Indeed, one can prove that a(2^(2t-1))=4t. Thus if n=2^(2t-1), where, for any m > 0, t=2^(m-1) then 2n is a multiple of a(n) while 2n+1 is a Fermat number which, as is well known, is not always a prime. It is an interesting problem to describe all composite numbers for which 2n is divisible by a(n). - Vladimir Shevelev, May 09 2008
For an algorithm of calculation of a(n) see author's comment in A179680. - Vladimir Shevelev, Jul 21 2010
From V. Raman, Sep 18 2012, Dec 10 2012: (Start)
If 2n+1 is prime, then the polynomial (x^(2n+1)+1)/(x+1) factors into 2n/a(n) polynomials of the same degree a(n) over GF(2).
If (x^(2n+1)+1)/(x+1) is irreducible over GF(2), then 2n+1 is prime, and 2 is a primitive root (mod 2n+1) (cf. A001122).
For all n > 0, a(n) is the degree of the largest irreducible polynomial factor for the polynomial (x^(2n+1)+1)/(x+1) over GF(2). (End)
a(n) is a factor of phi(2n+1) (A000010(2n+1)). - Douglas Boffey, Oct 21 2013
Conjecture: if p is an odd prime then a((p^3-1)/2) = p * a((p^2-1)/2). Because otherwise a((p^3-1)/2) < p * a((p^2-1)/2) iff a((p^3-1)/2) = a((p-1)/2) for a prime p. Equivalently p^3 divides 2^(p-1)-1, but no such prime p is known. - Thomas Ordowski, Feb 10 2014
A generalization of the previous conjecture: For each k>=2, if p is an odd prime then a(((p^(k+1))-1)/2) = p * a((p^k-1)/2). Computer testing of this generalized conjecture shows that there is no counterexample for k and p both up to 1000. - Ahmad J. Masad, Oct 17 2020
a(n) = a((N-1)/2), with odd N = 2*n+1 >= 3 (n >= 1), is also the primitive period length of (1/N) in binary notation: (1/N)2 = 0.repeat(a[1]a[2]...a[P(N)]), and P(N) = a((N-1)/2). E.g., N = 11 (n = 5), (1/11)_2 = 0.repeat(0001011101), with P(11) = 10 = a(5). Proof: Use a cyclic shift operation sigma (1 step to the left) on the cycle: sigma((1/N)_2) = .repeat(a[2]...a[P(N)]a[1]). Then one can prove for the composition sigma^[k] (k=0 is the identity map) written back in decimal notation the result (sigma^[k]((1/N)_2))_10 = (1/N)*2^k (mod N). E.g. N = 11, sigma^[2]((1/11)_2) = .repeat(0101110100), written in base 10 as 4/11, etc. Hence P(N) and the order of 2 modulo N coincide. - _Gary W. Adamson and Wolfdieter Lang, Oct 14 2020

			From _Vladimir Shevelev_, Oct 03 2017: (Start)
Our algorithm for the calculation of a(n) in the author's comment in A179680 (see also the Sage program below) could be represented in the form of a "finite continued fraction". For example let n = 8, 2*n+1 = 17. We have
    1 + 17
    ------- + 17
       2
    ------------- + 17
           2
    ------------------- + 17
              2
    -------------------------- = 1
                 32
Here the denominators are the A006519 of the numerators: A006519(1+17) = 2, A006519(9+17) = 2, A006519(13+17) = 2, A006519(15+17) = 32. Summing the exponents of these powers of 2, we obtain the required result: a(8) = 1 + 1 + 1 + 5 = 8. Indeed, we have (((1*32 - 17)*2 - 17)*2 - 17)*2 - 17 = 1. So 32*2*2*2 - 1 == 0 (mod 17), 2^8 - 1 == 0 (mod 17). In the general case, note that all "partial fractions" (which indeed are integers) are odd residues modulo 2*n+1 in the interval [1, 2*n-1]. It is easy to prove that the first 1 appears not later than in the n-th step. (End)

E. Bach and Jeffrey Shallit, Algorithmic Number Theory, I.
T. Folger, "Shuffling Into Hyperspace," Discover, 1991 (vol 12, no 1), pages 66-67.
M. Gardner, "Card Shuffles," Mathematical Carnival chapter 10, pages 123-138. New York: Vintage Books, 1977.
L. Lunelli and M. Lunelli, Tavola di congruenza a^n == 1 mod K per a=2,5,10, Atti Sem. Mat. Fis. Univ. Modena 10 (1960/61), 219-236 (1961).
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 146, Exer. 21.3
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche, Manon Stipulanti, and Jia-Yan Yao, Doubling modulo odd integers, generalizations, and unexpected occurrences, arXiv:2504.17564 [math.NT], 2025.
M. Baake, U. Grimm and J. Nilsson, Scaling of the Thue-Morse diffraction measure, arXiv preprint arXiv:1311.4371 [math-ph], 2013.
D. Bayer and P. Diaconis, Trailing the dovetail shuffle to its lair, Ann. Appl. Prob. 2 (2) (1992) 294-313.
Matthew Brand, Choosing 1 of N with and without lucky numbers, arXiv:1808.07994 [math.NT], 2018.
J. Brillhart, J. S. Lomont and P. Morton, Cyclotomic properties of the Rudin-Shapiro polynomials, J. Reine Angew. Math.288 (1976), 37--65. See Table 2. MR0498479 (58 #16589).
Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, arXiv:1412.8533 [math.CO], 2014.
Steve Butler, Persi Diaconis and R. L. Graham, The mathematics of the flip and horseshoe shuffles, The American Mathematical Monthly 123.6 (2016): 542-556.
A. J. C. Cunningham, On Binal Fractions, Math. Gaz., 4 (71) (1908), circa p. 266.
P. Diaconis, R. L. Graham, and W. M. Kantor, The mathematics of perfect shuffles, Adv. Appl. Math. 4 (2) (1983) 175-196
M. J. Gardner and C. A. McMahan, Riffling casino checks, Math. Mag., 50 (1) (1977), 38-41.
S. W. Golomb, Permutations by cutting and shuffling, SIAM Rev., 3 (1961), 293-297.
Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv preprint arXiv:1608.00862 [math.GM], 2016.
Torleiv Klove, On covering sets for limited-magnitude errors, Cryptogr. Commun. 8 (3) (2016) 415-433
V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems [in Russian], Problemy Peredachi Informatsii, 43 (No. 3, 2007), 39-53.
V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems, Problems of Information Transmission, September 2007, Volume 43, Issue 3, pp 199-212 (translated from Russian)
Yuan-Hsun Lo, Kenneth W. Shum, Wing Shing Wong and Yijin Zhang, Multichannel Conflict-Avoiding Codes of Weights Three and Four, arXiv:2009.11754 [cs.IT], 2020.
Jarkko Peltomäki and Aleksi Saarela, Standard words and solutions of the word equation X_1^2 ... X_n^2 = (X_1 ... X_n)^2, Journal of Combinatorial Theory, Series A (2021) Vol. 178, 105340. See also arXiv:2004.14657 [cs.FL], 2020.
Vladimir Shevelev, Gilberto Garcia-Pulgarin, Juan Miguel Velasquez-Soto and John H. Castillo, Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers, arXiv preprint arXiv:1206:0606 [math.NT], 2012.
Vladimir Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers, J. Integer Seq. 15 (2012) Article 12.7.7.
Eric Weisstein's World of Mathematics, Riffle Shuffle
Eric Weisstein's World of Mathematics, In-Shuffle
Eric Weisstein's World of Mathematics, Out-Shuffle
Eric Weisstein's World of Mathematics, Multiplicative Order
Wikipedia, Riffle Shuffle

Cf. A003571, A003573, A217469, A070667-A070683, A053447, A053451, A292239, A292265.
Cf. A024222, A006694 (number of cyclotomic cosets).
Cf. A014664 (order of 2 mod n-th prime).
Cf. A001122 (primes for which 2 is a primitive root).
Cf. A216838 (primes for which 2 is not a primitive root).
Cf. A000010, A000225, A005420, A006519, A007733, A025192, A048675, A056239, A179680.
Bisections give A274298, A274299.
Partial sums: A359147.

List([0..100],n->OrderMod(2,2*n+1)); # Muniru A Asiru, Feb 01 2019

import Data.List (findIndex)
import Data.Maybe (fromJust)
a002326 n = (+ 1) $ fromJust $
            findIndex ((== 0) . (`mod` (2 * n + 1))) $ tail a000225_list
-- Reinhard Zumkeller, Apr 22 2013

[ 1 ] cat [ Modorder(2, 2*n+1): n in [1..72] ]; // Klaus Brockhaus, Dec 03 2008

a := n -> `if`(n=0, 1, numtheory:-order(2, 2*n+1)):
seq(a(n), n=0..72);

Table[MultiplicativeOrder[2, 2*n + 1], {n, 0, 100}] (* Robert G. Wilson v, Apr 05 2011 *)

a(n)=if(n<0,0,znorder(Mod(2,2*n+1))) /* Michael Somos, Mar 31 2005 */

from sympy import n_order
[n_order(2, 2*n+1) for n in range(73)] # Hermann Stamm-Wilbrandt, Jul 27 2021

# From Peter Luschny, Oct 06 2017: (Start)
[Mod(2,n).multiplicative_order() for n in (0..145) if gcd(n,2) == 1]
# Algorithm from Vladimir Shevelev as described in A179680 and presented in Example.
def A002326VS(n):
    s, m, N = 0, 1, 2*n + 1
    while True:
        k = N + m
        v = valuation(k, 2)
        s += v
        m = k >> v
        if m == 1: break
    return s
[A002326VS(n) for n in (0..72)] # (End)

a((3^n-1)/2) = A025192(n). - Vladimir Shevelev, May 09 2008
Bisection of A007733: a(n) = A007733(2*n+1). - Max Alekseyev, Jun 11 2009
a((b(n)-1)/2) = n for odd n and even n such that b(n/2) != b(n), where b(n) = A005420(n). - Thomas Ordowski, Jan 11 2014
Note that a(2^n-1) = n+1 and a(2^n) = 2*(n+1). - Thomas Ordowski, Jan 16 2014
a(n) = A056239(A292239(n)) = A048675(A292265(n)). - Antti Karttunen, Oct 04 2017

More terms from David W. Wilson, Jan 13 2000

More terms from Benoit Cloitre, Apr 11 2003

A053735 Sum of digits of (n written in base 3).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 4, 3, 4, 5, 2, 3, 4, 3, 4, 5, 4, 5, 6, 1, 2, 3, 2, 3, 4, 3, 4, 5, 2, 3, 4, 3, 4, 5, 4, 5, 6, 3, 4, 5, 4, 5, 6, 5, 6, 7, 2, 3, 4, 3, 4, 5, 4, 5, 6, 3, 4, 5, 4, 5, 6, 5, 6, 7, 4, 5, 6, 5, 6, 7, 6, 7, 8, 1, 2, 3, 2, 3, 4, 3, 4, 5, 2, 3, 4, 3, 4, 5, 4, 5, 6, 3, 4, 5, 4, 5, 6

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2}, 1->{1,2,3}, 2->{2,3,4}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 2 + 0 + 2 = 4 because 20 is written as 202 base 3.
From _Omar E. Pol_, Feb 20 2010: (Start)
This can be written as a triangle with row lengths A025192 (see the example in the entry A000120):
0,
1,2,
1,2,3,2,3,4,
1,2,3,2,3,4,3,4,5,2,3,4,3,4,5,4,5,6,
1,2,3,2,3,4,3,4,5,2,3,4,3,4,5,4,5,6,3,4,5,4,5,6,5,6,7,2,3,4,3,4,5,4,5,6,3,...
where the k-th row contains a(3^k+i) for 0<=i<2*3^k and converges to A173523 as k->infinity. (End) [Changed conjectures to statements in this entry. - _Franklin T. Adams-Watters_, Jul 02 2015]
G.f. = x + 2*x^2 + x^3 + 2*x^4 + 3*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + x^9 + 2*x^10 + ...

T. D. Noe, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS, Vol. 12 (2009), Article 09.5.6.
F. M. Dekking, The Thue-Morse Sequence in Base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
Michael Gilleland, Some Self-Similar Integer Sequences.
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See pp. 11, 13.
Jan-Christoph Puchta and Jürgen Spilker, Altes und Neues zur Quersumme, Math. Semesterber, Vol. 49 (2002), pp. 209-226; preprint.
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Vladimir Shevelev, Compact integers and factorials, Acta Arith., Vol. 126, No. 3 (2007), pp. 195-236 (cf. p.205).
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A065363, A007089, A173523. See A134451 for iterations.
Cf. A003137, A138530.
Sum of digits of n written in bases 2-16: A000120, this sequence, A053737, A053824, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Related base-3 sequences: A006047, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1), A286585, A286632, A289813, A289814.

a053735 = sum . a030341_row
-- Reinhard Zumkeller, Feb 21 2013, Feb 19 2012

m=1; for u=0:104; sol(m)=sum(dec2base(u,3)-'0'); m=m+1;end
sol; % Marius A. Burtea, Jan 17 2019

[&+Intseq(n,3):n in [0..104]]; // Marius A. Burtea, Jan 17 2019

seq(convert(convert(n,base,3),`+`),n=0..100); # Robert Israel, Jul 02 2015

Table[Plus @@ IntegerDigits[n, 3], {n, 0, 100}] (* or *)
Nest[Join[#, # + 1, # + 2] &, {0}, 6] (* Robert G. Wilson v, Jul 27 2006 and modified Jul 27 2014 *)

{a(n) = if( n<1, 0, a(n\3) + n%3)}; /* Michael Somos, Mar 06 2004 */

A053735(n)=sumdigits(n,3) \\ Requires version >= 2.7. Use sum(i=1,#n=digits(n,3),n[i]) in older versions. - M. F. Hasler, Mar 15 2016

(define (A053735 n) (let loop ((n n) (s 0)) (if (zero? n) s (let ((d (mod n 3))) (loop (/ (- n d) 3) (+ s d)))))) ;; For R6RS standard. Use modulo instead of mod in older Schemes like MIT/GNU Scheme. - Antti Karttunen, Jun 03 2017

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(3n) = a(n), a(3n + 1) = a(n) + 1, a(3n + 2) = a(n) + 2.
a(n) = n - 2*Sum_{k>0} floor(n/3^k) = n - 2*A054861(n). (End)
a(n) = A062756(n) + 2*A081603(n). - Reinhard Zumkeller, Mar 23 2003
G.f.: (Sum_{k >= 0} (x^(3^k) + 2*x^(2*3^k))/(1 + x^(3^k) + x^(2*3^k)))/(1 - x). - Michael Somos, Mar 06 2004, corrected by Franklin T. Adams-Watters, Nov 03 2005
In general, the sum of digits of (n written in base b) has generating function (Sum_{k>=0} (Sum_{0 <= i < b} i*x^(i*b^k))/(Sum_{i=0..b-1} x^(i*b^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005
First differences of A094345. - Vladeta Jovovic, Nov 08 2005
a(A062318(n)) = n and a(m) < n for m < A062318(n). - Reinhard Zumkeller, Feb 26 2008
a(n) = A138530(n,3) for n > 2. - Reinhard Zumkeller, Mar 26 2008
a(n) <= 2*log_3(n+1). - Vladimir Shevelev, Jun 01 2011
a(n) = Sum_{k>=0} A030341(n, k). - Philippe Deléham, Oct 21 2011
G.f. satisfies G(x) = (x+2*x^2)/(1-x^3) + (1+x+x^2)*G(x^3), and has a natural boundary at |x|=1. - Robert Israel, Jul 02 2015
a(n) = A056239(A006047(n)). - Antti Karttunen, Jun 03 2017
a(n) = A000120(A289813(n)) + 2*A000120(A289814(n)). - Antti Karttunen, Jul 20 2017
a(0) = 0; a(n) = a(n - 3^floor(log_3(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 3*log(3)/2 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A007583 a(n) = (2^(2*n + 1) + 1)/3.

Original entry on oeis.org

1, 3, 11, 43, 171, 683, 2731, 10923, 43691, 174763, 699051, 2796203, 11184811, 44739243, 178956971, 715827883, 2863311531, 11453246123, 45812984491, 183251937963, 733007751851, 2932031007403, 11728124029611, 46912496118443, 187649984473771, 750599937895083

Offset: 0

Simon Plouffe

Let u(k), v(k), w(k) be the 3 sequences defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)-w(k), v(k+1)=u(k)-v(k)+w(k), w(k+1)=-u(k)+v(k)+w(k); let M(k)=Max(u(k),v(k),w(k)); then a(n)=M(2n)=M(2n-1). - Benoit Cloitre, Mar 25 2002
Also the number of words of length 2n generated by the two letters s and t that reduce to the identity 1 by using the relations ssssss=1, tt=1 and stst=1. The generators s and t along with the three relations generate the dihedral group D6=C2xD3. - Jamaine Paddyfoot (jay_paddyfoot(AT)hotmail.com) and John W. Layman, Jul 08 2002
Binomial transform of A025192. - Paul Barry, Apr 11 2003
Number of walks of length 2n+1 between two adjacent vertices in the cycle graph C_6. Example: a(1)=3 because in the cycle ABCDEF we have three walks of length 3 between A and B: ABAB, ABCB and AFAB. - Emeric Deutsch, Apr 01 2004
Numbers of the form 1 + Sum_{i=1..m} 2^(2*i-1). - Artur Jasinski, Feb 09 2007
Prime numbers of the form 1+Sum[2^(2n-1)] are in A000979. Numbers x such that 1+Sum[2^(2n-1)] is prime for n=1,2,...,x is A127936. - Artur Jasinski, Feb 09 2007
Related to A024493(6n+1), A131708(6n+3), A024495(6n+5). - Paul Curtz, Mar 27 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-6, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1)*charpoly(A,2). - Milan Janjic, Feb 21 2010
Number of toothpicks in the toothpick structure of A139250 after 2^n stages. - Omar E. Pol, Feb 28 2011
Numbers whose binary representation is "10" repeated (n-1) times with "11" appended on the end, n >= 1. For example 171 = 10101011 (2). - Omar E. Pol, Nov 22 2012
a(n) is the smallest number for which A072219(a(n)) = 2*n+1. - Ramasamy Chandramouli, Dec 22 2012
An Engel expansion of 2 to the base b := 4/3 as defined in A181565, with the associated series expansion 2 = b + b^2/3 + b^3/(3*11) + b^4/(3*11*43) + .... Cf. A007051. - Peter Bala, Oct 29 2013
The positive integer solution (x,y) of 3*x - 2^n*y = 1, n>=0, with smallest x is (a(n/2), 2) if n is even and (a((n-1)/2), 1) if n is odd. - Wolfdieter Lang, Feb 15 2014
The smallest positive number that requires at least n additions and subtractions of powers of 2 to be formed. See Puzzling StackExchange link. - Alexander Cooke Jul 16 2023

H. W. Gould, Combinatorial Identities, Morgantown, 1972, (1.77), page 10.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..170
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Cecilia Bebeacua, Toufik Mansour, Alexander Postnikov, and Simone Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
Greg Bell, Austin Lawson, Neil Pritchard, and Dan Yasaki, On locally infinite Cayley graphs of the integers, arXiv preprint arXiv:1711.00809 [math.GT], 2017. See lambda_2.
Phillip G. Bradford, Efficient Exact Paths For Dyck and semi-Dyck Labeled Path Reachability, arXiv:1802.05239 [cs.DS], 2018.
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D, arXiv:1507.04803 [math.CO], 2015.
Ernesto Estrada and José A. de la Pena, From Integer Sequences to Block Designs via Counting Walks in Graphs, arXiv preprint arXiv:1302.1176 [math.CO], 2013.
Ernesto Estrada and José A. de la Pena, Integer sequences from walks in graphs, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 35.
Sungpyo Hong and Jin Ho Kwak, Regular fourfold coverings with respect to the identity automorphism, J. Graph Theory, 17 (1993), 621-627.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 893
Dmitry Kamenetsky, A magic grasshopper, Puzzling StackExchange, 2023.
Wolfdieter Lang, On Collatz' Words, Sequences and Trees, arXiv preprint arXiv:1404.2710 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.11.7.
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
Nathan Mihm, Optimal Addition-Subtraction Chains, Bachelor Honors Thesis, Univ. Minnesota-Twin Cities (2025). See p. 5.
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1.
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. also A006054, A006356, A005578, A075680.
Partial sums of A081294.
Cf. A002450, A004171, A007051, A083065, A083066, A083884.
Cf. A000978, A000979, A124400, A124401, A127936, A127955, A127956, A127957, A127958.
Cf. location of records in A007302.

List([0..25], n-> (2^(2*n+1) + 1)/3); # G. C. Greubel, Dec 25 2019

a007583 = (`div` 3) . (+ 1) . a004171
-- Reinhard Zumkeller, Jan 09 2013

[(2^(2*n+1) + 1)/3: n in [0..30] ]; // Vincenzo Librandi, Apr 28 2011

a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-1 od: seq(a[n], n=0..23); # Zerinvary Lajos, Feb 22 2008, with correction by K. Spage, Aug 20 2014
A007583 := proc(n)
    (2^(2*n+1)+1)/3 ;
end proc: # R. J. Mathar, Feb 19 2015

(* From Michael De Vlieger, Aug 22 2016 *)
Table[(2^(2n+1) + 1)/3, {n, 0, 23}]
Table[1 + 2Sum[4^k, {k, 0, n-1}], {n, 0, 23}]
NestList[4# -1 &, 1, 23]
Table[Sum[Binomial[n+k, 2k]/2^(k-n), {k, 0, n}], {n, 0, 23}]
CoefficientList[Series[(1-2x)/(1-5x+4x^2), {x, 0, 23}], x] (* End *)

a(n)=sum(k=-n\3,n\3,binomial(2*n+1,n+1+3*k))

a=1; for(n=1,23, print1(a,", "); a=bitor(a,3*a)) \\ K. Spage, Aug 20 2014

Vec((1-2*x)/(1-5*x+4*x^2) + O(x^30)) \\ Altug Alkan, Dec 08 2015

apply( {A007583(n)=2<<(2*n)\/3}, [0..25]) \\ M. F. Hasler, Nov 30 2021

[(2^(2*n+1) + 1)/3 for n in (0..25)] # G. C. Greubel, Dec 25 2019

a(n) = 2*A002450(n) + 1.
From Wolfdieter Lang, Apr 24 2001: (Start)
a(n) = Sum_{m = 0..n} A060920(n, m) = A002450(n+1) - 2*A002450(n).
G.f.: (1-2*x)/(1-5*x+4*x^2). (End)
a(n) = Sum_{k = 0..n} binomial(n+k, 2*k)/2^(k - n).
a(n) = 4*a(n-1) - 1, n > 0.
From Paul Barry, Mar 17 2003: (Start)
a(n) = 1 + 2*Sum_{k = 0..n-1} 4^k;
a(n) = A001045(2n+1). (End)
a(n) = A020988(n-1) + 1 = A039301(n+1) - 1 = A083584(n-1) + 2. - Ralf Stephan, Jun 14 2003
a(0) = 1; a(n+1) = a(n) * 4 - 1. - Regis Decamps (decamps(AT)users.sf.net), Feb 04 2004 (correction to lead index by K. Spage, Aug 20 2014)
a(n) = Sum_{i + j + k = n; 0 <= i, j, k <= n} (n+k)!/i!/j!/(2*k)!. - Benoit Cloitre, Mar 25 2004
a(n) = 5*a(n-1) - 4*a(n-2). - Emeric Deutsch, Apr 01 2004
a(n) = 4^n - A001045(2*n). - Paul Barry, Apr 17 2004
a(n) = 2*(A001045(n))^2 + (A001045(n+1))^2. - Paul Barry, Jul 15 2004
a(n) = left and right terms in M^n * [1 1 1] where M = the 3X3 matrix [1 1 1 / 1 3 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A002450(n+1) a(n)] E.g. a(3) = 43 since M^n * [1 1 1] = [43 85 43] = [a(3) A002450(4) a(3)]. - Gary W. Adamson, Dec 18 2004
a(n) = A072197(n) - A020988(n). - Creighton Dement, Dec 31 2004
a(n) = A139250(2^n). - Omar E. Pol, Feb 28 2011
a(n) = A193652(2*n+1). - Reinhard Zumkeller, Aug 08 2011
a(n) = Sum_{k = -floor(n/3)..floor(n/3)} binomial(2*n, n+3*k)/2. - Mircea Merca, Jan 28 2012
a(n) = 2^(2*(n+1)) - A072197(n). - Vladimir Pletser, Apr 12 2014
a(n) == 2*n + 1 (mod 3). Indeed, from Regis Decamps' formula (Feb 04 2004) we have a(i+1) - a(i) == -1 (mod 3), i= 0, 1, ..., n - 1. Summing, we have a(n) - 1 == -n (mod 3), and the formula follows. - Vladimir Shevelev, May 20 2015
For n > 0 a(n) = A133494(0) + 2 * (A133494(n) + Sum_{x = 1..n - 1}Sum_{k = 0..x - 1}(binomial(x - 1, k)*(A133494(k+1) + A133494(n-x+k)))). - J. Conrad, Dec 06 2015
a(n) = Sum_{k = 0..2n} (-2)^k == 1 + Sum_{k = 1..n} 2^(2k-1). - Bob Selcoe, Aug 21 2016
E.g.f.: (1 + 2*exp(3*x))*exp(x)/3. - Ilya Gutkovskiy, Aug 21 2016
A075680(a(n)) = 1, for n > 0. - Ralf Stephan, Jun 17 2025

A193722 Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion.

Original entry on oeis.org

1, 1, 2, 1, 5, 6, 1, 8, 21, 18, 1, 11, 45, 81, 54, 1, 14, 78, 216, 297, 162, 1, 17, 120, 450, 945, 1053, 486, 1, 20, 171, 810, 2295, 3888, 3645, 1458, 1, 23, 231, 1323, 4725, 10773, 15309, 12393, 4374, 1, 26, 300, 2016, 8694, 24948, 47628, 58320, 41553, 13122

Offset: 0

Clark Kimberling, Aug 04 2011

Suppose that p = p(n)*x^n + p(n-1)*x^(n-1) + ... + p(1)*x + p(0) is a polynomial and that Q is a sequence of polynomials
...
q(k,x)=t(k,0)*x^k+t(k,1)*x^(k-1)+...+t(k,k-1)*x+t(k,k),
...
for k=0,1,2,...  The Q-upstep of p is the polynomial given by
...
U(p) = p(n)*q(n+1,x) + p(n-1)*q(n,x) + ... + p(0)*q(1,x); note that q(0,x) does not appear.
...
Now suppose that P=(p(n,x)) and Q=(q(n,x)) are sequences of polynomials, where n indicates degree.  The fusion of P by Q, denoted by P**Q, is introduced here as the sequence W=(w(n,x)) of polynomials defined by w(0,x)=1 and w(n+1,x)=U(p(n,x)).
...
Strictly speaking, ** is an operation on sequences of polynomials.  However, if P and Q are regarded as numerical triangles (e.g., coefficients of polynomials), then ** can be regarded as an operation on numerical triangles.  In this case, row (n+1) of P**Q, for n >= 0, is given by the matrix product P(n)*QQ(n), where P(n)=(p(n,n)...p(n,n-1)......p(n,1), p(n,0)) and QQ(n) is the (n+1)-by-(n+2) matrix given by
...
q(n+1,0) .. q(n+1,1)........... q(n+1,n) .... q(n+1,n+1)
0 ......... q(n,0)............. q(n,n-1) .... q(n,n)
0 ......... 0.................. q(n-1,n-2) .. q(n-1,n-1)
...
0 ......... 0.................. q(2,1) ...... q(2,2)
0 ......... 0 ................. q(1,0) ...... q(1,1);
here, the polynomial q(k,x) is taken to be
q(k,0)*x^k + q(k,1)x^(k-1) + ... + q(k,k)*x+q(k,k-1); i.e., "q" is used instead of "t".
...
If s=(s(1),s(2),s(3),...) is a sequence, then the infinite square matrix indicated by
s(1)...s(2)...s(3)...s(4)...s(5)...
..0....s(1)...s(2)...s(3)...s(4)...
..0......0....s(1)...s(2)...s(3)...
..0......0.......0...s(1)...s(2)...
is the self-fusion matrix of s; e.g., A202453, A202670.
...
Example:  let p(n,x)=(x+1)^n and q(n,x)=(x+2)^n.  Then
  ...
w(0,x) = 1 by definition of W
w(1,x) = U(p(0,x)) = U(1) = p(0,0)*q(1,x) = 1*(x+2) = x+2;
w(2,x) = U(p(1,x)) = U(x+1) = q(2,x) + q(1,x) = x^2+5x+6;
w(3,x) = U(p(2,x)) = U(x^2+2x+1) = q(3,x) + 2q(2,x) + q(1,x) = x^3+8x^2+21x+18;
  ...
From these first 4 polynomials in the sequence P**Q, we can write the first 4 rows of P**Q when P, Q, and P**Q are regarded as triangles:
  1;
  1,  2;
  1,  5,  6;
  1,  8, 21, 18;
  ...
Generally, if P and Q are the sequences given by p(n,x)=(ax+b)^n and q(n,x)=(cx+d)^n, then P**Q is given by (cx+d)(bcx+a+bd)^n.
...
In the following examples, r(P**Q) is the mirror of P**Q, obtained by reversing the rows of P**Q.
...
..P...........Q.........P**Q.......r(P**Q)
(x+1)^n.....(x+1)^n.....A081277....A118800 (unsigned)
(x+1)^n.....(x+2)^n.....A193722....A193723
(x+2)^n.....(x+1)^n.....A193724....A193725
(x+2)^n.....(x+2)^n.....A193726....A193727
(x+2)^n.....(2x+1)^n....A193728....A193729
(2x+1)^n....(x+1)^n.....A038763....A136158
(2x+1)^n....(2x+1)^n....A193730....A193731
(2x+1)^n,...(x+1)^n.....A193734....A193735
...
Continuing, let u denote the polynomial x^n+x^(n-1)+...+x+1, and let Fibo[n,x] denote the n-th Fibonacci polynomial.
...
P.............Q.........P**Q.......r(P**Q)
Fib[n+1,x]...(x+1)^n....A193736....A193737
u.............u.........A193738....A193739
u**u..........u**u......A193740....A193741
...
Regarding A193722:
col 1 ..... A000012
col 2 ..... A016789
col 3 ..... A081266
w(n,n) .... A025192
w(n,n-1) .. A081038
...
Associated with "upstep" as defined above is "downstep" defined at A193842 in connection with fission.

			First six rows:
  1;
  1,   2;
  1,   5,   6;
  1,   8,  21,  18;
  1,  11,  45,  81,  54;
  1,  14,  78, 216, 297, 162;

Jinyuan Wang, Rows n=0..101 of triangle, flattened
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

Cf. A081277, A084938, A118800, A193649, A193723-A193741, A202453.

Flat(List([0..10], n-> List([0..n], k-> 3^(k-1)*( Binomial(n-1,k) + 2*Binomial(n,k) ) ))); # G. C. Greubel, Feb 18 2020

[3^(k-1)*( Binomial(n-1,k) + 2*Binomial(n,k) ): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 18 2020

fusion := proc(p, q, n) local d, k;
p(n-1,0)*q(n,x)+add(coeff(p(n-1,x),x^k)*q(n-k,x), k=1..n-1);
[1,seq(coeff(%,x,n-1-k), k=0..n-1)] end:
p := (n, x) -> (x + 1)^n; q := (n, x) -> (x + 2)^n;
A193722_row := n -> fusion(p, q, n);
for n from 0 to 5 do A193722_row(n) od; # Peter Luschny, Jul 24 2014

(* First program *)
z = 9; a = 1; b = 1; c = 1; d = 2;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193722 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193723 *)
(* Second program *)
Table[3^(k-1)*(Binomial[n-1,k] +2*Binomial[n,k]), {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2020 *)

T(n,k) = 3^(k-1)*(binomial(n-1,k) +2*binomial(n,k)); \\ G. C. Greubel, Feb 18 2020

def fusion(p, q, n):
    F = p(n-1,0)*q(n,x)+add(expand(p(n-1,x)).coefficient(x,k)*q(n-k,x) for k in (1..n-1))
    return [1]+[expand(F).coefficient(x,n-1-k) for k in (0..n-1)]
A193842_row = lambda k: fusion(lambda n,x: (x+1)^n, lambda n,x: (x+2)^n, k)
for n in range(7): A193842_row(n) # Peter Luschny, Jul 24 2014

Triangle T(n,k), read by rows, given by [1,0,0,0,0,0,0,0,...] DELTA [2,1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 04 2011
T(n,k) = 3*T(n-1,k-1) + T(n-1,k) with T(0,0)=T(1,0)=1 and T(1,1)=2. - Philippe Deléham, Oct 05 2011
T(n, k) = 3^(k-1)*( binomial(n-1,k) + 2*binomial(n,k) ). - G. C. Greubel, Feb 18 2020

A132019 Decimal expansion of Product_{k>=0} 1-1/(2*3^k).

Original entry on oeis.org

3, 8, 2, 6, 6, 3, 1, 9, 6, 6, 7, 9, 0, 3, 3, 0, 2, 3, 2, 8, 8, 9, 5, 5, 0, 3, 3, 5, 3, 3, 1, 9, 1, 3, 2, 2, 7, 9, 5, 3, 7, 7, 1, 9, 7, 3, 1, 2, 7, 6, 7, 1, 1, 8, 0, 5, 5, 1, 4, 9, 5, 3, 5, 4, 6, 7, 8, 6, 8, 7, 5, 2, 4, 4, 0, 8, 2, 7, 5, 9, 9, 2, 7, 0, 3, 5, 3, 6, 4, 7, 1, 8, 8, 7, 4, 2, 5, 1, 6, 5, 6, 4, 6

Offset: 0

Hieronymus Fischer, Aug 13 2007

			0.3826631966790330232889550...

Cf. A098844, A067080, A132026, A132027, A000217, A025192.

digits = 103; NProduct[1-1/(2*3^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/2, 1/3], 10, 120][[1]] (* Amiram Eldar, May 08 2023 *)

Equals lim inf_{n->oo} Product_{k=0..floor(log_3(n))} floor(n/3^k)*3^k/n.
Equals lim inf_{n->oo} A132027(n)/n^(1+floor(log_3(n)))*3^(1/2*(1+floor(log_3(n)))*floor(log_3(n))).
Equals lim inf_{n->oo} A132027(n)/n^(1+floor(log_3(n)))*3^A000217(floor(log_3(n))).
Equals (1/2)*exp(-Sum_{n>0} 3^(-n)*Sum_{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132027(n)/A132027(n+1).
Equals Product_{n>=1} (1 - 1/A025192(n)). - Amiram Eldar, May 08 2023

A016089 Numbers n such that n divides n-th Lucas number A000032(n).

Original entry on oeis.org

1, 6, 18, 54, 162, 486, 1458, 1926, 4374, 5778, 13122, 17334, 39366, 52002, 118098, 156006, 206082, 354294, 468018, 618246, 1062882, 1404054, 1854738, 2471058, 3188646, 4212162, 5564214, 7413174, 9565938, 12636486, 16692642, 22050774

Offset: 1

Robert G. Wilson v

nonn

Note that if n divides A000032(n) and p is an odd prime divisor of A000032(n), then pn divides A000032(pn) and, furthermore, p^k*n divides A000032(p^k*n) for every integer k>=0.
In particular, since 6 divides A000032(6) = 2*3^2, A016089 includes all terms of the geometric progression 2*3^k for k>0 (see A099856); since 18 divides A000032(18) = 2*3^3*107, A016089 includes all terms of the form 2*107^m*3^k for k>1 and m>=0; etc.
Terms of A016089 starting with 18 are multiples of 18. There are no other terms of the form 18p where p is prime, except for p=3 and p=107. - Alexander Adamchuk, May 11 2007

Lars Blomberg, Table of n, a(n) for n = 1..91
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 75.
C. Smyth, The terms in Lucas Sequences divisible by their indices, JIS 13 (2010) #10.2.4.

Cf. A000032, A000204, A025192, A008776.

Cf. A099856, A072378 = numbers n such that 12n divides Fibonacci(12n), A023172 = numbers n such that n divides Fibonacci(n).

a = 1; b = 3; Do[c = a + b; a = b; b = c; If[Mod[c, n] == 0, Print[n]], {n, 3, 2, 10^6}]

is(n)=(Mod([0,1;1,1],n)^n*[2;1])[1,1]==0 \\ Charles R Greathouse IV, Nov 04 2016

Extended and revised by Max Alekseyev, May 13 2007, May 15 2008, May 16 2008

A025177 Triangular array, read by rows: first differences in n,n direction of trinomial array A027907.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 2, 4, 4, 4, 2, 1, 1, 3, 7, 10, 12, 10, 7, 3, 1, 1, 4, 11, 20, 29, 32, 29, 20, 11, 4, 1, 1, 5, 16, 35, 60, 81, 90, 81, 60, 35, 16, 5, 1, 1, 6, 22, 56, 111, 176, 231, 252, 231, 176, 111, 56, 22, 6, 1, 1, 7, 29, 84, 189, 343, 518, 659, 714, 659, 518, 343

Offset: 0

Clark Kimberling

The Motzkin transforms of the rows starting (1, 2), (1, 3) and (1, 4), extended by zeros after their last element, are apparently in A026134, A026109 and A026110. - R. J. Mathar, Dec 11 2008

			               1
            1  0  1
         1  1  2  1  1
      1  2  4  4  4  2  1
   1  3  7 10 12 10  7  3  1
1  4 11 20 29 32 29 20 11  4  1

Georg Fischer, Table of n, a(n) for n = 0..2499 [rows 0..49; first 676 terms from _G. C. Greubel_]

Columns include A025178, A025179, A025180, A025181, A025182.

Cf. A024996, A025192 (row sums).

A025177 := proc(n,k)
    option remember;
    if k < 0 or k > 2*n then
        0;
    elif n = 0 then
        1 ;
    elif n = 1 then
        op(k+1,[1,0,1]) ;
    else
        procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc:
seq(seq(A025177(n,k),k=0..2*n),n=0..20)  ; # R. J. Mathar, Feb 25 2015

nmax = 10; CoefficientList[CoefficientList[Series[(1 - y*x)/(1 - x*(1 + y + y^2)), {x, 0, nmax}, {y, 0, 2*nmax}], x], y] // Flatten (* G. C. Greubel, May 22 2017; amended by Georg Fischer, Jun 24 2020 *)

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, if( n==1, [1,0,1][k+1], if( n==2, [1,1,2,1,1][k+1], T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)))))};

T(n,k)=polcoeff(Ser(polcoeff(Ser((1-y*z)/(1-z*(1+y+y^2)),y),k,y),z),n,z)

{T(n, k) = if( k<0 || k>2*n, 0, if( n==0, 1, polcoeff( (1 + x + x^2)^n, k) - polcoeff( (1 + x + x^2)^(n-1), k-1)))};

g=matrix(33,65);
for(n=0,32,for(k=0,2*n,g[n+1,k+1]=0));
g[1,1]=1;
g[2,1]=1;g[2,2]=0;g[2,3]=1;
g[3,1]=1;g[3,2]=1;g[3,3]=2;g[3,4]=1;g[3,5]=1;
for(n=0,2,for(k=0,2*n,print(n," ",k," ",g[n+1,k+1])))
for(n=3,32,g[n+1,1]=1;print(n," 1 1");g[n+1,2]=n-1;print(n," 2 ",n-1);for(k=2,2*n,g[n+1,k+1]=g[n,k-1]+g[n,k]+g[n,k+1];print(n," ",k," ",g[n+1,k+1])))
\\ Michael B. Porter, Feb 02 2010

T(n, k) = T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), starting with [1], [1, 0, 1].

G.f.: (1-y*z)/[1-z*(1+y+y^2)].

Edited by Ralf Stephan, Jan 09 2005

Offset corrected by R. J. Mathar, Feb 25 2015

A081277 Square array of unsigned coefficients of Chebyshev polynomials of the first kind.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 8, 4, 1, 7, 18, 20, 8, 1, 9, 32, 56, 48, 16, 1, 11, 50, 120, 160, 112, 32, 1, 13, 72, 220, 400, 432, 256, 64, 1, 15, 98, 364, 840, 1232, 1120, 576, 128, 1, 17, 128, 560, 1568, 2912, 3584, 2816, 1280, 256, 1, 19, 162, 816, 2688, 6048, 9408, 9984, 6912

Offset: 0

Paul Barry, Mar 16 2003

Rows include A011782, A001792, A001793, A001794, A006974.
Formatted as a triangular array, this is [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] (see construction in A084938 ). - Philippe Deléham, Aug 09 2005
Antidiagonal sums are in A025192. - Philippe Deléham, Dec 04 2006
Binomial transform of n-th row of the triangle (followed by zeros) = n-th row of the A142978 array and n-th column of triangle A104698. - Gary W. Adamson, Jul 17 2008
When formatted as a triangle, A038763=fusion of polynomial sequences (x+1)^n and (x+1)^n; see A193722 for the definition of fusion of two polynomial sequences or triangular arrays.  Row n of A038763, as a triangle, consists of coefficients of the product (x+1)*(x+2)^n. - Clark Kimberling, Aug 04 2011

			Rows begin
  1, 1,  2,   4,   8, ...
  1, 3,  8,  20,  48, ...
  1, 5, 18,  56, 160, ...
  1, 7, 32, 120, 400, ...
  1, 9, 50, 220, 840, ...
  ...
As a triangle:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  8,  4;
  1,  7, 18, 20,  8;

Robert Davis, Greg Simay, Further Combinatorics and Applications of Two-Toned Tilings, arXiv:2001.11089 [math.CO], 2020.

Cf. A079628.
Cf. A142978, A104698.
Cf. A167580 and A167591. - Johannes W. Meijer, Nov 23 2009
Cf. A053120 (antidiagonals give signed version) and A124182 (skewed version). - Mathias Zechmeister, Jul 26 2022

(* Program generates triangle A081277 as the self-fusion of Pascal's triangle *)
z = 8; a = 1; b = 1; c = 1; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A081277 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* abs val of A118800 *)
Factor[w[6, x]]
(* Clark Kimberling, Aug 04 2011 *)

T(n, k) = (n+2k)*binomial(n+k-1, k-1)*2^(n-1)/k, k > 0.
T(n, 0) defined by g.f. (1-x)/(1-2x). Other rows are defined by (1-x)/(1-2x)^n.
T(n, 0) = 0 if n < 0, T(0, k) = 0 if k < 0, T(0, 0) = T(1, 0) = 1, T(n, k) = T(n, k-1) + 2*T(n-1, k); for example, 160 = 48 + 2*56 for n = 4 and k = 2. -Philippe Deléham, Aug 12 2005
G.f. of the triangular interpretation: (-1+x*y)/(-1+2*x*y+x). - R. J. Mathar, Aug 11 2015

A038044 Shifts left under transform T where Ta is a DCONV a.

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 40, 80, 168, 340, 698, 1396, 2844, 5688, 11456, 22948, 46072, 92144, 184696, 369392, 739536, 1479232, 2959860, 5919720, 11842696, 23685473, 47376634, 94753940, 189519576, 379039152, 758102900, 1516205800

Offset: 1

Christian G. Bower

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms

Positions of odd terms are given by A003095. Other self-convolved sequences: A000108, A007460 - A007464, A025192, A061922, A062177.
Column k=1 of A144324 and A144823. - Alois P. Heinz, Nov 04 2012
Cf. A038040.
Cf. A000010.

import Data.Function (on)
a038044 n = a038044_list !! (n-1)
a038044_list = 1 : f 1 [1] where
   f x ys = y : f (x + 1) (y:ys) where
     y = sum $ zipWith ((*) `on` a038044) divs $ reverse divs
         where divs = a027750_row x
-- Reinhard Zumkeller, Jan 21 2014

with(numtheory); EIGENbyDIRCONV := proc(upto_n) local n,a,j,i,s,m; a := [1]; for i from 1 to upto_n do s := 0; m := convert(divisors(i),set); n := nops(m); for j from 1 to n do s := s+(a[m[j]]*a[m[(n-j)+1]]); od; a := [op(a),s]; od; RETURN(a); end;

dc[b_, c_] := Module[{p}, p[n_] := p[n] = Sum[b[d]*c[n/d], {d, If[n<0, {}, Divisors[n]]}]; p]; A[n_, k_] := Module[{f, b, t}, b[1] = dc[f, f]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], b[t-1]]]; f = Function[m, If[m == 1, 1, b[k][m-1]]]; f[n]]; a[n_] := A[n, 1]; Array[a, 40] (* Jean-François Alcover, Mar 20 2017, after A144324 *)

From Benoit Cloitre, Aug 29 2004: (Start)
a(n+1) = Sum_{d|n} a(d)*a(n/d), a(1) = 1.
a(prime(k)+1) = 2*a(prime(k));
a(n) is asymptotic to c*2^n where c=0.353030198... (End)
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 01 2019 [modified by Ilya Gutkovskiy, May 09 2019]
a(n+1) = Sum_{k=1..n} a(gcd(n,k))*a(n/gcd(n,k))/phi(n/gcd(n,k)) where phi = A000010. - Richard L. Ollerton, May 19 2021

cp's OEIS Frontend

A008776 Pisot sequences E(2,6), L(2,6), P(2,6), T(2,6).

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A002326 Multiplicative order of 2 mod 2n+1.

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A053735 Sum of digits of (n written in base 3).

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A007583 a(n) = (2^(2*n + 1) + 1)/3.

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A193722 Triangular array: the fusion of (x+1)^n and (x+2)^n; see Comments for the definition of fusion.

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