A004663 -id:A004663 - cp's OEIS frontend

A004642 Powers of 2 written in base 3.

1, 2, 11, 22, 121, 1012, 2101, 11202, 100111, 200222, 1101221, 2210212, 12121201, 102020102, 211110211, 1122221122, 10022220021, 20122210112, 111022121001, 222122012002, 1222021101011, 10221112202022, 21220002111121, 120210012000012, 1011120101000101, 2100010202000202

Offset: 0

N. J. A. Sloane

When n is odd, a(n) ends in 1, and when n is even, a(n) ends in 2, since 2^n is congruent to 1 mod 3 when n is odd and to 2 mod 3 when n is even. - Alonso del Arte Dec 11 2009
Sloane (1973) conjectured a(n) always has a 0 between the most and least significant digits if n > 15 (see A102483 and A346497).
Erdős (1978) conjectured that for n > 8 a(n) has at least one 2 (see link to Terry Tao's blog). - Dmitry Kamenetsky, Jan 10 2017

N. J. A. Sloane, The Persistence of a Number, J. Recr. Math. 6 (1973), 97-98.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Yagub N. Aliyev, Digits of powers of 2 in ternary numeral system, Notes on Number Theory and Discrete Mathematics, Vol. 29, No. 3 (2023), 474-485.
Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 67-70.
Donald L. Kreher and Douglas R. Stinson, On min-base palindromic representations of powers of 2, arXiv:2401.07351 [math.NT], 2024. See Table 4 p. 10.
Jeffrey C. Lagarias, Ternary Expansions of Powers of 2, Journal of the London Mathematical Society, Vol. 79, No. 3 (2009), pp. 562-588; arXiv preprint, arXiv:math/0512006 [math.DS], 2005-2008.
Terry Tao, The Collatz Conjecture, Littlewood-Offord theory, and powers of 2 and 3, 2011.
Eric Weisstein's World of Mathematics, Ternary.

Cf. A000079: powers of 2 written in base 10.
Cf. A004643, ..., A004655: powers of 2 written in base 4, 5, ..., 16.
Cf. A004656, A004658, A004659, ..., A004663: powers of 3 written in base 2, 4, 5, ..., 9.

[Seqint(Intseq(2^n, 3)): n in [0..30]]; // G. C. Greubel, Sep 10 2018

Table[FromDigits[IntegerDigits[2^n, 3]], {n, 25}] (* Alonso del Arte Dec 11 2009 *)

a(n)=fromdigits(digits(2^n,3)) \\ M. F. Hasler, Jun 23 2018

A096881 Expansion of g.f. (1 + 4x)/(1 - 17x^2).

Original entry on oeis.org

1, 4, 17, 68, 289, 1156, 4913, 19652, 83521, 334084, 1419857, 5679428, 24137569, 96550276, 410338673, 1641354692, 6975757441, 27903029764, 118587876497, 474351505988, 2015993900449, 8063975601796, 34271896307633, 137087585230532, 582622237229761, 2330488948919044

Offset: 0

Paul Barry, Jul 14 2004

Index entries for linear recurrences with constant coefficients, signature (0,17).

Cf. A004663, A026383, A016116.

I:=[1,4,17]; [n le 3 select I[n] else 17*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 26 2016

CoefficientList[Series[(1+4x)/(1-17x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {0,17},{1,4},30] (* Harvey P. Dale, Jan 21 2012 *)

Vec((1+4*x)/(1-17*x^2) + O(x^40)) \\ Michel Marcus, Jan 26 2016

a(n) = 3*a(n-1) + 4*a(n-2) + 17^floor((n-2)/2).
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2), k)*4^(n-2*k).
a(n) = 17*a(n-2), n>1. - Harvey P. Dale, Jan 21 2012
E.g.f.: cosh(sqrt(17)*x) + 4*sinh(sqrt(17)*x)/sqrt(17). - Stefano Spezia, Mar 31 2023

More terms from Stefano Spezia, Mar 31 2023

A096882 Expansion of g.f. (1 + 7x)/(1 - 50x^2).

Original entry on oeis.org

1, 7, 50, 350, 2500, 17500, 125000, 875000, 6250000, 43750000, 312500000, 2187500000, 15625000000, 109375000000, 781250000000, 5468750000000, 39062500000000, 273437500000000, 1953125000000000, 13671875000000000, 97656250000000000, 683593750000000000, 4882812500000000000

Offset: 0

Paul Barry, Jul 14 2004

Index entries for linear recurrences with constant coefficients, signature (0,50).

Cf. A004663, A026383, A016116, A096881.

a[n_]:=Sum[Binomial[Floor[n/2],k]7^(n-2k),{k,0,Floor[n/2]}]; Array[a,25,0] (* Stefano Spezia, Mar 31 2023 *)
LinearRecurrence[{0,50},{1,7},30] (* Harvey P. Dale, Sep 20 2024 *)

a(n) = 6*a(n-1) + 7*a(n-2) + 50^floor((n-2)/2).
a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2), k)*7^(n-2*k).
E.g.f.: cosh(5*sqrt(2)*x) + 7*sinh(5*sqrt(2)*x)/(5*sqrt(2)). - Stefano Spezia, Mar 31 2023

More terms from Stefano Spezia, Mar 31 2023

A096883 Expansion of (1+10x)/(1-101x^2).

Original entry on oeis.org

1, 10, 101, 1010, 10201, 102010, 1030301, 10303010, 104060401, 1040604010, 10510100501, 105101005010, 1061520150601, 10615201506010, 107213535210701, 1072135352107010, 10828567056280801, 108285670562808010

Offset: 0

Paul Barry, Jul 14 2004

Index entries for linear recurrences with constant coefficients, signature (0,101).

Cf. A004663, A026383, A016116, A096881, A096882.

a(n)=9a(n-1)+10a(n-2)+101^floor((n-2)/2); a(n)=sum{k=0..floor(n/2), binomial(floor(n/2), k)10^(n-2k) }.

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A004642 Powers of 2 written in base 3.

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A096881 Expansion of g.f. (1 + 4*x)/(1 - 17*x^2).

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A096882 Expansion of g.f. (1 + 7*x)/(1 - 50*x^2).

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A096883 Expansion of (1+10x)/(1-101x^2).

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A096881 Expansion of g.f. (1 + 4x)/(1 - 17x^2).

A096882 Expansion of g.f. (1 + 7x)/(1 - 50x^2).