A048878 -id:A048878 - cp's OEIS frontend

A134451 Ternary digital root of n.

0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

Reinhard Zumkeller, Oct 27 2007

Continued fraction expansion of sqrt(3) - 1. - N. J. A. Sloane, Dec 17 2007. Cf. A040001, A048878/A002530.
Minimum number of terms required to express n as a sum of odd numbers.
Shadow transform of even numbers A005843. - Michel Marcus, Jun 06 2013
From Jianing Song, Nov 01 2022: (Start)
For n > 0, a(n) is the minimal gap of distinct numbers coprime to n. Proof: denote the minimal gap by b(n). For odd n we have A058026(n) > 0, hence b(n) = 1. For even n, since 1 and -1 are both coprime to n we have b(n) <= 2, and that b(n) >= 2 is obvious.
The maximal gap is given by A048669. (End)

			n=42: A007089(42) = '1120', A053735(42) = 1+1+2+0 = 4,
A007089(4)='11', A053735(4)=1+1=2: therefore a(42) = 2.
0.732050807568877293527446341... = 0 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, May 31 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Digital Root.
Eric Weisstein's World of Mathematics, Ternary.

Cf. A000010, A055034, A134452, A160390 (decimal expansion).
Apart from a(0) the same as A040001.
Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1).

a134451 = until (< 3) a053735
-- Reinhard Zumkeller, May 12 2011

A134451:=n->((n+1) mod 2)+2*signum(n)-1; seq(A134451(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

Table[Mod[n + 1, 2] + 2 Sign[n] - 1, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 06 2013 *)

a(n) = if(!n, 0, 2 - n%2) \\ Andrew Howroyd, Dec 08 2024

a(n) = n if n <= 2, otherwise a(A053735(n)).
a(A005408(n)) = 1; a(A005843(n)) = 2 for n>0;
a(n) = 0 if n=0, otherwise A000034(n-1).
a(n) = ((n+1) mod 2) + 2*sign(n) - 1. - Wesley Ivan Hurt, Dec 06 2013
Multiplicative with a(2^e) = 2, a(p^e) = 1 for odd prime p. - Andrew Howroyd, Aug 06 2018
a(0) = A055034(1) / A000010(1), a(n) = A000010(n+1) / A055034(n+1), n>1. - Torlach Rush, Oct 29 2019
Dirichlet g.f.: zeta(s)*(1+1/2^s). - Amiram Eldar, Jan 01 2023

A110527 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.

Original entry on oeis.org

0, 1, 8, 29, 128, 537, 2280, 9653, 40896, 173233, 733832, 3108557, 13168064, 55780809, 236291304, 1000946021, 4240075392, 17961247585, 76085065736, 322301510525, 1365291107840, 5783465941881, 24499154875368

Offset: 0

Creighton Dement, Jul 24 2005

A048878(n) = a(n) + a(n+1). Compare with A110526.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5,1).

Cf. A110526, A110528, A033887, A001076, A049661, A033887.

seriestolist(series(-x*(1+5*x)/((1+x)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 1lesseq[(- 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj')(+ .5'i + .5i' + .5'jj' + .5'kk')], apart from initial term.

LinearRecurrence[{3,5,1},{0,1,8},30] (* Harvey P. Dale, Feb 12 2015 *)

x='x+O('x^50); concat(0, Vec(-x*(1+5*x)/((1+x)*(x^2+4*x-1)))) \\ G. C. Greubel, Aug 30 2017

G.f.: -x*(1+5*x)/((1+x)*(x^2+4*x-1)).
a(n) = (-1)^n + 3*A001076(n) - A015448(n). - Ehren Metcalfe, Nov 18 2017
a(n) = (-1)^n + 2*A110526(n) + A110679(n-2) for n >= 2. - Yomna Bakr and Greg Dresden, May 25 2024

A153884 a(n) = ((7 + sqrt(5))^n - (7 - sqrt(5))^n)/(2*sqrt(5)).

Original entry on oeis.org

1, 14, 152, 1512, 14480, 136192, 1269568, 11781504, 109080064, 1008734720, 9322763264, 86134358016, 795679428608, 7349600247808, 67884508610560, 627000709644288, 5791091556155392, 53487250561826816, 494013479394738176

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009

Fifth binomial transform of A048878.

lim_{n -> infinity} a(n)/a(n-1) = 7 + sqrt(5) = 9.236067977499789696....

G. C. Greubel, Table of n, a(n) for n = 1..1000
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Index entries for linear recurrences with constant coefficients, signature (14,-44).

Cf. A002163 (decimal expansion of sqrt(5)), A048878.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((7+r)^n-(7-r)^n)/(2*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 04 2009

I:=[1,14]; [n le 2 select I[n] else 14*Self(n-1)-44*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 01 2016

Join[{a=1,b=14},Table[c=14*b-44*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
LinearRecurrence[{14, -44}, {1, 14}, 25] (* or *) Table[((7 + sqrt(5))^n - (7 - sqrt(5))^n)/(2*sqrt(5)) , {n,0,25}] (* G. C. Greubel, Aug 31 2016 *)

Vec(x/(1-14*x+44*x^2) + O(x^99)) \\ Altug Alkan, Sep 01 2016

From Philippe Deléham, Jan 03 2009: (Start)
a(n) = 14*a(n-1) - 44*a(n-2) for n>1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 14*x + 44*x^2). (End)
E.g.f.: sinh(sqrt(5)*x)*exp(7*x)/sqrt(5). - Ilya Gutkovskiy, Sep 01 2016

Extended beyond a(7) by Klaus Brockhaus, Jan 04 2009

Edited by Klaus Brockhaus, Oct 11 2009

cp's OEIS Frontend

A134451 Ternary digital root of n.

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A110527 a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.

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Maple

Mathematica

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Formula

A153884 a(n) = ((7 + sqrt(5))^n - (7 - sqrt(5))^n)/(2*sqrt(5)).

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A110527 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.