A037576 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,3.

1, 7, 29, 119, 477, 1911, 7645, 30583, 122333, 489335, 1957341, 7829367, 31317469, 125269879, 501079517, 2004318071, 8017272285, 32069089143, 128276356573, 513105426295, 2052421705181, 8209686820727, 32838747282909

Offset

1,2

Comments

Partial sums of A255465. - Klaus Purath, Mar 18 2021

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Rule 190

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Formula

G.f.: x*(1+3*x)/((1-x)*(1-4*x)*(1+x)). - Vincenzo Librandi, Jun 22 2012

a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3). - Vincenzo Librandi, Jun 22 2012

a(n) = (7*4^n + 3*(-1)^n - 10)/15. - Bruno Berselli, Jun 22 2012, corrected by Klaus Purath, Mar 18 2021.

a(n) = floor(7*4^n/15). - Karl V. Keller, Jr., Mar 09 2021

From Klaus Purath, Mar 18 2021: (Start)

a(n) = 16*a(n-2) - 3*(-1)^n + 10, assuming that a(0) = 0.

a(n) = 4*a(n-1) + 2 + (-1)^n.

a(n) = 5*a(n-1) - 4*a(n-2) + 2*(-1)^n, n > 2. (End)

Mathematica

CoefficientList[Series[(1+3*x)/((x-1)*(4*x-1)*(1+x)), {x, 0, 30}], x] (*or*) LinearRecurrence[{4, 1, -4}, {1, 7, 29}, 40] (* Vincenzo Librandi, Jun 22 2012 *)

Prog

(Magma) I:=[1, 7, 29]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-2)-4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012
(PARI) my(x='x+O('x^99)); Vec(x*(1+3*x)/((1-x)*(1-4*x)*(1+x))) \\ Altug Alkan, Sep 21 2018
(Python) print([7*4**n//15 for n in range(1, 30)]) # Karl V. Keller, Jr., Mar 09 2021

Crossrefs

Cf. A007090 (numbers in base 4), A037582 (decimal), A265688 (binary), A118111.

Sequence in context: A118171 A072261 A066744 * A327587 A055427 A048876

Adjacent sequences: A037573 A037574 A037575 * A037577 A037578 A037579

Keyword

nonn,base,easy

Author

Clark Kimberling

Status

approved