A037496 -id:A037496 - cp's OEIS frontend

A037497 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,0,2.

1, 4, 18, 73, 292, 1170, 4681, 18724, 74898, 299593, 1198372, 4793490, 19173961, 76695844, 306783378, 1227133513, 4908534052, 19634136210, 78536544841, 314146179364, 1256584717458, 5026338869833, 20105355479332, 80421421917330

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,1,-4).

Other bases: A037496, A037498, A037499, A037500, A037501, A037502, A037503.

I:=[1, 4, 18, 73]; [n le 4 select I[n] else 4*Self(n-1)+Self(n-3)-4*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012

LinearRecurrence[{4, 0, 1, -4}, {1, 4, 18, 73}, 40] (* or *) CoefficientList[Series[(1 + 2 x^2)/((1 - x)(1 - 4 x) (1 + x + x^2)),{x,0,40}],x] (* Vincenzo Librandi, Jun 22 2012 *)

print([2*4**n//7 for n in range(1, 25)]) # Karl V. Keller, Jr., Sep 22 2020

From Vincenzo Librandi, Jun 22 2012: (Start)
G.f.: x*(1+2*x^2)/((1-x)*(1-4*x)*(1+x+x^2)).
a(n) = 4*a(n-1) + a(n-3) - 4*a(n-4). (End)
a(n) = floor(2*4^n/7). - Karl V. Keller, Jr., Sep 22 2020

A037503 Decimal expansion of a(n) is given by the first n terms of the periodic sequence with initial period 1,0,2.

Original entry on oeis.org

1, 10, 102, 1021, 10210, 102102, 1021021, 10210210, 102102102, 1021021021, 10210210210, 102102102102, 1021021021021, 10210210210210, 102102102102102, 1021021021021021

Offset: 1

Clark Kimberling

Index entries for linear recurrences with constant coefficients, signature (10,0,1,-10).

Other bases: A037496, A037497, A037498, A037499, A037500, A037501, A037502.

Vec(x*(1+2*x^2)/((x-1)*(10*x-1)*(1+x+x^2)) + O(x^25)) \\ Jinyuan Wang, Apr 14 2020

G.f.: x*(1+2*x^2) / ( (x-1)*(10*x-1)*(1+x+x^2) ). - R. J. Mathar, Aug 12 2013

a(n) = 10*a(n-1) + a(n-3) - 10*a(n-4). - Wesley Ivan Hurt, Sep 22 2020

A369635 Numbers in whose base 3-representation every two consecutive digits and every three consecutive digits are distinct.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 11, 15, 19, 21, 34, 46, 59, 65, 102, 140, 177, 202, 308, 420, 532, 606, 925, 1261, 1598, 1820, 2775, 3785, 4794, 5467, 8327, 11355, 14383, 16401, 24982, 34066, 43151, 49205, 74946, 102200, 129453, 147622, 224840, 306600, 388360, 442866

Offset: 1

Clark Kimberling, Feb 26 2024

In other words, the ternary expansion of the number does not contain any string xx or xxx.
The first eleven terms of this sequence comprise the base-3 xenodrome, A023798.
Ordered union of {0}, A037496, A037504, A037512, and A037520.

			The base-3 representation of 7 is 21, in which every two consecutive digits are distinct, so 7 is a term of the sequence.
The base-3 representation of 532 is 201201, in which every 3 consecutive digits are distinct, so 532 is a term of the sequence.

Cf. A023798, A037496, A037504, A037512, and A037520.

s1 = LinearRecurrence[{3, 0, 1, -3}, {0, 1, 3, 11}, 30]  (* A037496 *)
s2 = LinearRecurrence[{3, 0, 1, -3}, {0, 1, 5, 15}, 30]  (* A037504 *)
s3 = LinearRecurrence[{3, 0, 1, -3}, {0, 2, 6, 19}, 30]  (* A037512 *)
s4 = LinearRecurrence[{3, 0, 1, -3}, {0, 7, 21, 65}, 30] (* A037520 *)
s = Union[s1, s2, s3, s4]

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A037497 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,0,2.

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A037503 Decimal expansion of a(n) is given by the first n terms of the periodic sequence with initial period 1,0,2.

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A369635 Numbers in whose base 3-representation every two consecutive digits and every three consecutive digits are distinct.

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Mathematica