A033140 -id:A033140 - cp's OEIS frontend

A037521 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 2, 1, 0.

2, 9, 36, 146, 585, 2340, 9362, 37449, 149796, 599186, 2396745, 9586980, 38347922, 153391689, 613566756, 2454267026, 9817068105, 39268272420, 157073089682, 628292358729, 2513169434916, 10052677739666, 40210710958665

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).

[Floor(4^(n+1)/7) : n in [1..30]]; // Vincenzo Librandi, Jun 25 2011

seq(floor(4^(n+1)/7),n=1..30); # Mircea Merca, Dec 26 2010

Table[FromDigits[PadRight[{}, n, {2, 1, 0}], 4], {n, 30}] (* Harvey P. Dale, Jul 03 2017 *)

a(n) = floor(4^(n+1)/7). - Mircea Merca, Dec 26 2010
G.f.: x*(2+x) / ( (1-x)*(1-4*x)*(1+x+x^2) ).
a(n) = 4*a(n-1) + a(n-3) - 4*a(n-4).
a(n) = 4^(n+1)/7 - 1/3 - (-1)^(n mod 3)*A167380(4 + (n mod 3))/21 = 2*A033140(n) + A033140(n-1). - R. J. Mathar, Jan 08 2011

A037604 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

Original entry on oeis.org

1, 6, 27, 109, 438, 1755, 7021, 28086, 112347, 449389, 1797558, 7190235, 28760941, 115043766, 460175067, 1840700269, 7362801078, 29451204315, 117804817261, 471219269046, 1884877076187, 7539508304749, 30158033218998

Offset: 1

Clark Kimberling

Convolution of A000302 with A010882. - Philippe Deléham, Mar 24 2013

Michael De Vlieger, Table of n, a(n) for n = 1..1661
Index entries for linear recurrences with constant coefficients, signature (4,0,1,-4).
Index entries for 2-automatic sequences.

Cf. A000302, A010882.

Rest@ CoefficientList[Series[x (1 + 2 x + 3 x^2)/((1 - x) (1 - 4 x) (1 + x + x^2)), {x, 0, 23}], x] (* Michael De Vlieger, Mar 19 2021 *)
Table[FromDigits[PadRight[{},n,{1,2,3}],4],{n,30}] (* or *) LinearRecurrence[{4,0,1,-4},{1,6,27,109},30] (* Harvey P. Dale, May 07 2023 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -4,1,0,4]^(n-1)*[1;6;27;109])[1,1] \\ Charles R Greathouse IV, Feb 13 2017

print([3*4**n//7 for n in range(1,24)]) # Karl V. Keller, Jr., Mar 18 2021

G.f.: x*(1+2x+3*x^2)/((1-x)*(1-4x)*(1+x+x^2)). - Philippe Deléham, Mar 24 2013
a(n) = 4*a(n-1) + a(n-3) -4*a(n-4) for n > 5, a(1) = 1, a(2) = 6, a(3) = 27, a(4) = 109, a(5) = 438. - Philippe Deléham, Mar 24 2013
a(n+2) = 6*A033140(n) + A191597(n+2). - Philippe Deléham, Mar 24 2013
A007090(a(n)) = A037610(n). - R. J. Mathar, Apr 27 2015
a(n) = floor(3*4^n/7). - Karl V. Keller, Jr., Mar 18 2021

A368345 a(n) = Sum_{k=0..n} 4^(n-k) * floor(k/3).

Original entry on oeis.org

0, 0, 0, 1, 5, 21, 86, 346, 1386, 5547, 22191, 88767, 355072, 1420292, 5681172, 22724693, 90898777, 363595113, 1454380458, 5817521838, 23270087358, 93080349439, 372321397763, 1489285591059, 5957142364244, 23828569456984, 95314277827944, 381257111311785

Offset: 0

Seiichi Manyama, Dec 22 2023

Index entries for linear recurrences with constant coefficients, signature (5,-4,1,-5,4).

Partial sums of A033140.
Column k=4 of A368343.
Cf. A097138.

a(n, m=3, k=4) = (k^(n+1)\(k^m-1)-(n+1)\m)/(k-1);

a(n) = a(n-3) + (4^(n-2) - 1)/3.
a(n) = 1/3 * Sum_{k=0..n} floor(4^k/21) = Sum_{k=0..n} floor(4^k/63).
a(n) = 5*a(n-1) - 4*a(n-2) + a(n-3) - 5*a(n-4) + 4*a(n-5).
G.f.: x^3/((1-x) * (1-4*x) * (1-x^3)).
a(n) = (floor(4^(n+1)/63) - floor((n+1)/3))/3.
E.g.f.: exp(-x/2)*(exp(3*x/2)*(4*exp(3*x) - 7 - 21*x) + 3*cos(sqrt(3)*x/2) + 9*sqrt(3)*sin(sqrt(3)*x/2))/189. - Stefano Spezia, Jun 07 2025

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A037521 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 2, 1, 0.

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A037604 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,2,3.

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A368345 a(n) = Sum_{k=0..n} 4^(n-k) * floor(k/3).

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