A094014 -id:A094014 - cp's OEIS frontend

A094015 Expansion of (1+4x)/(1-8x^2).

1, 4, 8, 32, 64, 256, 512, 2048, 4096, 16384, 32768, 131072, 262144, 1048576, 2097152, 8388608, 16777216, 67108864, 134217728, 536870912, 1073741824, 4294967296, 8589934592, 34359738368, 68719476736, 274877906944

Offset: 0

Paul Barry, Apr 21 2004

Row sums of triangle A135838. - Gary W. Adamson, Dec 01 2007

Row sums of triangle A152842. - Reinhard Zumkeller, May 01 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,8).

Cf. A010685, A094014, A135838, A152842.

a094015 = sum . a152842_row  -- Reinhard Zumkeller, May 01 2014

[2*8^Floor((n-1)/2)*(3+(-1)^n): n in [0..30]]; // G. C. Greubel, Nov 22 2021

a:=n->mul(3-(-1)^j,j=1..n):seq(a(n),n=0..25); # Zerinvary Lajos, Dec 13 2008

Table[8^Floor[n/2]*Mod[4^n, 5], {n, 0, 30}] (* G. C. Greubel, Nov 22 2021 *)

[8^(n//2)*(4^n%5) for n in (0..30)] # G. C. Greubel, Nov 22 2021

a(n) = 2^(3*n/2)*(1 + sqrt(2) + (-1)^n*(1 - sqrt(2)))/2.
a(n) = (1/4)*(3 + (-1)^n)*8^floor((n+1)/2). - Paul Barry, Jul 14 2004
a(n) = (1 + sqrt(2))*(2*sqrt(2))^n/2 + (1 - sqrt(2))*(-2*sqrt(2))^n/2. Third binomial transform is A002315 (NSW numbers). - Paul Barry, Jul 17 2004
a(n) = 2^A007494(n). - Paul Barry, Aug 18 2007
a(n) = A016116(n+1)*A000079(n). - R. J. Mathar, Jul 08 2009
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = 8^floor(n/2)*A010685(n). - G. C. Greubel, Nov 22 2021

A113836 a(n) = Sum[2^(A001651(i-1)-1), {i,1,n}].

Original entry on oeis.org

1, 3, 11, 27, 91, 219, 731, 1755, 5851, 14043, 46811, 112347, 374491, 898779, 2995931, 7190235, 23967451, 57521883, 191739611, 460175067, 1533916891, 3681400539, 12271335131, 29451204315, 98170681051, 235609634523

Offset: 1

Artur Jasinski, Jan 27 2006

nonn

From Reinhard Zumkeller, Feb 22 2010: (Start)
For n>1: a(n)=A173593(3*n-5): terms of A173593 ending with digits '11' in binary representation;
for n>0: a(n)=A033129(3*n-1); a(n+1)-a(n)=ABS(A094014(n)). (End)

			a(2) = 2^(A001651(0)-1) + 2^(A001651(1)-1) = 2^0 + 2^1 = 3

Cf. A001651.

a = {}; s = 0; For[n = 1, n < 40, n++, If[Length[Intersection[{Mod[n, 3]}, {1,2}]] > 0, s = s + 2^(n - 1); AppendTo[a, s]]]; a

Empirical g.f.: x*(2*x+1) / ((x-1)*(8*x^2-1)). - Colin Barker, Sep 01 2013

Edited by Stefan Steinerberger, Jul 23 2007

A135536 a(n) = 8*a(n-2), with a(0) = 7, a(1) = 14.

Original entry on oeis.org

7, 14, 56, 112, 448, 896, 3584, 7168, 28672, 57344, 229376, 458752, 1835008, 3670016, 14680064, 29360128, 117440512, 234881024, 939524096, 1879048192, 7516192768, 15032385536, 60129542144, 120259084288, 481036337152

Offset: 0

Paul Curtz, Feb 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,8).

Table[(7/4)*( (2 + Sqrt[2]) + (-1)^n*(2 - Sqrt[2]) )*(Sqrt[2])^(3*n), {n,0,25}] (* or *) LinearRecurrence[{0,8},{7,14}, 25] (* G. C. Greubel, Oct 18 2016 *)

a(n)=([0,1; 8,0]^n*[7;14])[1,1] \\ Charles R Greathouse IV, Oct 18 2016

a(n) = b(3*n) + b(3*n + 1) + b(3*n + 2), where b(n) = A135530(n) [previous name].
a(n) = 7 * abs(A094014(n)).
O.g.f.: 7*(1 + 2*x)/(1 - 8*x^2). - R. J. Mathar, Feb 23 2008
From G. C. Greubel, Oct 18 2016: (Start)
a(n) = (7/4)*( (2 + sqrt(2)) + (-1)^n*(2 - sqrt(2)) )*(sqrt(2))^(3*n).
a(n) = 8*a(n-2).
E.g.f.: (7/2)*( 2*cosh(2*sqrt(2)*x) + sqrt(2)*sinh(2*sqrt(2)*x) ). (End)

More terms from R. J. Mathar, Feb 23 2008

New name from G. C. Greubel, Oct 18 2016

A235202 Numbers written in an alternating binary-then-quaternary base.

Original entry on oeis.org

1, 10, 11, 20, 21, 30, 31, 100, 101, 110, 111, 120, 121, 130, 131, 1000, 1001, 1010, 1011, 1020, 1021, 1030, 1031, 1100, 1101, 1110, 1111, 1120, 1121, 1130, 1131, 2000, 2001, 2010, 2011, 2020, 2021, 2030, 2031, 2100

Offset: 1

Jeremy Gardiner, Jan 04 2014

Mixed-radix number representation produced by a serial counter with generating sequence (1, 3, 1, 3, ...) = A010684.
Places reading from the right have values (1, 2, 8, 16, 64, 128, ...) = unsigned A094014.
Conjecture: This sequence interpreted as quaternary (base 4) numbers gives A126001 (hence a simplified scheme for computing that sequence).

			a(15) = 131 since 15 = 1*1+3*2+1*8.

Jeremy Gardiner, Table of n, a(n) for n = 1..1024

Cf. A109827 (Numbers written in an alternating binary-then-ternary base).

A368043 Triangle read by rows: T(n, k) = 2^(n + k).

Original entry on oeis.org

1, 2, 4, 4, 8, 16, 8, 16, 32, 64, 16, 32, 64, 128, 256, 32, 64, 128, 256, 512, 1024, 64, 128, 256, 512, 1024, 2048, 4096, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144

Offset: 0

Peter Luschny, Dec 09 2023

			[0]  [  1]
[1]  [  2,   4]
[2]  [  4,   8,  16]
[3]  [  8,  16,  32,    64]
[4]  [ 16,  32,  64,   128,  256]
[5]  [ 32,  64,  128,  256,  512, 1024]
[6]  [ 64, 128,  256,  512, 1024, 2048,  4096]
[7]  [128, 256,  512, 1024, 2048, 4096,  8192, 16384]
[8]  [256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536]

Cf. A000079 (T(n,0)), A004171 (T(n,n-1)), A000302 (T(n,n)), A171476 (row sums), A003683 (alternating row sums), A134353 (antidiagonal sums), A001018 (T(2n, n)), A094014 (T(n, n/2)), A002697.

Array[2^Range[#,2#]&,10,0] (* Paolo Xausa, Dec 09 2023 *)

from functools import cache
@cache
def T_row(n: int) -> list[int]:
    if n == 0: return [1]
    row = T_row(n - 1) + [0]
    for k in range(n): row[k] *= 2
    row[n] = row[n - 1] * 2
    return row
for n in range(11): print(T_row(n))

G.f.: 1/((1 - 2*x)*(1 - 4*x*y)). - Stefano Spezia, Dec 09 2023

cp's OEIS Frontend

A094015 Expansion of (1+4*x)/(1-8*x^2).

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A113836 a(n) = Sum[2^(A001651(i-1)-1), {i,1,n}].

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A135536 a(n) = 8*a(n-2), with a(0) = 7, a(1) = 14.

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A235202 Numbers written in an alternating binary-then-quaternary base.

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A368043 Triangle read by rows: T(n, k) = 2^(n + k).

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A094015 Expansion of (1+4x)/(1-8x^2).