A159741 -id:A159741 - cp's OEIS frontend

A175165 a(n) = 32*(2^n - 1).

0, 32, 96, 224, 480, 992, 2016, 4064, 8160, 16352, 32736, 65504, 131040, 262112, 524256, 1048544, 2097120, 4194272, 8388576, 16777184, 33554400, 67108832, 134217696, 268435424, 536870880

Offset: 0

Reinhard Zumkeller, Feb 28 2010

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n - 1): A000225 (m=1), A000918 (m=2), A068156 (m=3), A028399 (m=4), A068293 (m=6), A159741 (m=8), A175164 (m=16), this sequence (m=32), A175166 (m=64).

Cf. A173787.

I:=[0,32]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

32(2^Range[0,30] -1) (* or *) LinearRecurrence[{3,-2},{0,32},30] (* Harvey P. Dale, Mar 23 2015 *)

def A175165(n): return (1<Chai Wah Wu, Jun 27 2023

[32*(2^n -1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = 2^(n+5) - 32.
a(n) = A173787(n+5, 5).
a(n) = 3*a(n-1) - 2*a(n-2); a(0)=0, a(1)=32. - Vincenzo Librandi, Dec 28 2010
From G. C. Greubel, Jul 08 2021: (Start)
G.f.: 32*x/((1-x)*(1-2*x)).
E.g.f.: 32*(exp(2*x) - exp(x)). (End)

A161770 n 1's followed by three 0's.

Original entry on oeis.org

1000, 11000, 111000, 1111000, 11111000, 111111000, 1111111000, 11111111000, 111111111000, 1111111111000, 11111111111000, 111111111111000, 1111111111111000, 11111111111111000, 111111111111111000, 1111111111111111000, 11111111111111111000, 111111111111111111000

Offset: 1

Jaroslav Krizek, Jun 18 2009

Sequence A159741 written in base 2.

Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A105279, A124166, A159741.

Table[FromDigits[PadLeft[{0,0,0},n,1]],{n,4,20}] (* Harvey P. Dale, Aug 07 2023 *)

a(n) = 1000*(10^n - 1)/9 = 1000*A002275(n).
G.f.: 1000*x/((10*x-1)*(x-1)).
From Elmo R. Oliveira, Jun 18 2025: (Start)
E.g.f.: 1000*exp(x)*(exp(9*x) - 1)/9.
a(n) = 100*A105279(n) = 10*A124166(n).
a(n) = 11*a(n-1) - 10*a(n-2) for n > 2. (End)

Edited by Charles R Greathouse IV, Oct 12 2009

A246168 a(n) = 2^n - 10.

Original entry on oeis.org

-9, -8, -6, -2, 6, 22, 54, 118, 246, 502, 1014, 2038, 4086, 8182, 16374, 32758, 65526, 131062, 262134, 524278, 1048566, 2097142, 4194294, 8388598, 16777206, 33554422, 67108854, 134217718, 268435446, 536870902, 1073741814, 2147483638

Offset: 0

Vincenzo Librandi, Aug 18 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form 2^n-k: A000079 (k=0), A000225 (k=1), A000918 (k=2), A036563 (k=3), A028399 (k=4), A168616 (k=5), A131130 (k=6), A048490 (k=7), A159741 (k=8), A185346 (k=9), this sequence (k=10).

Magma
```
[2^n-10: n in [0..40]];
```

Table[2^n - 10, {n, 0, 35}] (* or *) CoefficientList[Series[(-9 + 19 x)/(1 - 3 x + 2 x^2), {x, 0, 35}], x]
LinearRecurrence[{3,-2},{-9,-8},50] (* Harvey P. Dale, Jan 11 2024 *)

vector(50, n, 2^(n-1)-10) \\ Derek Orr, Aug 18 2014

G.f.: (-9+19*x)/(1-3*x+2*x^2).
a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = A000079(n) - 10.
From Elmo R. Oliveira, Dec 21 2023: (Start)
a(n) = 2*a(n-1) + 10 for n>0.
E.g.f.: exp(x)*(exp(x) - 10). (End)

A036982 a(n)=[ aa(n-1)+b ]/p^r, where a=2.001, b=3.2, p=2 and p^r is the highest power of p dividing [ aa(n-1)+b ].

Original entry on oeis.org

1, 5, 13, 29, 61, 125, 253, 509, 1021, 1023, 1025, 1027, 1029, 1031, 1033, 1035, 1037, 1039, 1041, 1043, 1045, 1047, 1049, 1051, 1053, 1055, 1057, 1059, 1061, 1063, 1065, 1067, 1069, 1071, 1073, 1075, 1077, 1079, 1081, 1083, 1085, 1087, 1089, 1091, 1093

Offset: 0

Yasutoshi Kohmoto

nonn

A cyclic sequence. The period is 6767.

If an array is made of columns of -nacci sequences, fibo-, tribo- etc. all starting w. 1,1,2 etc, the nw to se diagonals can be extended by computation. The above minus the first digit is diagonal 4. See A159741 for details. [From Al Hakanson (hawkuu(AT)gmail.com), Apr 20 2009]

Cf. A029580.

More terms from James Sellers, Aug 08 2000

A172252 a(n) = 4*2^n - 9.

Original entry on oeis.org

-1, 7, 23, 55, 119, 247, 503, 1015, 2039, 4087, 8183, 16375, 32759, 65527, 131063, 262135, 524279, 1048567, 2097143, 4194295, 8388599, 16777207, 33554423, 67108855, 134217719, 268435447, 536870903, 1073741815, 2147483639, 4294967287, 8589934583, 17179869175, 34359738359

Offset: 1

Artur Jasinski, Jan 29 2010

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A159741, A185346, A206853.

Table[4 2^n - 9, {n, 1, 100}]
LinearRecurrence[{3,-2},{-1,7},30] (* Harvey P. Dale, May 27 2021 *)

my(x='x+O('x^34)); Vec(x*(10*x-1)/((x-1)*(2*x-1))) \\ Elmo R. Oliveira, Jun 15 2025

a(n) = 2*a(n-1) + 9, a(1)= -1. - Vincenzo Librandi, Mar 20 2011
For n >= 3, a(n) = 8<+>(n+2), where operation <+> is defined in A206853. - Vladimir Shevelev, Feb 17 2012
From Elmo R. Oliveira, Jun 15 2025: (Start)
G.f.: x*(10*x-1)/((x-1)*(2*x-1)).
E.g.f.: 5 + exp(x)*(4*exp(x) - 9).
a(n) = A185346(n+2) = 4*A000225(n) - 5.
a(n) = A159741(n-1) - 1 for n > 1. (End)

More terms from Elmo R. Oliveira, Jun 15 2025

A341419 a(0) = 1, a(1) = 1, a(2^(n-1)..2^n-1) = fwht(0..2^(n-2)). Here "fwht" is the fast Walsh-Hadamard transform with natural ordering and without multiplication of any factors.

Original entry on oeis.org

1, 1, 2, 0, 4, 2, 0, -2, 8, 6, 8, -2, 0, -2, -8, -2, 16, 14, 24, -2, 32, 14, -8, -18, 0, -2, -8, -2, -32, -18, -8, 14, 32, 30, 56, -2, 96, 46, -8, -50, 128, 94, 120, -34, -32, -50, -136, -18, 0, -2, -8, -2, -32, -18, -8, 14, -128, -98, -136, 30, -32, 14, 120, 46, 64, 62

Offset: 0

Thomas Scheuerle, Mar 24 2021

This sequence is a rough integer-valued approximation to one of the nontrivial solutions to f(n) = a*fwht(f(n)).

Thomas Scheuerle, Table of n, a(n) for n = 0..16383

Cf. A159741, A175165.

function a = A341419(max_n)
a(1) = 1;
a(2) = 1;
    while length(a) < max_n
        w = fwht(a,[],'hadamard')*length(a);
        %w = myfwht(a); % own implementation for documentation purpose
        a = [a w];
    end
end
function w = myfwht(in)
    h = 1;
    while h < length(in)
        for i = 1:h*2:length(in)
            for j = i:i+h-1
                x = in(j);
                y = in(j+h);
                in(j) = x+y;
                in(j+h) = x-y;
            end
        end
        h = h*2;
    end
    w = in;
end

a(2^n) = 2^n.
a(2^n + 1) = 2^n-2 for n > 0.
a(2^n + 2) = 8*(2^(n-2) - 1) = A159741(n-2) for n > 1.
a(2^n + 3) = -2 for n > 1.
a(2^n + 4) = 32*(2^(n-3) - 1) = A175165(n-3) for n > 2.
a(2^n + 5) = 2*(2^n - 9) for n > 2.
a(2^n + 6) = -8 for n > 2.
a(2^n + 7) = -2*(8 * 2^(n-3) - 7) for n > 2.
a(2^n + 8) = 64*(2^(n-3) - 2) for n > 3.

cp's OEIS Frontend

A175165 a(n) = 32*(2^n - 1).

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A161770 n 1's followed by three 0's.

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A246168 a(n) = 2^n - 10.

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A036982 a(n)=[ a*a(n-1)+b ]/p^r, where a=2.001, b=3.2, p=2 and p^r is the highest power of p dividing [ a*a(n-1)+b ].

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A172252 a(n) = 4*2^n - 9.

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Mathematica

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Extensions

A341419 a(0) = 1, a(1) = 1, a(2^(n-1)..2^n-1) = fwht(0..2^(n-2)). Here "fwht" is the fast Walsh-Hadamard transform with natural ordering and without multiplication of any factors.

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Formula

A036982 a(n)=[ aa(n-1)+b ]/p^r, where a=2.001, b=3.2, p=2 and p^r is the highest power of p dividing [ aa(n-1)+b ].