A140660 -id:A140660 - cp's OEIS frontend

A024037 a(n) = 4^n - n.

1, 3, 14, 61, 252, 1019, 4090, 16377, 65528, 262135, 1048566, 4194293, 16777204, 67108851, 268435442, 1073741809, 4294967280, 17179869167, 68719476718, 274877906925, 1099511627756, 4398046511083, 17592186044394, 70368744177641, 281474976710632, 1125899906842599

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).

Cf. numbers of the form k^n - n: A000325 (k=2), A024024 (k=3), this sequence (k=4), A024050 (k=5), A024063 (k=6), A024076 (k=7), A024089 (k=8), A024102 (k=9), A024115 (k=10), A024128 (k=11), A024141 (k=12).

Cf. A140660 (first differences).

[4^n - n: n in [0..35]]; // Vincenzo Librandi, May 13 2011

I:=[1, 3, 14]; [n le 3 select I[n] else 6*Self(n-1)-9*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 16 2013

Table[4^n - n, {n, 0, 30}] (* or *) CoefficientList[Series[(1 - 3 x + 5 x^2) / ((1 - 4 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 16 2013 *)

a(n)=4^n-n \\ Charles R Greathouse IV, Sep 24 2015

From Vincenzo Librandi, Jun 16 2013: (Start)
G.f.: (1 - 3*x + 5*x^2)/((1 - 4*x)*(1 - x)^2).
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3). (End)
E.g.f.: exp(x)*(exp(3*x) - x). - Elmo R. Oliveira, Sep 10 2024

A164346 a(n) = 3 * 4^n.

Original entry on oeis.org

3, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248, 211106232532992, 844424930131968

Offset: 0

Klaus Brockhaus, Aug 13 2009

Binomial transform of A000244 without initial 1.
Second binomial transform of A007283.
Third binomial transform of A010701.
Inverse binomial transform of A005053 without initial 1.
First differences of A024036. - Omar E. Pol, Feb 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000302 (powers of 4), A000244 (powers of 3), A007283 (3*2^n), A010701 (all 3's), A005053, A002001, A096045, A140660 (3*4^n+1), A002023 (6*4^n), A002063(9*4^n), A056120, A084509.

Magma
```
[ 3*4^n: n in [0..22] ];
```

3 4^Range[0,30]  (* Harvey P. Dale, Mar 11 2011 *)

A164346(n)=3*4^n \\ M. F. Hasler, Jul 28 2015

def A164346(n): return 3<<(n<<1) # Chai Wah Wu, Aug 30 2024

a(n) = 4*a(n-1) for n > 1; a(0) = 3.
G.f.: 3/(1-4*x).
a(n) = A002001(n+1). a(n) = A096045(n)+2. a(n) = A140660(n)-1.
a(n) = A002023(n)/2. a(n) = A002063(n)/3. a(n) = A056120(n+3)/9.
Apparently a(n) = A084509(n+3)/2.
a(n) = A110594(n+1), n>1. - R. J. Mathar, Aug 17 2009
a(n) = 3*A000302(n). - Omar E. Pol, Feb 18 2013
a(n) = A000079(2*n) + A000079(2*n+1). - M. F. Hasler, Jul 28 2015
E.g.f.: 3*exp(4*x). - G. C. Greubel, Sep 15 2017

A136412 a(n) = (5*4^n + 1)/3.

Original entry on oeis.org

2, 7, 27, 107, 427, 1707, 6827, 27307, 109227, 436907, 1747627, 6990507, 27962027, 111848107, 447392427, 1789569707, 7158278827, 28633115307, 114532461227, 458129844907, 1832519379627, 7330077518507, 29320310074027

Offset: 0

Paul Curtz, Mar 31 2008

An Engel expansion of 4/5 to the base b := 4/3 as defined in A181565, with the associated series expansion 4/5 = b/2 + b^2/(2*7) + b^3/(2*7*27) + b^4/(2*7*27*107) + .... Cf. A199115 and A140660. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Sequences of the form (m*4^n + 1)/3: A007583 (m=2), this sequence (m=5), A199210 (m=11), A199210 (m=11), A206373 (m=14).

Cf. A007302, A140660, A181565, A199115.

a136412 = (`div` 3) . (+ 1) . (* 5) . (4 ^)
-- Reinhard Zumkeller, Jun 17 2012

[(5*4^n+1)/3: n in [0..30]]; // Vincenzo Librandi, Nov 04 2011

LinearRecurrence[{5,-4}, {2,7}, 31] (* G. C. Greubel, Jan 19 2023 *)

a(n)=(5*4^n+1)/3 \\ Charles R Greathouse IV, Oct 07 2015

[(5*4^n+1)/3 for n in range(31)] # G. C. Greubel, Jan 19 2023

a(n) = 4*a(n-1) - 1.
a(n) = A199115(n)/3.
O.g.f.: (2-3*x)/((1-x)*(1-4*x)). - R. J. Mathar, Apr 04 2008
a(n) = 5*a(n-1) - 4*a(n-2). - Vincenzo Librandi, Nov 04 2011
E.g.f.: (1/3)*(5*exp(4*x) + exp(x)). - G. C. Greubel, Jan 19 2023

Formula in definition and more terms from R. J. Mathar, Apr 04 2008

A140683 a(n) = 3(-1)^(n+1)2^n - 1.

Original entry on oeis.org

-4, 5, -13, 23, -49, 95, -193, 383, -769, 1535, -3073, 6143, -12289, 24575, -49153, 98303, -196609, 393215, -786433, 1572863, -3145729, 6291455, -12582913, 25165823, -50331649, 100663295, -201326593, 402653183, -805306369, 1610612735, -3221225473

Offset: 0

Paul Curtz, Jul 11 2008

sign

Alternated reading of negative of A140660 and A140529.
The binomial transform yields -4 followed by the negative of A140657.
The inverse binomial transform yields essentially a signed version of A000244. - R. J. Mathar, Aug 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1, 2).

[3*(-1)^(n+1)*2^n-1: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011

Table[3(-1)^(n+1)2^n-1,{n,0,40}] (* or *) LinearRecurrence[{-1,2},{-4,5},40] (* Harvey P. Dale, May 26 2011 *)

a(2n) = -A140660(n). a(2n+1) = A140529(n).
a(n+1) - a(n) = (-1)^n*A005010(n). a(2n) + a(2n+1) = A096045(n).
a(n) = A140590(n+1) - 2*A140590(n).
O.g.f: (4-x)/((x-1)(2x+1)). - R. J. Mathar, Aug 02 2008
a(n) = -a(n-1) + 2*a(n-2); a(0)=-4, a(1)=5. - Harvey P. Dale, May 26 2011

Edited and extended by R. J. Mathar, Aug 02 2008

A199115 a(n) = 5*4^n+1.

Original entry on oeis.org

6, 21, 81, 321, 1281, 5121, 20481, 81921, 327681, 1310721, 5242881, 20971521, 83886081, 335544321, 1342177281, 5368709121, 21474836481, 85899345921, 343597383681, 1374389534721, 5497558138881, 21990232555521, 87960930222081

Offset: 0

Vincenzo Librandi, Nov 04 2011

An Engel expansion of 4/5 to the base 4 as defined in A181565, with the associated series expansion 4/5 = 4/6 + 4^2/(6*21) + 4^3/(6*21*81) + 4^4/(6*21*81*321) + ... . Cf. A136412 and A140660. - Peter Bala, Oct 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A136412, A140660, A199115.

Magma
```
[5*4^n+1: n in [0..30]];
```

5*4^Range[0,30]+1 (* or  *) LinearRecurrence[{5,-4},{6,21},30] (* Harvey P. Dale, Oct 19 2024 *)

a(n) = 3*A136412(n).
a(n) = 4*a(n-1)-3.
a(n) = 5*a(n-1)-4*a(n-2).
G.f.: 3*(2-3*x)/((1-x)*(1-4*x)). - Bruno Berselli, Nov 04 2011

A140788 a(n) = 6*4^n + 2.

Original entry on oeis.org

8, 26, 98, 386, 1538, 6146, 24578, 98306, 393218, 1572866, 6291458, 25165826, 100663298, 402653186, 1610612738, 6442450946, 25769803778, 103079215106, 412316860418, 1649267441666, 6597069766658, 26388279066626, 105553116266498, 422212465065986, 1688849860263938

Offset: 0

Paul Curtz, Jul 14 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A140660.

[6*4^n+2: n in [0..30]]; // Vincenzo Librandi, Aug 08 2011

6*4^Range[0,30]+2 (* or *) LinearRecurrence[{5,-4},{8,26},30] (* Harvey P. Dale, Apr 15 2019 *)

a(n) = 2*A140660(n).
From R. J. Mathar, Feb 07 2009: (Start)
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: 2*(4-7*x)/((1-x)*(1-4*x)). (End)
E.g.f.: 2*exp(x)*(3*exp(3*x) + 1). - Elmo R. Oliveira, Apr 02 2025

More terms from R. J. Mathar, Feb 07 2009

A197918 Pythagorean primes p such that for all primes q < p, p XOR q is not equal to p - q.

Original entry on oeis.org

2, 5, 17, 41, 73, 97, 137, 193, 257, 521, 577, 641, 1033, 1153, 2081, 2113, 4129, 7681, 8353, 8737, 9281, 10369, 10753, 12289, 16417, 17921, 18433, 21569, 25601, 32801, 32833, 36353, 37889, 38921, 39041, 40961, 50177, 53377, 65537, 131617, 133121, 136193, 139273, 139297, 139393, 147457, 163841

Offset: 1

Brad Clardy, Oct 24 2011

It is conjectured that with the exception of the first three terms (2,5,17) all of the terms are a subset of all primes p such that p XOR 22 = p + 22.

If the inequality in the definition is replaced with equality the result are the Mersenne primes A000668, which is equivalent to for all primes q

This sequence is apparently a subset of A081091 Primes of the form 2^i + 2^j + 1, i>j>0, with the added conditions that j <> 1 or 2, and if j can be written as 2n then i cannot be 2n+1. This removes A123250 Primes of form 2^n + 5 (or 2^n + 2^2 +1) for n>0, primes from A140660 3*4^n + 1 (or 2^(2n+1) + 2^(2n) + 1) for n>0, and A057733 Primes of form 2^n + 3 (2^n + 2^1 + 1) for n>1.

			5 is a Pythagorean prime (1^2 + 2^2) and a member since ((5 XOR 2) <> (5 - 2)) and ((5 XOR 3) <> (5 - 3)).
13 is a Pythagorean prime (2^2 + 3^2) however it is not a member because 5, a prime less than 13, (13 XOR 5) = (13 - 5).

Charles R Greathouse IV, Table of n, a(n) for n = 1..303

Cf. A000668, A057733, A081091, A123250, A140660, A214643.

XOR := func;
i:=0; k:=0; pn:=0;
for n:= 5 to 10000 by 4 do
       if IsPrime(n)  then  pn:=n;  end if;
       if (pn eq n) then k:=0;
           for j in [2 .. n-2] do
                if IsPrime(j)  then pj:=j;
                     if (XOR(pn,pj) ne pn-pj) then i+:=1;
                         else k+:=1;
                     end if;
                end if;
           end for;
       end if;
       if ((i ne 0) and (k eq 0))  then pn; end if;
       i:=0; k:=0;
end for;

forprime(p=2,1e6,if(p%4-3==0,next);forprime(q=2,p-1,if(bitxor(p,q)==p-q, next(2)));print1(p", ")) \\ Charles R Greathouse IV, Jul 31 2012

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cp's OEIS Frontend

A024037 a(n) = 4^n - n.

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A164346 a(n) = 3 * 4^n.

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A136412 a(n) = (5*4^n + 1)/3.

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A140683 a(n) = 3*(-1)^(n+1)*2^n - 1.

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A199115 a(n) = 5*4^n+1.

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A140788 a(n) = 6*4^n + 2.

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A197918 Pythagorean primes p such that for all primes q < p, p XOR q is not equal to p - q.

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A140683 a(n) = 3(-1)^(n+1)2^n - 1.