A129868 -id:A129868 - cp's OEIS frontend

A156589 a(n) = 4^n - 2^n - 1.

-1, 1, 11, 55, 239, 991, 4031, 16255, 65279, 261631, 1047551, 4192255, 16773119, 67100671, 268419071, 1073709055, 4294901759, 17179738111, 68719214591, 274877382655, 1099510579199, 4398044413951, 17592181850111, 70368735789055

Offset: 0

M. F. Hasler, Feb 10 2009

Sequence A098845 lists indices of primes, i.e., a(n) prime <=> n=A098845(k) for some k.
Starting with n=2, binary numbers of the form (n-1)0(n) where n is the index and the number of 1's. It can also be formed by appending a 1 to the right of each term of A129868.
1/a(n) = Sum_{m>0} A000045(m)*2^(-n(m+1)) for n > 0. E.g., 1/a(4) = 0.0000 0001 0001 0010 0011 0101 1000 ... in binary. - Lee A. Newberg, Apr 12 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A000045, A098845, A129868.

[4^n-2^n-1: n in [0..30]]; // Vincenzo Librandi, Apr 13 2018

Table[4^n - 2^n - 1, {n, 0, 25}] (* Vincenzo Librandi, Apr 13 2018 *)

PARI
```
vector(99,n,4^n-2^n-1)
```

G.f.: ( 1-8*x+10*x^2 ) / ( (-1+x)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Oct 21 2014

A170940 4^n-2^n-2.

Original entry on oeis.org

0, 10, 54, 238, 990, 4030, 16254, 65278, 261630, 1047550, 4192254, 16773118, 67100670, 268419070, 1073709054, 4294901758, 17179738110, 68719214590, 274877382654, 1099510579198, 4398044413950, 17592181850110, 70368735789054, 281474959933438, 1125899873288190

Offset: 1

N. J. A. Sloane, Feb 13 2010

nonn

a(n) is also the number whose binary representation is the concatenation of n-1 1's, 0, n-1 1's and 0 (See example). [From Omar E. Pol, Mar 16 2010]

			Contribution from _Omar E. Pol_, Mar 16 2010: (Start)
n ...... a(n) written in base 2 ..... a(n)
1 ................ 0 ................ 0
2 ............... 1010 .............. 10
3 .............. 110110 ............. 54
4 ............. 11101110 ............ 238
5 ............ 1111011110 ........... 990
6 ........... 111110111110 .......... 4030
7 .......... 11111101111110 ......... 16254
8 ......... 1111111011111110 ........ 65278
9 ........ 111111110111111110 ....... 261630
10 ...... 11111111101111111110 ...... 1047550
(End)

Index entries for linear recurrences with constant coefficients, signature (7, -14, 8).

Cf. A170926.

Cf. A006516, A138148, A173521. [From Omar E. Pol, Mar 16 2010]

a(n)= 7*a(n-1) -14*a(n-2) +8*a(n-3) = 2*A129868(n-1). G.f.: 2*x^2*(-5+8*x)/((x-1) * (2*x-1) * (4*x-1)). [From R. J. Mathar, Feb 14 2010]

a(n) = 2*(A006516(n)-1) [From Omar E. Pol, Mar 16 2010]

A220236 Binary palindromic numbers with only two 0 bits, both in the middle.

Original entry on oeis.org

9, 51, 231, 975, 3999, 16191, 65151, 261375, 1047039, 4191231, 16771071, 67096575, 268410879, 1073692671, 4294868991, 17179672575, 68719083519, 274877120511, 1099510054911, 4398043365375, 17592179752959, 70368731594751, 281474951544831, 1125899856510975

Offset: 1

Alex Ratushnyak, Dec 08 2012

Binary expansion is 1001, 110011, 11100111, 1111001111, ...

Last digit of the decimal representation follows the pattern 9, 1, 1, 5, 9, 1, 1, 5, 9, ...

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A129868.

Table[2^(2n + 2) - 2^(n + 1) - 2^n - 1, {n, 25}] (* Alonso del Arte, Dec 08 2012 *)
LinearRecurrence[{7,-14,8},{9,51,231},30] (* Harvey P. Dale, Jan 24 2019 *)

for n in range(1,77):
    print (2**(2*n+2)-2**n-2**(n+1)-1),

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

a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3). G.f.: 3*x*(4*x-3) / ((x-1)*(2*x-1)*(4*x-1)). - Colin Barker, May 31 2013

A357574 a(n) is the maximum number of pairs that sum to a power of 2 in a set of n consecutive odd numbers.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 9, 11, 13, 15, 17, 19, 21, 24, 26, 29, 31, 34, 36, 39, 41, 44, 46, 49, 51, 54, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 98, 101, 104, 107, 110, 113, 116, 119, 122, 125, 128, 131, 134, 137, 140, 143, 146, 150, 153, 157, 160, 164, 167

Offset: 1

Thomas Scheuerle, Oct 04 2022

nonn

An optimal set delivering a(n) pairs summing to powers of 2 can be formed by n-A357409(n) first negative odd numbers and A357409(n) first positive odd numbers, that is, by the odd numbers in the interval [-2*(n-A357409(n))+1, 2*A357409(n)-1].
a(n) is in many cases equal to A347301(n) but there are some deviations: a(3) = 2 but A347301(3) = 3, a(29) = 62 but A347301(29) = 61, a(31) = 68 but A347301(32) = 67, a(33) = 74 but A347301(33) = 73, ... . Hence it appears that a(n) may be used as an improved lower bound for A352178(n) in many cases.
Conjecture: a(n+1) - a(n) = k, if n is even then A129868(k-1) < n < A129868(k), if n is odd then A020515(k) <= n < A020515(k+1).

			a(5) = 5 because A357409(5) = 4, for which the corresponding set {-1, 1, 3, 5, 7} produces 5 powers of 2: 1+3, 1+7, 3+5, 3-1, 5-1.

Thomas Scheuerle, a(n+1) - a(n) for n = 1..500
Max A. Alekseyev, On computing sets of integers with maximum number of pairs summing to powers of 2, arXiv:2303.02872 [math.CO], 2023.

Cf. A020515, A129868, A274089, A347301, A352178, A357409.

function a = A357574( max_n )
    a(1) = 0; q = [];
    for n = 1:max_n
        c = 0;
        for k = 0:n
            s = (2*([0:n]-k))+1;
            r = countpowtwo(s);
            if c < r
                c = r;
                q = s;
            end
        end
        a(n+1) = c;
    end
end
function c = countpowtwo(s)
    M = repmat(s, [length(s), 1]);
    M = M+M';
    M(M<=0) = 7;
    M = bitand(M, M-1);
    M = M + eye(size(M));
    c = length(find(M == 0))/2;
end

a(n) <= A352178(n).

a(n) >= n-1. This would be the maximum value that could be attained for a set of only positive odd numbers and size n.

Edited by Max Alekseyev, Mar 09 2023

A187560 a(n) = 4^(n+1)-2^n-1.

Original entry on oeis.org

2, 13, 59, 247, 1007, 4063, 16319, 65407, 261887, 1048063, 4193279, 16775167, 67104767, 268427263, 1073725439, 4294934527, 17179803647, 68719345663, 274877644799, 1099511103487, 4398045462527, 17592183947263, 70368739983359, 281474968322047, 1125899890065407

Offset: 0

Brad Clardy, Mar 25 2011

For n>0, binary numbers of the form (n+1)0 n, where n is the index value and the number of 1's. This can be formed by appending a leading 1 to the terms of A129868. It is also A156589 written in bit-reverse order.

			Binary values of the first 7 terms are 10, 1101, 111011, 11110111, 1111101111, 111111011111, 11111110111111.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A171499.

Table[4^(n+1)-2^n-1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{2,13,59},30] (* Harvey P. Dale, Feb 25 2013 *)

a(n)=4^(n+1)-2^n-1 \\ Charles R Greathouse IV, Nov 01 2015

a(n) = 4^(n+1)-2^n-1 = A171499(n)-1.
G.f.: ( -2+x+4*x^2 ) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Apr 09 2011
a(0)=2, a(1)=13, a(2)=59, a(n)=7*a(n-1)-14*a(n-2)+8*a(n-3). - Harvey P. Dale, Feb 25 2013
E.g.f.: exp(x)*(4*exp(3*x) - exp(x) - 1). - Stefano Spezia, Apr 11 2025

Terms a(21) and beyond from Andrew Howroyd, Feb 25 2018

A267090 Triangle read by rows: Fill an n X n square with 1's, except for 0's on the two main diagonals. Then T(n,k) is decimal equivalent of the k-th row (0<=k<=n).

Original entry on oeis.org

0, 0, 0, 2, 5, 2, 6, 9, 9, 6, 14, 21, 27, 21, 14, 30, 45, 51, 51, 45, 30, 62, 93, 107, 119, 107, 93, 62, 126, 189, 219, 231, 231, 219, 189, 126, 254, 381, 443, 471, 495, 471, 443, 381, 254, 510, 765, 891, 951, 975, 975, 951, 891, 765, 510

Offset: 0

Kival Ngaokrajang, Jan 10 2016

Inspired by A137932 and A042948.
Conjectures:
(i) The first column is A000225/2.
(ii) For even-n, T(n,n/2) = A129868.
(iii) For odd-n, T(n,(n-1)/2) = T(n,(n+1)/2) = A220236.

			Triangle begins:
n\k  0   1   2   3   4   5   6   7   8 ...
0    0
1    0   0
2    2   5   2
3    6   9   9   6
4   14  21  27  21  14
5   30  45  51  51  45  30
6   62  93 107 119 107  93  62
7  126 189 219 231 231 219 189 126
8  254 381 443 471 495 471 443 381 254
...

Kival Ngaokrajang, Illustration of initial terms, T(n,k) for n = 0..10, k = 0..10

Cf. A000225, A042948, A129868, A137932, A267089.

A267812 Decimal representation of the n-th iteration of the "Rule 217" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 1, 27, 119, 495, 2015, 8127, 32639, 130815, 523775, 2096127, 8386559, 33550335, 134209535, 536854527, 2147450879, 8589869055, 34359607295, 137438691327, 549755289599, 2199022206975, 8796090925055, 35184367894527, 140737479966719, 562949936644095

Offset: 0

Robert Price, Jan 20 2016

nonn

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A267810.

Cf. A129868, A267685.

rule=217; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}] (* Decimal Representation of Rows *)

Conjectures from Colin Barker, Jan 22 2016: (Start)
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>4.
G.f.: (1-6*x+34*x^2-64*x^3+32*x^4) / ((1-x)*(1-2*x)*(1-4*x)).
(End)

A279260 Numbers which are cyclops palindromic in their binary reflected Gray code representation.

Original entry on oeis.org

0, 6, 18, 90, 330, 1386, 5418, 21930, 87210, 349866, 1397418, 5593770, 22366890, 89483946, 357903018, 1431677610, 5726579370, 22906579626, 91625794218, 366504225450, 1466014804650, 5864063412906, 23456245263018, 93824997829290, 375299957762730, 1501199898159786, 6004799458421418

Offset: 0

Indranil Ghosh, Jan 17 2017

Cyclops palindromic numbers in base 2 are numbers with middle bit 0, having equal number of 1's on both side of the 0. There is a single 0 bit in the middle and the total number of bits is odd. The middle '0' represents the eye of a cyclops.

a(n) mod 6 = 0.

			90 is in the sequence because the binary reflected Gray code representation of 90 is '1110111' which is a cyclops palindromic binary number.

Indranil Ghosh, Table of n, a(n) for n = 0..1000
Indranil Ghosh, Proof of 6|{(-2*(1+((-2)^n)-(2^(2*n+1))))/3}
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video, (2015)

Cf. A014550, A129868, A134808, A138148.

Partial sums of A071930.

a(n)=(-2*(1+((-2)^n)-(2^(2*n+1))))/3 \\ Charles R Greathouse IV, Jun 29 2018

def a(n):
    return (-2*(1+((-2)**n)-(2**(2*n+1))))/3

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

cp's OEIS Frontend

A156589 a(n) = 4^n - 2^n - 1.

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A170940 4^n-2^n-2.

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A220236 Binary palindromic numbers with only two 0 bits, both in the middle.

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A357574 a(n) is the maximum number of pairs that sum to a power of 2 in a set of n consecutive odd numbers.

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A187560 a(n) = 4^(n+1)-2^n-1.

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A267090 Triangle read by rows: Fill an n X n square with 1's, except for 0's on the two main diagonals. Then T(n,k) is decimal equivalent of the k-th row (0<=k<=n).

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A267812 Decimal representation of the n-th iteration of the "Rule 217" elementary cellular automaton starting with a single ON (black) cell.

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Mathematica

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A279260 Numbers which are cyclops palindromic in their binary reflected Gray code representation.

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