A248646 -id:A248646 - cp's OEIS frontend

A005009 a(n) = 7*2^n.

7, 14, 28, 56, 112, 224, 448, 896, 1792, 3584, 7168, 14336, 28672, 57344, 114688, 229376, 458752, 917504, 1835008, 3670016, 7340032, 14680064, 29360128, 58720256, 117440512, 234881024, 469762048, 939524096, 1879048192, 3758096384

Offset: 0

N. J. A. Sloane

The first differences are the sequence itself. - Alexandre Wajnberg & Eric Angelini, Sep 07 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), this sequence (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Row sums of (6, 1)-Pascal triangle A093563 and of (1, 6)-Pascal triangle A096956, n>=1.
Cf. A000120, A014311, A118416, A173787, A212191.

a005009 = (* 7) . (2 ^) -- Reinhard Zumkeller, May 03 2012

[7*2^n:n in [0..50]]; // Vincenzo Librandi, Sep 20 2011

7*2^Range[0,50] (* Vladimir Joseph Stephan Orlovsky, Mar 14 2011 *)
NestList[2#&,7,30] (* Harvey P. Dale, Aug 10 2024 *)

a(n)=7<Charles R Greathouse IV, Dec 22 2011

[7*2^n for n in range(51)] # G. C. Greubel, Jan 05 2023

G.f.: 7/(1-2*x).
a(n) = A118416(n+1,4) for n > 3. - Reinhard Zumkeller, Apr 27 2006
a(n) = 2*a(n-1), for n > 0, with a(0)=7 . - Philippe Deléham, Nov 23 2008
a(n) = 7 * A000079(n). - Omar E. Pol, Dec 16 2008
a(n) = A173787(n+3,n). - Reinhard Zumkeller, Feb 28 2010
Intersection of A014311 and A212191: all terms and their squares are the sum of exactly three distinct powers of 2, A000120(a(n)) = A000120(a(n)^2) = 3. - Reinhard Zumkeller, May 03 2012
G.f.: 2/x/G(0) - 1/x + 9, where G(k)= 1 + 1/(1 - x*(7*k+2)/(x*(7*k+9) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
E.g.f.: 7*exp(2*x). - Stefano Spezia, May 15 2021

A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

Original entry on oeis.org

6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 1

Enoch Haga, Jan 24 2004

Sequence arising in Farideh Firoozbakht's solution to Prime Puzzle 251 - 23 is the only pointer prime (A089823) not containing digit "1".

The monotonic increasing value of successive product of digits strongly suggests that in successive n the digit 1 must be present.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Tanya Khovanova, Recursive Sequences
Carlos Rivera, Puzzle 251, Pointer primes, The Prime Puzzles and Problems Connection.
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).
Cf. A089823, A091628, A091630, A091631, A091632.
Similar to A003945, A042950, A058764, A087009, A081808.

[3*2^n : n in [1..40]]; // Wesley Ivan Hurt, Jul 17 2020

3*2^Range[1, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)

[3*2^n for n in range(1,51)] # G. C. Greubel, Jan 05 2023

a(n) = 3 * 2^n = product of digits of A091628(n).
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 6*2^(n-1).
a(n) = 2*a(n-1), with a(1) = 6.
G.f.: 6*x/(1-2*x). (End)
E.g.f.: 3*(exp(2*x) - 1). - G. C. Greubel, Jan 05 2023

Edited and extended by Ray Chandler, Feb 07 2004

A110287 a(n) = 17*2^n.

Original entry on oeis.org

17, 34, 68, 136, 272, 544, 1088, 2176, 4352, 8704, 17408, 34816, 69632, 139264, 278528, 557056, 1114112, 2228224, 4456448, 8912896, 17825792, 35651584, 71303168, 142606336, 285212672, 570425344, 1140850688, 2281701376, 4563402752, 9126805504, 18253611008

Offset: 0

Alexandre Wajnberg, Sep 07 2005

The first differences are the sequence itself. Doubling the terms gives the same sequence (beginning one step further).

17 times powers of 2. - Omar E. Pol, Dec 17 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..235
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A003945 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), this sequence (m=8), A110288 (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).

Cf. A007283.

[17*2^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

17*2^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[2#&,17,30] (* Harvey P. Dale, Jul 21 2014 *)

a(n)=17*2^n \\ Charles R Greathouse IV, Oct 07 2015

[17*2^n for n in range(51)] # G. C. Greubel, Jan 05 2023

G.f.: 17/(1-2*x). - Philippe Deléham, Nov 23 2008
a(n) = 17*A000079(n). - Omar E. Pol, Dec 17 2008
a(n) = 2*a(n-1) (with a(0)=17). - Vincenzo Librandi, Dec 26 2010
a(n) = A173786(n+4, n) for n>3. - Reinhard Zumkeller, Feb 28 2010
E.g.f.: 17*exp(2*x). - G. C. Greubel, Jan 05 2023

Edited by Omar E. Pol, Dec 16 2008

A110288 a(n) = 19*2^n.

Original entry on oeis.org

19, 38, 76, 152, 304, 608, 1216, 2432, 4864, 9728, 19456, 38912, 77824, 155648, 311296, 622592, 1245184, 2490368, 4980736, 9961472, 19922944, 39845888, 79691776, 159383552, 318767104, 637534208, 1275068416, 2550136832, 5100273664, 10200547328, 20401094656

Offset: 0

Alexandre Wajnberg, Sep 07 2005

The first differences are the sequence itself. Doubling the terms gives the same sequence (beginning one step further).

19 times powers of 2. - Omar E. Pol, Dec 17 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..235
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2).

Sequences of the form (2*m+1)*2^n: A000079 (m=0), A007283 (m=1), A020714 (m=2), A005009 (m=3), A005010 (m=4), A005015 (m=5), A005029 (m=6), A110286 (m=7), A110287 (m=8), this sequence (m=9), A175805 (m=10), A248646 (m=11), A164161 (m=12), A175806 (m=13), A257548 (m=15).

[19*2^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

19*2^Range[0, 60] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
NestList[2#&,19,30] (* Harvey P. Dale, May 11 2018 *)

[19*2^n for n in range(41)] # G. C. Greubel, Jan 04 2023

G.f.: 19/(1-2*x). - Philippe Deléham, Nov 23 2008
a(n) = A000079(n)*19. - Omar E. Pol, Dec 17 2008
E.g.f.: 19*exp(2*x). - G. C. Greubel, Jan 04 2023

Edited by Omar E. Pol, Dec 16 2008

A256494 Expansion of -x^2(x^3+x-1) / ((x-1)(x+1)(2x-1)*(x^2+1)).

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 13, 26, 51, 103, 205, 410, 819, 1639, 3277, 6554, 13107, 26215, 52429, 104858, 209715, 419431, 838861, 1677722, 3355443, 6710887, 13421773, 26843546, 53687091, 107374183, 214748365, 429496730, 858993459, 1717986919, 3435973837, 6871947674, 13743895347, 27487790695, 54975581389, 109951162778

Offset: 1

Armands Strazds, Mar 30 2015

Previous name was: Golden Book's Level Leap Sequence.

x-positions a(n) of transition from phase 1 (I I) to 2 (/\) for the Golden Book’s y-position n.

Colin Barker, Table of n, a(n) for n = 1..1000
Armands Strazds, The Golden Book, 1990. [broken link]
Index entries for linear recurrences with constant coefficients, signature (2,0,0,1,-2).

Cf. A248646, A001045.

I:=[0,1,1,2,3,7]; [n le 6 select I[n] else 2*Self(n-1)+Self(n-4)-2*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 25 2015

Join[{0}, LinearRecurrence[{2, 0, 0, 1, - 2}, {1, 1, 2, 3, 7}, 50]] (* Vincenzo Librandi, Dec 25 2015 *)

concat(0, Vec(-x^2*(x^3+x-1)/((x-1)*(x+1)*(2*x-1)*(x^2+1)) + O(x^100))) \\ Colin Barker, Apr 09 2015

$r = array(1, -1, 0, -1);
$a[0] = 0;
for ($n = 1; $n < 40; $n++) {
$a[$n] = 2 * $a[$n - 1] + $r[($n - 1) % 4];
}
echo(implode(", ", $a));

a(n) = 2 * a(n - 1) + r((n - 1) % 4); r = array(1, -1, 0, -1).
From Colin Barker, Apr 09 2015: (Start)
a(n) = 2*a(n-1)+a(n-4)-2*a(n-5) for n>5.
a(n) = (5+5*(-1)^n-(1+2*i)*(-i)^n-(1-i*2)*i^n+2^(1+n))/20 for n>0 where i=sqrt(-1).
G.f.: -x^2*(x^3+x-1) / ((x-1)*(x+1)*(2*x-1)*(x^2+1)). (End)
E.g.f.: (5*cosh(x) + cosh(2*x) - cos(x) + sinh(2*x) - 2*sin(x) - 5)/10. - Stefano Spezia, May 18 2025

New name (using g.f. from Colin Barker) from Joerg Arndt, Dec 26 2015

A258055 Concatenation of the decimal representations of the lengths (increased by 1) of the runs of zeros between successive ones in the binary representation of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 11, 0, 3, 2, 21, 1, 12, 11, 111, 0, 4, 3, 31, 2, 22, 21, 211, 1, 13, 12, 121, 11, 112, 111, 1111, 0, 5, 4, 41, 3, 32, 31, 311, 2, 23, 22, 221, 21, 212, 211, 2111, 1, 14, 13, 131, 12, 122, 121, 1211, 11, 113, 112, 1121, 111, 1112, 1111

Offset: 0

Armands Strazds, May 17 2015

Originally called the "Golden Book's ZI-sequence" by the author.
The ZI-sequence is related to the binary numbers sequence with 10 ^ n substituted by the respective exponent increased by 1 (i.e., 10 as 2, 100 as 3, etc.) and the least significant bit discarded, e.g., binary 1011 converts to ZI 21.
a(n) = 0 when no successive ones exist in the binary representation of n, i.e., when n=0 and when n is a power of 2. - Giovanni Resta, Aug 31 2015

			Example for n=6: binary 110 => split into 10^m components: 1 (10^0) and 10 (10^1) => 1; the least significant bit, and thus the whole last component, here 10, is discarded.
840 in binary is 1100101000. The runs of zeros between successive ones have length 0, 2 and 1, hence a(840) = 132. - _Giovanni Resta_, Aug 31 2015

A. Strazds, The Golden Book [broken link]

Cf. A248646, A256494. See also A261300 for another version.

a[0] = 0; a[n_] := FromDigits@ Flatten[ IntegerDigits /@ Most[ Length /@ (Split[ Flatten[ IntegerDigits[n, 2] /. 1 -> {1, 0}]][[2 ;; ;; 2]]) ]]; Table[a@ n, {n, 0, 100}] (* Giovanni Resta, Aug 31 2015 *)

function dec2zi ($d) {
$b = base_convert($d, 10, 2); $b = str_split($b);
$i = $z = 0; $r = "";
foreach($b as $v) {
if (!$v) {
$i++;
} else {
if ($i > 0) {
$r .= $i + $v; $i = 0;
} else {
if ($z > 0) {
$r .= $v; $z = 0;
}
$z++; }}}
return $r == "" ? 0 : $r; }

A275536 Differences of the exponents of the adjacent distinct powers of 2 in the binary representation of n (with -1 subtracted from the least exponent present) are concatenated as decimal digits in reverse order.

Original entry on oeis.org

1, 2, 11, 3, 12, 21, 111, 4, 13, 22, 112, 31, 121, 211, 1111, 5, 14, 23, 113, 32, 122, 212, 1112, 41, 131, 221, 1121, 311, 1211, 2111, 11111, 6, 15, 24, 114, 33, 123, 213, 1113, 42, 132, 222, 1122, 312, 1212, 2112, 11112

Offset: 1

Armands Strazds, Aug 01 2016

A preferable representation is a sequence of arrays, since multi-digit items are possible: [1],[2],[1,1],[3],[1,2],[2,1],[1,1,1],[4],[1,3],[2,2],[1,1,2],[3,1],[1,2,1],[2,1,1],[1,1,1,1],[5],[1,4],[2,3],[1,1,3],[3,2],[1,2,2],[2,1,2],[1,1,1,2],[4,1],[1,3,1],[2,2,1],[1,1,2,1],[3,1,1],[1,2,1,1],[2,1,1,1],[1,1,1,1,1],[6],[1,5],[2,4],[1,1,4],[3,3],[1,2,3],[2,1,3],[1,1,1,3],[4,2],[1,3,2],[2,2,2],[1,1,2,2],[3,1,2],[1,2,1,2],[2,1,1,2],[1,1,1,1,2]. 0 is not allowed as a digit.
a(512) is the first term which cannot be expressed unambiguously in decimal. - Charles R Greathouse IV, Aug 02 2016
The first two terms which are equal (because of the ambiguity inherent in using decimal, or more generally any finite base) are a(3) = a(1024) = 11. a(3) corresponds to the array [1,1] while a(1024) corresponds to [11]. - Charles R Greathouse IV, Mar 19 2017

			5 = 2^2 + 2^0, so the representation is [2-0, 0-(-1)] = [2, 1] so a(5) = 12.
6 = 2^2 + 2^1, so the representation is [2-1, 1-(-1)] = [1, 2] so a(6) = 21.
18 = 2^4 + 2^1, so the representation is [4-1, 1-(-1)] = [3, 2] so a(18) = 23.

Charles R Greathouse IV, Table of n, a(n) for n = 1..511

Cf. A069800, A261300, A248646, A256494, A258055, A004086, A004719, A071160.

a(n)=my(v=List(),k); while(n, k=valuation(n,2)+1; n>>=k; listput(v,k)); fromdigits(Vec(v)) \\ Charles R Greathouse IV, Aug 02 2016

function dec2delta($k) {
  $p = -1;
  while ($k > 0) {
    $k -= $c = pow(2, floor(log($k, 2)));
    if ($p > -1) $d[] = $p - floor(log($c, 2));
    $p = floor(log($c, 2));
  }
  $d[] = $p + 1;
  return array_reverse($d);
}
function delta2dec($d) {
  $k = 0;
  $e = -1;
  foreach ($d AS $v) {
    if ($v > 0) {
      $e += $v;
      $k += pow(2, $e);
    }
  }
  return $k;
}

For n=1..511, a(n) = A004086(A004719(A071160(n))) [In other words, terms of A071160 with 0-digits deleted and the remaining digits reversed.] - Antti Karttunen, Sep 03 2016

cp's OEIS Frontend

A005009 a(n) = 7*2^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

SageMath

Formula

A091629 Product of digits associated with A091628(n). Essentially the same as A007283.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A110287 a(n) = 17*2^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A110288 a(n) = 19*2^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A256494 Expansion of -x^2*(x^3+x-1) / ((x-1)*(x+1)*(2*x-1)*(x^2+1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PHP

Formula

Extensions

A258055 Concatenation of the decimal representations of the lengths (increased by 1) of the runs of zeros between successive ones in the binary representation of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PHP

A256494 Expansion of -x^2(x^3+x-1) / ((x-1)(x+1)(2x-1)*(x^2+1)).