A005009 -id:A005009 - cp's OEIS frontend

A029746 Numbers of the form 2^k or 7*2^k.

1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 128, 224, 256, 448, 512, 896, 1024, 1792, 2048, 3584, 4096, 7168, 8192, 14336, 16384, 28672, 32768, 57344, 65536, 114688, 131072, 229376, 262144, 458752, 524288, 917504, 1048576, 1835008, 2097152

Offset: 0

N. J. A. Sloane

nonn

Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079, A005009, A029744, A029748.

Sort[Flatten[{#,7#}&/@(2^Range[0,21])]] (* or *) CoefficientList[Series[ (1+2x+2x^2+3x^3)/(1-2x^2),{x,0,50}],x]  (* Harvey P. Dale, Apr 23 2011 *)

a(n)=if(n<0,0,if(n<2,2^n,if(n%2,7/2,2)*2^(n\2)))

G.f.: (1+2x+2x^2+3x^3)/(1-2x^2). - Michael Somos, Nov 05 2002

Sum_{n>=0} 1/a(n) = 16/7. - Amiram Eldar, Jan 17 2022

A164285 a(n) = 7*2^n + 3.

Original entry on oeis.org

10, 17, 31, 59, 115, 227, 451, 899, 1795, 3587, 7171, 14339, 28675, 57347, 114691, 229379, 458755, 917507, 1835011, 3670019, 7340035, 14680067, 29360131, 58720259, 117440515, 234881027, 469762051, 939524099, 1879048195, 3758096387

Offset: 0

Vincenzo Librandi, Aug 12 2009

Contains the primes 17, 31, 59, 227, 57347, 114691, 14680067, 7516192771,..

			At n=0, a(0)=7*2^0+3=10. At n=1, a(1)=7*2^1+3=17.

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

[7*2^n+3: n in [0..40]]; // Vincenzo Librandi, Sep 12 2013

CoefficientList[Series[(10 - 13 x) / ((2 x - 1) (x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 12 2013 *)
LinearRecurrence[{3,-2},{10,17},40] (* Harvey P. Dale, Aug 29 2017 *)

x='x+O('x^50); Vec((10-13*x)/((2*x-1)*(x-1))) \\ G. C. Greubel, Sep 12 2017

a(n) = 2*a(n-1) - 3.
a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = A005009(n) + 3, a(0)=10.
G.f.: (10-13*x)/((2*x-1)*(x-1)).
E.g.f.: 7*exp(2*x) + 3*exp(x). - G. C. Greubel, Sep 12 2017

Offset corrected by R. J. Mathar, Aug 19 2009

A168309 Period 2: repeat 4,-3.

Original entry on oeis.org

4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3

Offset: 1

Klaus Brockhaus, Nov 22 2009

Interleaving of A010709 and -3*A000012.
Binomial transform of 4 followed by a signed version of A005009.
Inverse binomial transform of 4 followed by A000079.
a(n+1) - a(n) = 7*(-1)^n.
A168230 without initial term 0 gives partial sums.
Nonsimple continued fraction expansion of 2+2*sqrt(2/3) = 3.6329931618... - R. J. Mathar, Mar 08 2012

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010709 (all 4's sequence), A000012 (all 1's sequence), A010727 (all 7's sequence), A168230, A005009 (7*2^n), A000079 (powers of 2).

&cat[ [4, -3]: n in [1..42] ];
[ n eq 1 select 4 else -Self(n-1)+1: n in [1..84] ];

LinearRecurrence[{0,1},{4, -3}, 50] (* or *) Table[(1 - 7*(-1)^n)/2,{n,0,25}] (* G. C. Greubel, Jul 17 2016 *)
PadRight[{},120,{4,-3}] (* Harvey P. Dale, Oct 20 2018 *)

a(n) = (1 - 7*(-1)^n)/2.
a(n) = -a(n-1) + 1 for n > 1; a(1) = 4.
a(n) = a(n-2) for n > 2; a(1) = 4, a(2) = -3.
G.f.: x*(4 - 3*x)/((1-x)*(1+x)).
E.g.f.: (1/2)*(-1 + exp(x))*(7 + exp(x))*exp(-x). - G. C. Greubel, Jul 17 2016

A171160 a(n) = a(n-1) + 2*a(n-2) with a(0)=3, a(1)=4.

Original entry on oeis.org

3, 4, 10, 18, 38, 74, 150, 298, 598, 1194, 2390, 4778, 9558, 19114, 38230, 76458, 152918, 305834, 611670, 1223338, 2446678, 4893354, 9786710, 19573418, 39146838, 78293674, 156587350, 313174698, 626349398, 1252698794, 2505397590, 5010795178, 10021590358

Offset: 0

Paul Curtz, Dec 04 2009

J. Mulder, Table of n, a(n) for n = 0..2999
Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A001045, A078008, A175805.
Cf. A062510, A083582, (-1)^n*A140966.
Cf. A092297, A154879.
Cf. A062092, A083595.

f[n_]:=2/(n+1);x=5;Table[x=f[x];Numerator[x],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
LinearRecurrence[{1,2},{3,4},40] (* Harvey P. Dale, Sep 04 2013 *)

Vec(-(x+3)/((x+1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Feb 10 2015

a(n) = (1/3)*(2*(-1)^n + 7*2^n), with n>=0. - Paolo P. Lava, Dec 14 2009
G.f.: -(x+3) / ((x+1)*(2*x-1)). - Colin Barker, Feb 10 2015
From Paul Curtz, Jun 03 2022: (Start)
a(n) = A078008(n) + A078008(n+1) + A078008(n+2).
a(n) = 2^(n+1) + A078008(n).
a(n) = A001045(n+3) - A001045(n).
(a(n) + a(n+1) = a(n+2) - a(n) = A005009(n).)
a(n) + a(n+3) = A175805(n).
a(n) = A062510(n) + A083582(n-1) with A083582(-1) = 3.
a(n) = A092297(n) + A154879(n). (End)
a(n) = 2*A062092(n-1), for n>0; 2*a(n) = A083595(n+1). - Paul Curtz, Jun 08 2022

Edited by N. J. A. Sloane, Dec 05 2009
More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010
More terms from Max Alekseyev, Apr 24 2010

A321483 a(n) = 7*2^n + (-1)^n.

Original entry on oeis.org

8, 13, 29, 55, 113, 223, 449, 895, 1793, 3583, 7169, 14335, 28673, 57343, 114689, 229375, 458753, 917503, 1835009, 3670015, 7340033, 14680063, 29360129, 58720255, 117440513, 234881023, 469762049, 939524095, 1879048193, 3758096383, 7516192769, 15032385535

Offset: 0

Paul Curtz, Nov 11 2018

Difference table:
   8, 13, 29, 55, 113, 223, 449, ...
   5, 16, 26, 58, 110, 226, 446, 898, ...
  11, 10, 32, 52, 116, 220, 452, 892, 1796, ...
  -1, 22, 20, 64, 104, 232, 440, 904, 1784, 3592, ...
      -2, 44, 40, 128, 208, 464, 880, 1808, 3568, 7184, ...
etc.
Every diagonal is a sequence of the form k*2^m.
a(n) is divisible by
.  5 if n is a term of A004767,
. 11 if n is a term of A016885,
. 13 if n is a term of A017533.

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

Cf. A000079, A001045, A005029, A010710, A014551, A062092, A062510, A070366, A175805, A199207, A206372.
Cf. A033999, A005009.
Subsequence of A001651, A160543, A160544.

a[n_] := 7*2^n + (-1)^n ; Array[a, 32, 0] (* Amiram Eldar, Nov 12 2018 *)
CoefficientList[Series[E^-x + 7 E^(2 x), {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Nov 12 2018 *)
LinearRecurrence[{1,2},{8,13},40] (* Harvey P. Dale, Mar 18 2022 *)

Vec((8 + 5*x) / ((1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 11 2018

O.g.f.: (8 + 5*x) / ((1 + x)*(1 - 2*x)). - Colin Barker, Nov 11 2018
E.g.f.: exp(-x) + 7*exp(2*x). - Stefano Spezia, Nov 12 2018
a(n) = a(n-1) + 2*a(n-2).
a(n) = 2*a(n-1) + 3*(-1)^n for n>0, a(0)=8.
a(2*k) = 7*4^k + 1, a(2*k+1) = 14*4^k - 1.
a(n) = A014551(n) + A014551(n-1) + A014551(n-2).
a(n) = 2^(n+3) - 3*A001045(n).
a(n) mod 9 = A070366(n+3).
a(n) + a(n+1) = 21*2^n.

Two terms corrected, and more terms added by Colin Barker, Nov 11 2018

A340717 Lexicographically earliest sequence of nonnegative integers with as many distinct values as possible such that for any n >= 0, a(rev(n)) = a(n) (where rev(n) = A030101(n) corresponds to the binary reversal of n).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 6, 4, 7, 1, 8, 5, 9, 3, 10, 6, 11, 2, 9, 6, 12, 4, 11, 7, 13, 1, 14, 8, 15, 5, 16, 9, 17, 3, 16, 10, 18, 6, 19, 11, 20, 2, 15, 9, 21, 6, 18, 12, 22, 4, 17, 11, 22, 7, 20, 13, 23, 1, 24, 14, 25, 8, 26, 15, 27, 5, 28, 16

Offset: 0

Rémy Sigrist, Jan 17 2021

The condition "with as many distinct values as possible" means here that for any distinct m and n, provided the orbits of m and n under the map x -> rev(x) do not merge, then a(m) <> a(n).

			The first terms, alongside rev(n), are:
  n   a(n)  rev(n)
  --  ----  ------
   0     0       0
   1     1       1
   2     1       1
   3     2       3
   4     1       1
   5     3       5
   6     2       3
   7     4       7
   8     1       1
   9     5       9
  10     3       5
  11     6      13
  12     2       3
  13     6      11
  14     4       7
  15     7      15

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Colored scatterplot of the first 2^16 terms (where the color is function of A007814(n), the 2-adic valuation of n)
Rémy Sigrist, PARI program for A340717

See A340716 for similar sequences.

Cf. A000079, A005009, A005010, A007283, A007814, A020714, A030101, A340718.

PARI
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See Links section.
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a(2*n) = a(n).
a(n) = 1 iff n is a power of 2.
a(n) = 2 iff n belongs to A007283.
a(n) = 3 iff n belongs to A020714.
a(n) = 4 iff n belongs to A005009.
a(n) = 5 iff n belongs to A005010.
a(A340718(n)) = n (and this is the first occurrence of n in the sequence).

A362804 Numbers k such that the set of divisors {d | k, BitOr(k, d) = k} has an integer harmonic mean.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 45, 48, 56, 60, 64, 90, 96, 112, 120, 128, 180, 192, 224, 240, 256, 360, 384, 448, 480, 496, 512, 720, 768, 896, 960, 992, 1024, 1440, 1536, 1792, 1920, 1984, 2048, 2880, 3072, 3584, 3840, 3968, 4096, 5760, 6144, 7168, 7680

Offset: 1

Amiram Eldar, May 04 2023

Equivalently, the set of divisors can be defined by {d | k, BitAnd(k, d) = d}.
Analogous to harmonic (or Ore) numbers (A001599) where the divisors d of k are restricted by BitOr(k, d) = k or BitAnd(k, d) = d.
If k is a term then so is 2*k. The primitive terms are in A362805. Thus, this sequence includes all the powers of 2 (A000079), all the numbers of the form 3*2^m and 15*2^m for m >= 1, and all the numbers of the form 7*2^m for m >= 2.
All the even perfect numbers (A000396) are terms: if k = 2^(p-1)*(2^p-1) is a perfect number (where p is a Mersenne exponent, A000043), then the only divisors of k such that BitOr(k, d) = k are 2^(p-1) and k itself, and the harmonic mean of 2^(p-1) and 2^(p-1)*(2^p-1) is 2^p - 1.
Are 1 and 45 the only odd terms in this sequence?

Amiram Eldar, Table of n, a(n) for n = 1..406

Cf. A000043, A000396, A246600, A246601.
Subsequences: A000079, A007283 \ {3}, A005009 \ {7, 14}, A110286 \ {15}, A362805.
Similar sequences: A001599, A006086, A063947, A286325, A319745.

q[n_] := IntegerQ[HarmonicMean[Select[Divisors[n], BitAnd[n, #] == # &]]]; Select[Range[10^4], q]

div(n) = select(x->(bitor(x, n) == n), divisors(n));
is(n) = {my(d = div(n)); denominator(#d/sum(i = 1, #d, 1/d[i])) == 1;}

A213948 Triangle, by rows, generated from the INVERT transforms of (1, 1, 2, 4, 8, 16, ...).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 4, 4, 1, 7, 10, 8, 8, 1, 12, 24, 20, 16, 16, 1, 20, 52, 56, 40, 32, 32, 1, 33, 112, 144, 112, 80, 64, 64, 1, 54, 238, 344, 320, 224, 160, 128, 128, 1, 88, 496, 828, 848, 640, 448, 320, 256, 256

Offset: 1

Gary W. Adamson, Jun 25 2012

Row sums = A001519, the odd-indexed Fibonacci terms. The triangle is a companion to A213947, having row sums of the even-indexed Fibonacci terms.

			First few rows of the triangle are:
  1;
  1,   1;
  1,   2,   2;
  1,   4,   4,   4;
  1,   7,  20,   8,   8;
  1,  12,  24,  20,  16,  16;
  1,  20,  52,  56,  40,  32,  32;
  1,  33, 112, 144, 112,  80,  64,  64;
  1,  54, 238, 344, 320, 224, 160, 128, 128;
  1,  88, 496, 828, 848, 640, 448, 320, 256, 256;
  ...

Cf. A001519, A213947, A000071 (2nd column), A020714 (subdiagonal), A005009 (subdiagonal).

read("transforms") ;
A213948i := proc(n,k)
    if n = 1 then
        L := [1,seq(0,i=0..k)] ;
    else
        L := [1,seq(2^i,i=0..n-2),seq(0,i=0..k)] ;
    end if;
    INVERT(L) ;
    op(k,%) ;
end proc:
A213948 := proc(n,k)
    if k = 1 then
        1;
    else
        A213948i(k,n)-A213948i(k-1,n) ;
    end if;
end proc: # R. J. Mathar, Jun 30 2012

Create an array in which the n-th row is the INVERT transform of the first n terms in the sequence (1, 1, 2, 4, 8, 16, 32, ...):
  1,    1,    1,    1,    1,    1,
  1,    2,    3,    5,    8,   13,    (essentially A000045)
  1,    2,    5,    9,   18,   37,    (essentially A077947)
  1,    2,    5,   13,   26,   57,
Terms of the n-th row of the triangle are the finite differences downwards the n-th column of this array.

A029749 Numbers of the form 2^k times 1, 5 or 7.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 32, 40, 56, 64, 80, 112, 128, 160, 224, 256, 320, 448, 512, 640, 896, 1024, 1280, 1792, 2048, 2560, 3584, 4096, 5120, 7168, 8192, 10240, 14336, 16384, 20480, 28672, 32768, 40960, 57344, 65536, 81920, 114688, 131072

Offset: 0

N. J. A. Sloane

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

Cf. A000079, A005009, A029746, A029748, A094958.

CoefficientList[Series[-(3 x^4 + 3 x^3 + 4 x^2 + 2 x + 1) / (2 x^3 - 1), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 20 2013 *)

From Colin Barker, Jul 19 2013: (Start)
a(n) = 2*a(n-3) for n>4.
G.f.: -(3*x^4 + 3*x^3 + 4*x^2 + 2*x + 1) / (2*x^3 - 1). (End)
Sum_{n>=0} 1/a(n) = 94/35. - Amiram Eldar, Jan 21 2022

More terms from Colin Barker, Jul 19 2013

A159021 a(0)=19; a(n) = a(n-1) + floor(sqrt(a(n-1))), n > 0.

Original entry on oeis.org

19, 23, 27, 32, 37, 43, 49, 56, 63, 70, 78, 86, 95, 104, 114, 124, 135, 146, 158, 170, 183, 196, 210, 224, 238, 253, 268, 284, 300, 317, 334, 352, 370, 389, 408, 428, 448, 469, 490, 512, 534, 557, 580, 604, 628, 653, 678, 704, 730, 757, 784, 812, 840, 868, 897, 926, 956, 986, 1017

Offset: 0

Philippe Deléham, Apr 02 2009

nonn

Row 3 in square array A159016.
This sequence contains infinitely many squares. - Philippe Deléham, Apr 04 2009
Conjecture: The squares are of the form (7*2^k)^2 (see A005009). - Vincenzo Librandi, Apr 10 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A028392, A005009.

NestList[#+Floor[Sqrt[#]]&,19,60] (* Harvey P. Dale, Jan 04 2013 *)

More terms from Vincenzo Librandi, Apr 10 2009

cp's OEIS Frontend

A029746 Numbers of the form 2^k or 7*2^k.

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A164285 a(n) = 7*2^n + 3.

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A168309 Period 2: repeat 4,-3.

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A171160 a(n) = a(n-1) + 2*a(n-2) with a(0)=3, a(1)=4.

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A321483 a(n) = 7*2^n + (-1)^n.

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A340717 Lexicographically earliest sequence of nonnegative integers with as many distinct values as possible such that for any n >= 0, a(rev(n)) = a(n) (where rev(n) = A030101(n) corresponds to the binary reversal of n).

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A362804 Numbers k such that the set of divisors {d | k, BitOr(k, d) = k} has an integer harmonic mean.

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A213948 Triangle, by rows, generated from the INVERT transforms of (1, 1, 2, 4, 8, 16, ...).