A147595 -id:A147595 - cp's OEIS frontend

A138144 Palindromes formed from the reflected decimal expansion of the concatenation of 1, 1 and infinite 0's.

1, 11, 111, 1111, 11011, 110011, 1100011, 11000011, 110000011, 1100000011, 11000000011, 110000000011, 1100000000011, 11000000000011, 110000000000011, 1100000000000011, 11000000000000011, 110000000000000011

Offset: 1

Omar E. Pol, Mar 29 2008

a(n) is also A147595(n) written in base 2. [From Omar E. Pol, Nov 08 2008]

			n .... a(n)
1 .... 1
2 .... 11
3 .... 111
4 .... 1111
5 .... 11011
6 .... 110011
7 .... 1100011
8 .... 11000011
9 .... 110000011
10 ... 1100000011

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

Cf. A000533.

Cf. A138118, A138119, A138120, A138145, A138146, A138721, A138826, A147595. [From Omar E. Pol, Nov 08 2008]

LinearRecurrence[{11,-10},{1,11,111,1111,11011},20] (* Harvey P. Dale, Aug 21 2016 *)

Vec(-x*(10*x^2-1)*(10*x^2+1)/((x-1)*(10*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

a(n) = 11+11*10^(n-2) for n>3. a(n) = 11*a(n-1)-10*a(n-2). G.f.: -x*(10*x^2-1)*(10*x^2+1) / ((x-1)*(10*x-1)). - Colin Barker, Sep 15 2013

Better definition. - Omar E. Pol, Nov 15 2008

A147596 a(n) is the number whose binary representation is A138145(n).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 119, 231, 455, 903, 1799, 3591, 7175, 14343, 28679, 57351, 114695, 229383, 458759, 917511, 1835015, 3670023, 7340039, 14680071, 29360135, 58720263, 117440519, 234881031, 469762055, 939524103, 1879048199, 3758096391

Offset: 1

Omar E. Pol, Nov 08 2008

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A138145, A145641, A147537, A147538, A147539.

Cf. A147540, A147590, A147595, A147597.

[1,3,7,15,31] cat [7*(1+2^(n-3)): n in [6..40]]; // G. C. Greubel, Oct 25 2022

Join[{1,3,7,15,31}, 7*(1+2^(Range[6, 40] -3))] (* G. C. Greubel, Oct 25 2022 *)

Vec(-x*(2*x^2-1)*(4*x^4+2*x^2+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Sep 15 2013

def A147596(n): return 7*(1+2^(n-3)) -(1/8)*(63*int(n==0) +62*int(n==1) +60*int(n ==2)) -(7*int(n==3) +6*int(n==4) +4*int(n==5))
[A147596(n) for n in range(1,40)] # G. C. Greubel, Oct 25 2022

a(n) = 7*(2^(n-3) + 1) if n >= 6. - Hagen von Eitzen, Jun 02 2009
From Colin Barker, Sep 15 2013: (Start)
a(n) = 3*a(n-1) - 2*a(n-2), for n >= 8.
G.f.: x*(1-2*x^2)*(1+2*x^2+4*x^4) / ((1-x)*(1-2*x)). (End)
E.g.f.: (7/8)*(8*exp(x) + exp(2*x)) - (1/8)*(63 + 62*x + 30*x^2) - 7*x^3/6 - x^4/4 - x^5/30. - G. C. Greubel, Oct 25 2022

More terms from Hagen von Eitzen, Jun 02 2009

A147597 a(n) is the number whose binary representation is A138146(n).

Original entry on oeis.org

1, 7, 31, 119, 455, 1799, 7175, 28679, 114695, 458759, 1835015, 7340039, 29360135, 117440519, 469762055, 1879048199, 7516192775, 30064771079, 120259084295, 481036337159, 1924145348615, 7696581394439, 30786325577735, 123145302310919, 492581209243655

Offset: 1

Omar E. Pol, Nov 08 2008

Bisection of A147596.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A138146, A145641, A147537, A147538, A147539.

Cf. A147540, A147590, A147595, A147596.

[1,7,31] cat [7*(1+4^(n-2)): n in [4..40]]; // G. C. Greubel, Oct 25 2022

Table[FromDigits[#, 2] &@ If[n < 4, ConstantArray[1, 2 n - 1], Join[#, ConstantArray[0, 2 n - 7], #]] &@ ConstantArray[1, 3], {n, 25}] (* or *)
Rest@ CoefficientList[Series[x (2 x + 1) (2 x - 1) (4 x^2 + 2 x + 1)/((4 x - 1) (1 - x)), {x, 0, 25}], x] (* Michael De Vlieger, Nov 25 2016 *)
LinearRecurrence[{5,-4},{1,7,31,119,455},30] (* Harvey P. Dale, Aug 04 2025 *)

Vec(x*(2*x+1)*(2*x-1)*(4*x^2+2*x+1)/((4*x-1)*(1-x)) + O(x^30)) \\ Colin Barker, Nov 25 2016

def A147597(n): return 7*(1+4^(n-2)) -(119/16)*int(n==0) -(31/4)*int(n==1) -7*int(n==2) -4*int(n==3)
[A147597(n) for n in range(1,41)] # G. C. Greubel, Oct 25 2022

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = 5*a(n-1) - 4*a(n-2) for n>5.
G.f.: x*(2*x+1)*(2*x-1)*(4*x^2+2*x+1)/((4*x-1)*(1-x)). (End)
a(n) = 7*4^(n-2) + 7 for n>3. - Colin Barker, Nov 25 2016
E.g.f.: (7/16)*(16*exp(x) + exp(4*x)) -(119/16) -31*x/4 -7*x^2/2 -2*x^3/3. - G. C. Greubel, Oct 25 2022

More terms from R. J. Mathar, Feb 05 2010

A224380 Table read by antidiagonals of numbers of form (2^n -1)*2^(m+2) + 3 where n>=1, m>=1.

Original entry on oeis.org

11, 19, 27, 35, 51, 59, 67, 99, 115, 123, 131, 195, 227, 243, 251, 259, 387, 451, 483, 499, 507, 515, 771, 899, 963, 995, 1011, 1019, 1027, 1539, 1795, 1923, 1987, 2019, 2035, 2043, 2051, 3075, 3587, 3843, 3971, 4035, 4067, 4083, 4091, 4099, 6147, 7171, 7683, 7939, 8067, 8131, 8163, 8179

Offset: 1

Brad Clardy, Apr 05 2013

The table has row labels 2^n - 1 and column labels 2^(m+2). The table entry is row*col + 3. A MAGMA program is provided that generates the numbers in a table format. The sequence is read along the antidiagonals starting from the top left corner. Using the lexicographic ordering of A057555  the sequence is:
A(n) = Table(i,j) with (i,j)=(1,1),(1,2),(2,1),(1,3),(2,2),(3,1)...
+3  |    8    16    32    64   128    256    512 ...
----|-------------------------------------------
1   |   11    19    35    67   131    259    515
3   |   27    51    99   195   387    771   1539
7   |   59   115   227   451   899   1795   3587
15  |  123   243   483   963  1923   3843   7683
31  |  251   499   995  1987  3971   7939  15875
63  |  507  1011  2019  4035  8067  16131  32259
127 | 1019  2035  4067  8131 16259  32515  65027
...
All of these numbers have the following property: let m be a member of A(n); if a sequence B(n) = all i such that i XOR (m - 1) = i - (m - 1), then the differences between successive members of B(n) is an alternating series of 1's and 3's with the last difference in the pattern m. The number of alternating 1's and 3's in the pattern is 2^(j+1) - 1, where j is the column index.
As an example consider A(1) which is 11, the sequence B(n) where i XOR 10 = i - 10 starts as 10, 11, 14, 15, 26, 27, 30, 31, 42, ... (A214864) with successive differences of 1, 3, 1, 11.
Main diagonal is A191341, the largest k such that k-1 and k+1 in binary representation have the same number of 1's and 0's

Brad Clardy, Table of n, a(n) for n = 1..1000

Cf. A057555(lexicographic ordering), A214864(example), A224195.
Rows: A062729(i=1), A147595(2 n>=5), A164285(3 n>=3).
Cols: A168616(j=1 n>=4).
Diagonal: A191341.

//program generates values in a table form,row labels of 2^i -1
for i:=1 to 10 do
    m:=2^i - 1;
    m, [ m*2^n +1 : n in [1..10]];
end for;
//program generates sequence in lexicographic ordering of A057555, read
//along antidiagonals from top. Primes in the sequence are marked with *.
for i:=2 to 18 do
    for j:=1 to i-1 do
       m:=2^j -1;
       k:=m*2^(2+i-j) + 3;
       if IsPrime(k) then k, "*";
          else k;
       end if;;
    end for;
end for;

a(n) = 2^(A057555(2*n - 1))*2^(A057555(2*n) + 2) + 3 for n>=1.

cp's OEIS Frontend

A138144 Palindromes formed from the reflected decimal expansion of the concatenation of 1, 1 and infinite 0's.

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A147596 a(n) is the number whose binary representation is A138145(n).

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A147597 a(n) is the number whose binary representation is A138146(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A224380 Table read by antidiagonals of numbers of form (2^n -1)*2^(m+2) + 3 where n>=1, m>=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Formula