A147538 -id:A147538 - cp's OEIS frontend

A225910 Square array read by antidiagonals: a(m,n) is the number of binary pattern classes in the (m,n)-rectangular grid, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 7, 6, 1, 1, 10, 24, 24, 10, 1, 1, 20, 76, 168, 76, 20, 1, 1, 36, 288, 1120, 1120, 288, 36, 1, 1, 72, 1072, 8640, 16576, 8640, 1072, 72, 1, 1, 136, 4224, 66816, 263680, 263680, 66816, 4224, 136, 1, 1, 272, 16576, 529920, 4197376, 8407040, 4197376, 529920, 16576, 272, 1

Offset: 0

Yosu Yurramendi, May 20 2013

In the square table A000012, A005418, and A225826 to A225834 are the first 11 rows (see example).
In the square table, m odd (see formula). The order of the recurrence equations is 4. Let it be (a1(m),a2(m),a3(m),a4(m)) the characterizing 4-plet of a(m). The sequence a1(m) belongs to A028403 (2^m+2^((m+1)/2)), -a2(m) to A147538 (2^m*(2^((m+1)/2)-1)) and a4(m) to A013824 (2^(2m)*2^((m+1)/2)). -a3(m) sequence formula is 2^m*(2^m+2^((m+1)/2)).
All the coefficients of x in generating functions from A225826 to A225834 belong to A113979.

			Array begins:
  1   1      1         1            1               1                  1 ...
  1   2      3         6           10              20                 36 ...
  1   3      7        24           76             288               1072 ...
  1   6     24       168         1120            8640              66816 ...
  1  10     76      1120        16576          263680            4197376 ...
  1  20    288      8640       263680         8407040          268517376 ...
  1  36   1072     66816      4197376       268517376        17180065792 ...
  1  72   4224    529920     67133440      8590786560      1099516870656 ...
  1 136  16576   4212736   1073790976    274882625536     70368756760576 ...
  1 272  66048  33632256  17180262400   8796137062400   4503599962914816 ...
  1 528 262912 268713984 274878693376 281475261923328 288230376957018112 ...
  ...

Alois P. Heinz, Antidiagonals n = 0..65, flattened
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.

Cf. A005418, A225826-A225834.

m even and n even:
a(m,n) = 2^(m*n/2-2)*(2^(m*n/2) + 3);
m even and n odd:
a(m,n) = 2^(m*n/2-1)*(2^(m*n/2-1) + 2^(m/2-1) + 1);
m odd and n even:
a(m,n) = 2^(m*n/2-1)*(2^(m*n/2-1) + 2^(n/2-1) + 1);
m odd and n odd:
a(m,n) = 2^((m*n-1)/2-1)*(2^((m*n-1)/2) + 2^((m-1)/2) + 2^((n-1)/2) + 1).
m even:
a(m,n) = 2^m*a(m,n-1) + 2^m*a(m,n-2) - (2^m)^2*a(m,n-3) with n>2, a(m,0)=1, a(m,1)=a(1,m), a(m,2)=a(2,m).
m odd:
a(m,n) = 2^m*a(m,n-1) + 2^m*a(m,n-2) - (2^m)^2*a(m,n-3) - 2^(((m+1)/2)*n-3)*(2^((m-1)/2)-1) with n>2, a(m,0)=1, a(m,1)=a(1,m), a(m,2)=a(2,m).
Only a(1,n) and a(2,n) (A005418 and A225826) sequences are needed to define the others.

A016152 a(n) = 4^(n-1)*(2^n-1).

Original entry on oeis.org

0, 1, 12, 112, 960, 7936, 64512, 520192, 4177920, 33488896, 268173312, 2146435072, 17175674880, 137422176256, 1099444518912, 8795824586752, 70367670435840, 562945658454016, 4503582447501312, 36028728299487232

Offset: 0

N. J. A. Sloane

Numbers whose binary representation is the concatenation of n digits 1 and 2(n-1) digits 0, for n>0. (See A147816.) - Omar E. Pol, Nov 13 2008
a(n) is the number of lattices L in Z^n such that the quotient group Z^n / L is C_8. - Álvar Ibeas, Nov 29 2015
a(n) is a maximum number of intercalates in a Latin square of order 2^n (see A092237). - Eduard I. Vatutin, Apr 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..140
Eduard I. Vatutin, Example of Latin squares of order 2^n with maximum number of intercalates.
Index entries for linear recurrences with constant coefficients, signature (12,-32).

Second column of triangle A075499.
Cf. A019677, A147538, A147816.
Cf. A065442, A092237.

[4^(n-1)*(2^n-1): n in [0..40]]; // Vincenzo Librandi, Apr 26 2011

Table[4^(n - 1) (2^n - 1), {n, 0, 19}] (* Michael De Vlieger, Nov 30 2015 *)

a(n)=4^(n-1)*(2^n-1) \\ Charles R Greathouse IV, Oct 07 2015

my(x='x+O('x^30)); concat(0, Vec(x/((1-4*x)*(1-8*x)))) \\ Altug Alkan, Dec 04 2015

[lucas_number1(n,12,32) for n in range(0, 20)] # Zerinvary Lajos, Apr 27 2009

From Barry E. Williams, Jan 17 2000: (Start)
a(n) = ((8^(n+1)) - 4^(n+1))/4.
a(n) = 12a(n-1) - 32a(n-2), n>0; a(0)=1. (End)
a(n) = (4^(n-1))*Stirling2(n+1, 2), n>=0, with Stirling2(n, m)=A008277(n, m).
a(n) = -4^(n-1) + 2*8^(n-1).
E.g.f. for a(n+1), n>=0: d^2/dx^2((((exp(4*x)-1)/4)^2)/2!) = -exp(4*x) + 2*exp(8*x).
G.f.: x/((1-4*x)*(1-8*x)).
((6+sqrt4)^n - (6-sqrt4)^n)/4 in Fibonacci form. Offset 1. a(3)=112. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008
a(n) + A160873(n) + A006096(n) = A006096(n+2), for n > 2. - Álvar Ibeas, Nov 29 2015
Sum_{n>0} 1/a(n) = 4*E - 16/3, where E is the Erdős-Borwein constant (A065442). - Peter McNair, Dec 19 2022

A138119 Concatenation of n digits 1 and 2*n-1 digits 0.

Original entry on oeis.org

10, 11000, 11100000, 11110000000, 11111000000000, 11111100000000000, 11111110000000000000, 11111111000000000000000, 11111111100000000000000000, 11111111110000000000000000000, 11111111111000000000000000000000, 11111111111100000000000000000000000

Offset: 1

Omar E. Pol, Apr 03 2008

a(n) has 3*n-1 digits.

a(n) is also A147538(n) written in base 2. - Omar E. Pol, Nov 08 2008.

			n ...... a(n)
1 ....... 10
2 ...... 11000
3 ..... 11100000
4 .... 11110000000
5 ... 11111000000000

Index entries for linear recurrences with constant coefficients, signature (1100,-100000).

Cf. A138118, A138145, A138146, A138147, A138719, A138720, A138721, A138722, A147538.

LinearRecurrence[{1100, -100000}, {10, 11000}, 15] (* Paolo Xausa, Feb 06 2024 *)

Vec(10*x/((100*x-1)*(1000*x-1)) + O(x^100)) \\ Colin Barker, Sep 16 2013

From Colin Barker, Sep 16 2013: (Start)
a(n) = 1100*a(n-1) - 100000*a(n-2).
G.f.: 10*x / ((100*x-1)*(1000*x-1)). (End)

A147595 a(n) is the number whose binary representation is A138144(n).

Original entry on oeis.org

1, 3, 7, 15, 27, 51, 99, 195, 387, 771, 1539, 3075, 6147, 12291, 24579, 49155, 98307, 196611, 393219, 786435, 1572867, 3145731, 6291459, 12582915, 25165827, 50331651, 100663299, 201326595, 402653187, 805306371, 1610612739, 3221225475

Offset: 1

Omar E. Pol, Nov 08 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A138144, A145641, A147537, A147538, A147539, A147540, A147590, A147596, A147597.

[1,3,7] cat [3*(1+2^(n-2)): n in [4..40]]; // G. C. Greubel, Oct 25 2022

LinearRecurrence[{3,-2},{1,3,7,15,27},40] (* Harvey P. Dale, Nov 30 2020 *)

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

[1,3,7]+[3*(1+2^(n-2)) for n in range(4,40)] # G. C. Greubel, Oct 25 2022

a(n) = A060013(n+2), n > 3. - R. J. Mathar, Feb 05 2010
a(n+4) = 3*(2^(n+2) + 1), n >= 0. - Brad Clardy, Apr 03 2013
From Colin Barker, Sep 15 2013: (Start)
a(n) = 3*(4 + 2^n)/4 for n>3.
a(n) = 3*a(n-1) - 2*a(n-2).
G.f.: x*(1-2*x^2)*(1+2*x^2) / ((1-x)*(1-2*x)). (End)
E.g.f.: (3/4)*(4*exp(x) + exp(2*x)) - (15/4) - 7*x/2 - 3*x^2/2 - x^3/3. - G. C. Greubel, Oct 25 2022

Extended by R. J. Mathar, Feb 05 2010

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

A181710 Near-perfect numbers (A181595) of the form m2^p, where m = 2^(p-1)(2^p-1) is a perfect number (A000396).

Original entry on oeis.org

24, 224, 15872, 1040384, 274844352512, 1125891316908032, 72057456598974464, 4951760154835678090382802944, 6129982163463555430774932117031404988667342368173719552

Offset: 1

Vladimir Shevelev, Nov 07 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..14

Subsequence of A181595 and A181702.

Cf. A000043, A000396, A147538, A181596, A000396, A181701, A181702.

With[{p = MersennePrimeExponent[Range[10]]}, 2^(2*p - 1)*(2^p - 1)] (* Amiram Eldar, Apr 29 2024 *)

a(n) = A147538(A000043(n)). - Amiram Eldar, Apr 29 2024

a(6)-a(9) from Amiram Eldar, Apr 29 2024

cp's OEIS Frontend

A225910 Square array read by antidiagonals: a(m,n) is the number of binary pattern classes in the (m,n)-rectangular grid, two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

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A016152 a(n) = 4^(n-1)*(2^n-1).

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A138119 Concatenation of n digits 1 and 2*n-1 digits 0.

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A147595 a(n) is the number whose binary representation is A138144(n).

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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).

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Author

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Programs

Magma

Mathematica

PARI

SageMath

Formula

Extensions

A181710 Near-perfect numbers (A181595) of the form m*2^p, where m = 2^(p-1)*(2^p-1) is a perfect number (A000396).

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A181710 Near-perfect numbers (A181595) of the form m2^p, where m = 2^(p-1)(2^p-1) is a perfect number (A000396).