A185346 -id:A185346 - cp's OEIS frontend

A185246 Number of disconnected 4-regular simple graphs on n vertices with girth at least 6.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 5, 0, 23, 0, 1301, 25, 495379, 13529

Offset: 0

Jason Kimberley, Feb 22 2011

Jason Kimberley, Disconnected regular graphs with girth at least 6
Jason Kimberley, Index of sequences counting disconnected k-regular simple graphs with girth at least g

4-regular simple graphs with girth at least 4: A058348 (connected), this sequence (disconnected), A185346 (not necessarily connected).
Disconnected 4-regular simple graphs with girth at least g: A033483 (g=3), A185244 (g=4), A185245 (g=5), this sequence (g=6).
Disconnected k-regular simple graphs with girth at least 6: A185216 (all k), A185206 (triangle); A185226 (k=2), A185236 (k=3), this sequence (k=4).

a(n) = A185346(n) - A058348(n) = Euler_transformation(A058348)(n) - A058348(n).

A220087 a(n) = 2^n - 27.

Original entry on oeis.org

-26, -25, -23, -19, -11, 5, 37, 101, 229, 485, 997, 2021, 4069, 8165, 16357, 32741, 65509, 131045, 262117, 524261, 1048549, 2097125, 4194277, 8388581, 16777189, 33554405, 67108837, 134217701, 268435429, 536870885, 1073741797, 2147483621, 4294967269

Offset: 0

Andreas Rieber, Dec 04 2012

sign

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A036563, A185346.

Table[2^n - 27, {n, 0, 40}] (* T. D. Noe, Dec 04 2012 *)
LinearRecurrence[{3,-2},{-26,-25},40] (* Harvey P. Dale, May 17 2018 *)

From Chai Wah Wu, Jan 17 2020: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
G.f.: (53*x - 26)/((x - 1)*(2*x - 1)). (End)
From Elmo R. Oliveira, Nov 08 2023: (Start)
a(n) = 2*a(n-1) + 27 with a(0) = -26.
E.g.f.: exp(2*x) - 27*exp(x). (End)

A367559 Square array T(n, k) = 2^k - n, read by ascending antidiagonals.

Original entry on oeis.org

1, 0, 2, -1, 1, 4, -2, 0, 3, 8, -3, -1, 2, 7, 16, -4, -2, 1, 6, 15, 32, -5, -3, 0, 5, 14, 31, 64, -6, -4, -1, 4, 13, 30, 63, 128, -7, -5, -2, 3, 12, 29, 62, 127, 256, -8, -6, -3, 2, 11, 28, 61, 126, 255, 512, -9, -7, -4, 1, 10, 27, 60, 125, 254, 511, 1024

Offset: 0

Paul Curtz, Nov 22 2023

			This sequence as square array T(n, k):
  n\k  0    1    2    3    4    5    6    7    8    9    10.
  ---------------------------------------------------------.
  0 :  1    2    4    8   16   32   64  128  256  512  1024.
  1 :  0    1    3    7   15   31   63  127  255  511  1023.
  2 : -1    0    2    6   14   30   62  126  254  510  1022.
  3 : -2   -1    1    5   13   29   61  125  253  509  1021.
  4 : -3   -2    0    4   12   28   60  124  252  508  1020.
  5 : -4   -3   -1    3   11   27   59  123  251  507  1019.
  6 : -5   -4   -2    2   10   26   58  122  250  506  1018.
  7 : -6   -5   -3    1    9   25   57  121  249  505  1017.
  8 : -7   -6   -4    0    8   24   56  120  248  504  1016.
  9 : -8   -7   -5   -1    7   23   55  119  247  503  1015.
  10: -9   -8   -6   -2    6   22   54  118  246  502  1014.

Paolo Xausa, Table of n, a(n) for n = 0..11475 (antidiagonals 0..150, flattened).

Cf. A000079, A000225, A000325, A000295.
Cf. A000325, A000918, A084634, A036563.
Cf. A168616, A185346, A246168.
Cf. A002262, A025581.

Table[2^k-n+k,{n,0,10},{k,0,n}] (* Paolo Xausa, Nov 28 2023 *)

T(n, k) = 2^k-n \\ Thomas Scheuerle, Nov 23 2023

G.f. of row n: 1/(1-2*x) - n/(1-x).
E.g.f. of row n: exp(2*x) - n*exp(x).
T(0, k) = 2^k = A000079(k).
T(1, k) = 2^k - 1 = A000225(k).
T(2, k) = 2^k - 2 = A000918(k).
T(3, k) = 2^k - 3 = A036563(k).
T(5, k) = 2^k - 5 = A168616(k).
T(9, k) = 2^k - 9 = A185346(k).
T(10, k) = 2^k - 10 = A246168(k).
T(n, k) = 3*T(n, k-1) - 2*T(n, k-2) for k > 1.
T(n+1, k) = T(n, k) + 1.
T(n, n) = 2^n - n = A000325(n).
Sum_{k = 0..n} T(n - k, k) = A084634(n).
a(n) = 2^A002262(n) - A025581(n).
G.f.: (1 - 2*x - y + 3*x*y)/((1 - x)^2*(1 - y)*(1 - 2*y)). - Stefano Spezia, Nov 27 2023

A246168 a(n) = 2^n - 10.

Original entry on oeis.org

-9, -8, -6, -2, 6, 22, 54, 118, 246, 502, 1014, 2038, 4086, 8182, 16374, 32758, 65526, 131062, 262134, 524278, 1048566, 2097142, 4194294, 8388598, 16777206, 33554422, 67108854, 134217718, 268435446, 536870902, 1073741814, 2147483638

Offset: 0

Vincenzo Librandi, Aug 18 2014

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

Sequences of the form 2^n-k: A000079 (k=0), A000225 (k=1), A000918 (k=2), A036563 (k=3), A028399 (k=4), A168616 (k=5), A131130 (k=6), A048490 (k=7), A159741 (k=8), A185346 (k=9), this sequence (k=10).

Magma
```
[2^n-10: n in [0..40]];
```

Table[2^n - 10, {n, 0, 35}] (* or *) CoefficientList[Series[(-9 + 19 x)/(1 - 3 x + 2 x^2), {x, 0, 35}], x]
LinearRecurrence[{3,-2},{-9,-8},50] (* Harvey P. Dale, Jan 11 2024 *)

vector(50, n, 2^(n-1)-10) \\ Derek Orr, Aug 18 2014

G.f.: (-9+19*x)/(1-3*x+2*x^2).
a(n) = 3*a(n-1) - 2*a(n-2).
a(n) = A000079(n) - 10.
From Elmo R. Oliveira, Dec 21 2023: (Start)
a(n) = 2*a(n-1) + 10 for n>0.
E.g.f.: exp(x)*(exp(x) - 10). (End)

A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

Original entry on oeis.org

0, 2, -1, 8, 1, -3, 26, 7, -1, -7, 80, 25, 5, -5, -15, 242, 79, 23, 1, -13, -31, 728, 241, 77, 19, -7, -29, -63, 2186, 727, 239, 73, 11, -23, -61, -127, 6560, 2185, 725, 235, 65, -5, -55, -125, -255, 19682, 6559, 2183, 721, 227, 49, -37, -119, -253, -511, 59048, 19681, 6557, 2179, 713, 211, 17, -101, -247, -509, -1023

Offset: 0

K. G. Stier, Jan 22 2015

Table shows differences of a given power of 3 to the powers of 2 (columns), and differences of the powers of 3 to a given power of 2 (rows), respectively.

Note that positive terms (table's upper right area) and negative terms (lower left area) are separated by an imaginary line with slope -log(3)/log(2) = -1.5849625.. (see A020857). This "border zone" of the table is of interest in terms of how close powers of 3 and powers of 2 can get: i.e., those T(n,k) where k/n is a good rational approximation to log(3)/log(2), see A254351 for numerators k and respective A060528 for denominators n.

			Table begins
   0    2   8  26  80..
  -1    1   7  25  79..
  -3   -1   5  23  73..
  -7   -5   1  19  65..
  -15 -13  -7  11  49..
  ..   ..  ..  ..  ..

K. G. Stier, Table of n, a(n) for n = 0..5150

Row 0 (=3^n-1) is A024023.
Row 1 (=3^n-2) is A058481.
Row 2 (=3^n-4) is A168611.
Column 0 (=1-2^n) is (-1)A000225.
Column 1 (=3-2^n) is (-1)A036563.
Column 2 (=9-2^n) is (-1)A185346.
Column 3 (=27-2^n) is (-1)A220087.
0,0-Diagonal (=3^n-2^n) is A001047.
1,0-Diagonal (=3^n-2^(n-1)) for n>0 is A083313 or A064686.
0,1-Diagonal (=3^n-2^(n+1)) is A003063.
0,2-Diagonal (=3^n-2^(n+2)) is A214091.

Table[3^(n-k) - 2^k, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 18 2017 *)

for(i=0, 10, {
     for(j=0, i, print1((3^(i-j)-2^j),", "))
});

A172252 a(n) = 4*2^n - 9.

Original entry on oeis.org

-1, 7, 23, 55, 119, 247, 503, 1015, 2039, 4087, 8183, 16375, 32759, 65527, 131063, 262135, 524279, 1048567, 2097143, 4194295, 8388599, 16777207, 33554423, 67108855, 134217719, 268435447, 536870903, 1073741815, 2147483639, 4294967287, 8589934583, 17179869175, 34359738359

Offset: 1

Artur Jasinski, Jan 29 2010

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A159741, A185346, A206853.

Table[4 2^n - 9, {n, 1, 100}]
LinearRecurrence[{3,-2},{-1,7},30] (* Harvey P. Dale, May 27 2021 *)

my(x='x+O('x^34)); Vec(x*(10*x-1)/((x-1)*(2*x-1))) \\ Elmo R. Oliveira, Jun 15 2025

a(n) = 2*a(n-1) + 9, a(1)= -1. - Vincenzo Librandi, Mar 20 2011
For n >= 3, a(n) = 8<+>(n+2), where operation <+> is defined in A206853. - Vladimir Shevelev, Feb 17 2012
From Elmo R. Oliveira, Jun 15 2025: (Start)
G.f.: x*(10*x-1)/((x-1)*(2*x-1)).
E.g.f.: 5 + exp(x)*(4*exp(x) - 9).
a(n) = A185346(n+2) = 4*A000225(n) - 5.
a(n) = A159741(n-1) - 1 for n > 1. (End)

More terms from Elmo R. Oliveira, Jun 15 2025

A220088 a(n) = 2^n - 81.

Original entry on oeis.org

-80, -79, -77, -73, -65, -49, -17, 47, 175, 431, 943, 1967, 4015, 8111, 16303, 32687, 65455, 130991, 262063, 524207, 1048495, 2097071, 4194223, 8388527, 16777135, 33554351, 67108783, 134217647, 268435375, 536870831, 1073741743, 2147483567, 4294967215

Offset: 0

Andreas Rieber, Dec 04 2012

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A036563, A185346, A220087.

Table[2^n - 81, {n, 0, 40}] (* T. D. Noe, Dec 04 2012 *)

From Chai Wah Wu, Jan 17 2020: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
G.f.: (161*x - 80)/((x - 1)*(2*x - 1)). (End)
From Elmo R. Oliveira, Nov 11 2023: (Start)
a(n) = 2*a(n-1) + 81 with a(0) = -80.
E.g.f.: exp(2*x) - 81*exp(x). (End)

A220089 a(n) = 2^n - 243.

Original entry on oeis.org

-242, -241, -239, -235, -227, -211, -179, -115, 13, 269, 781, 1805, 3853, 7949, 16141, 32525, 65293, 130829, 261901, 524045, 1048333, 2096909, 4194061, 8388365, 16776973, 33554189, 67108621, 134217485, 268435213, 536870669, 1073741581, 2147483405, 4294967053

Offset: 0

Andreas Rieber, Dec 04 2012

sign

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A036563, A185346, A220087, A220088.

Table[2^n - 243, {n, 0, 40}] (* T. D. Noe, Dec 04 2012 *)

From Chai Wah Wu, Jan 17 2020: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
G.f.: (485*x - 242)/((x - 1)*(2*x - 1)). (End)
From Elmo R. Oliveira, Nov 11 2023: (Start)
a(n) = 2*a(n-1) + 243 with a(0) = -242.
E.g.f.: exp(2*x) - 243*exp(x). (End)

A294364 Linear recurrence with signature (1,1,-1,1,1), where the first terms are powers of 2 (1,2,4,8,16).

Original entry on oeis.org

1, 2, 4, 8, 16, 23, 37, 56, 94, 152, 250, 401, 649, 1046, 1696, 2744, 4444, 7187, 11629, 18812, 30442, 49256, 79702, 128957, 208657, 337610, 546268, 883880, 1430152, 2314031, 3744181, 6058208, 9802390, 15860600, 25662994, 41523593, 67186585, 108710174, 175896760, 284606936

Offset: 0

Jean-François Alcover and Paul Curtz, Oct 29 2017

The interest of this sequence mainly lies in the peculiarities of its array of successive differences, which begins:
1,    2,   4,   8,  16,  23,  37,  56,  94, ...
1,    2,   4,   8,   7,  14,  19,  38,  58, ...
1,    2,   4,  -1,   7,   5,  19,  20,  40, ...
1,    2,  -5,   8,  -2,  14,   1,  20,  13, ...
1,   -7,  13, -10,  16, -13,  19,  -7,  31, ...
-8,  20, -23,  26, -29,  32, -26,  38, -23, ...
28, -43,  49, -55,  61, -58,  64, -61,  67, ...
The main diagonal is A000079 (powers of 2).
The first upper subdiagonal is A254076.
The second upper subdiagonal (4, 8, 7, 14, 19, 38, ...) is not in the OEIS.
The third upper subdiagonal is A185346 (2^n-9).
Every row, once computed mod 9, is 6-periodic, repeating (1, 2, 4, 8, 7, 5) (A153130).

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

Cf. A000045, A000079, A153130, A185346, A254076.

LinearRecurrence[{1, 1, -1, 1, 1}, {1, 2, 4, 8, 16}, 40]

G.f.: (1+x+x^2+3*x^3+5*x^4) / (1-x-x^2+x^3-x^4-x^5).
a(n) = (9/2)*fibonacci(n) + (-1)^n - sqrt(3)*sin(n*Pi/3).
a(n) ~ (9/2)*fibonacci(n).

cp's OEIS Frontend

A185246 Number of disconnected 4-regular simple graphs on n vertices with girth at least 6.

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A220087 a(n) = 2^n - 27.

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A367559 Square array T(n, k) = 2^k - n, read by ascending antidiagonals.

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A246168 a(n) = 2^n - 10.

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A254027 Table T(n,k) = 3^n - 2^k read by antidiagonals.

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A172252 a(n) = 4*2^n - 9.

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A220088 a(n) = 2^n - 81.

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A220089 a(n) = 2^n - 243.

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A294364 Linear recurrence with signature (1,1,-1,1,1), where the first terms are powers of 2 (1,2,4,8,16).