A265077
Coordination sequence for (3,6,8) tiling of hyperbolic plane.
Original entry on oeis.org
1, 3, 6, 11, 20, 37, 66, 117, 208, 371, 662, 1179, 2100, 3741, 6666, 11877, 21160, 37699, 67166, 119667, 213204, 379853, 676762, 1205749, 2148216, 3827355, 6818982, 12148995, 21645180, 38563997, 68707298, 122411917, 218094408, 388566507, 692287030, 1233408755, 2197494812, 3915152565, 6975406506, 12427688349
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
- Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,1,-1).
Coordination sequences for triangular tilings of hyperbolic space:
A001630,
A007283,
A054886,
A078042,
A096231,
A163876,
A179070,
A265057,
A265058,
A265059,
A265060,
A265061,
A265062,
A265063,
A265064,
A265065,
A265066,
A265067,
A265068,
A265069,
A265070,
A265071,
A265072,
A265073,
A265074,
A265075,
A265076,
A265077.
-
I:=[1,3,6,11,20,37,66]; [n le 7 select I[n] else Self(n-1)+Self(n-2)+Self(n-4) + Self(n-5)-Self(n-6): n in [1..50]]; // Vincenzo Librandi, Dec 30 2015
-
CoefficientList[Series[(x^5 + x^4 + x^3 + x^2 + x + 1) (x + 1)/(x^6 - x^5 - x^4 - x^2 - x + 1), {x, 0, 60}], x] (* Vincenzo Librandi, Dec 30 2015 *)
-
Vec((x^5+x^4+x^3+x^2+x+1)*(x+1)/(x^6-x^5-x^4-x^2-x+1) + O(x^50)) \\ Michel Marcus, Dec 30 2015
A060482
New record highs reached in A060030.
Original entry on oeis.org
1, 2, 3, 5, 9, 13, 21, 29, 45, 61, 93, 125, 189, 253, 381, 509, 765, 1021, 1533, 2045, 3069, 4093, 6141, 8189, 12285, 16381, 24573, 32765, 49149, 65533, 98301, 131069, 196605, 262141, 393213, 524285, 786429, 1048573, 1572861, 2097149, 3145725, 4194301, 6291453
Offset: 1
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with
A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent:
A029744 (s(n));
A052955 (s(n)-1),
A027383 (s(n)-2),
A354788 (s(n)-3),
A347789 (s(n)-4),
A209721 (s(n)+1),
A209722 (s(n)+2),
A343177 (s(n)+3),
A209723 (s(n)+4);
A060482,
A136252 (minor differences from
A354788 at the start);
A354785 (3*s(n)),
A354789 (3*s(n)-7). The first differences of
A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences:
A016116,
A060546,
A117575,
A131572,
A152166,
A158780,
A163403,
A320770. The bisections of
A029744 are
A000079 and
A007283. -
N. J. A. Sloane, Jul 14 2022
-
LinearRecurrence[{1,2,-2},{1,2,3,5,9},50] (* Harvey P. Dale, Sep 11 2016 *)
-
{ for (n=1, 1000, if (n%2==0, m=n/2; a=2^(m + 1) - 3, m=(n - 1)/2; a=3*2^m - 3); if (n<3, a=n); write("b060482.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 05 2009
A136252
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).
Original entry on oeis.org
1, 3, 5, 9, 13, 21, 29, 45, 61, 93, 125, 189, 253, 381, 509, 765, 1021, 1533, 2045, 3069, 4093, 6141, 8189, 12285, 16381, 24573, 32765, 49149, 65533, 98301, 131069, 196605, 262141, 393213, 524285, 786429, 1048573, 1572861, 2097149, 3145725, 4194301, 6291453, 8388605
Offset: 0
Cf.
A007664 (Optimal 4-peg Tower of Hanoi).
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with
A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent:
A029744 (s(n));
A052955 (s(n)-1),
A027383 (s(n)-2),
A354788 (s(n)-3),
A347789 (s(n)-4),
A209721 (s(n)+1),
A209722 (s(n)+2),
A343177 (s(n)+3),
A209723 (s(n)+4);
A060482,
A136252 (minor differences from
A354788 at the start);
A354785 (3*s(n)),
A354789 (3*s(n)-7). The first differences of
A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences:
A016116,
A060546,
A117575,
A131572,
A152166,
A158780,
A163403,
A320770. The bisections of
A029744 are
A000079 and
A007283. -
N. J. A. Sloane, Jul 14 2022
-
a:=proc(n) options operator,arrow: 2^((1/2)*n-1)*(4+4*(-1)^n+3*sqrt(2)*(1-(-1)^n))-3 end proc: seq(a(n),n=0..40); # Emeric Deutsch, Mar 31 2008
-
LinearRecurrence[{1, 2, -2}, {1, 3, 5}, 100] (* G. C. Greubel, Feb 18 2017 *)
-
x='x+O('x^50); Vec((1+2*x)/((1-x)*(1-2*x^2))) \\ G. C. Greubel, Feb 18 2017
A152166
a(2*n) = 2^n; a(2*n+1) = -(2^(n+1)).
Original entry on oeis.org
1, -2, 2, -4, 4, -8, 8, -16, 16, -32, 32, -64, 64, -128, 128, -256, 256, -512, 512, -1024, 1024, -2048, 2048, -4096, 4096, -8192, 8192, -16384, 16384, -32768, 32768, -65536, 65536, -131072, 131072, -262144, 262144, -524288, 524288, -1048576, 1048576
Offset: 0
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with
A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent:
A029744 (s(n));
A052955 (s(n)-1),
A027383 (s(n)-2),
A354788 (s(n)-3),
A347789 (s(n)-4),
A209721 (s(n)+1),
A209722 (s(n)+2),
A343177 (s(n)+3),
A209723 (s(n)+4);
A060482,
A136252 (minor differences from
A354788 at the start);
A354785 (3*s(n)),
A354789 (3*s(n)-7). The first differences of
A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences:
A016116,
A060546,
A117575,
A131572,
A152166,
A158780,
A163403,
A320770. The bisections of
A029744 are
A000079 and
A007283. -
N. J. A. Sloane, Jul 14 2022
-
LinearRecurrence[{0, 2}, {1, -2}, 50] (* Paolo Xausa, Jul 19 2024 *)
A173786
Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.
Original entry on oeis.org
2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 68, 72, 80, 96, 128, 129, 130, 132, 136, 144, 160, 192, 256, 257, 258, 260, 264, 272, 288, 320, 384, 512, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048
Offset: 0
Triangle begins as:
2;
3, 4;
5, 6, 8;
9, 10, 12, 16;
17, 18, 20, 24, 32;
33, 34, 36, 40, 48, 64;
65, 66, 68, 72, 80, 96, 128;
129, 130, 132, 136, 144, 160, 192, 256;
257, 258, 260, 264, 272, 288, 320, 384, 512;
513, 514, 516, 520, 528, 544, 576, 640, 768, 1024;
1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048;
-
[2^n + 2^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 07 2021
-
Flatten[Table[2^n + 2^m, {n,0,10}, {m, 0, n}]] (* T. D. Noe, Jun 18 2013 *)
-
A173786(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); 1 << c + 1 << (n - binomial(c + 1, 2)); }; \\ Antti Karttunen, Feb 29 2024, after David A. Corneth's PARI-program in A048645
-
from math import isqrt, comb
def A173786(n):
a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
return (1<Chai Wah Wu, Jun 20 2025
-
flatten([[2^n + 2^k for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 07 2021
A117575
Expansion of (1-x^3)/((1-x)*(1+2*x^2)).
Original entry on oeis.org
1, 1, -1, -2, 2, 4, -4, -8, 8, 16, -16, -32, 32, 64, -64, -128, 128, 256, -256, -512, 512, 1024, -1024, -2048, 2048, 4096, -4096, -8192, 8192, 16384, -16384, -32768, 32768, 65536, -65536, -131072, 131072, 262144, -262144, -524288, 524288
Offset: 0
0/1, 1/1 1/1, 1/2, 0/2, -1/4, -1/4, -1/8, ...
1/1, 0/1, -1/2, -1/2, -1/4, 0/4, 1/8, 1/8, ...
-1/1, -1/2, 0/2, 1/4, 1/4, 1/8, 0/8, -1/16, ...
1/2, 1/2, 1/4, 0/4 -1/8, -1/8, -1/16, 0/16, ...
0/2, -1/4, -1/4, -1/8, 0/8, 1/16, 1/16, 1/32, ...
-1/4, 0/4, 1/8, 1/8, 1/16, 0/16, -1/32, -1/32, ...
1/4, 1/8, 0/8, -1/16, -1/16, -1/32, 0/32, 1/64, ...
-1/8, -1/8, -1/16, 0/16, 1/32, 1/32, 1/64, 0/64. - _Paul Curtz_, Oct 24 2012
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with
A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent:
A029744 (s(n));
A052955 (s(n)-1),
A027383 (s(n)-2),
A354788 (s(n)-3),
A347789 (s(n)-4),
A209721 (s(n)+1),
A209722 (s(n)+2),
A343177 (s(n)+3),
A209723 (s(n)+4);
A060482,
A136252 (minor differences from
A354788 at the start);
A354785 (3*s(n)),
A354789 (3*s(n)-7). The first differences of
A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences:
A016116,
A060546,
A117575,
A131572,
A152166,
A158780,
A163403,
A320770. The bisections of
A029744 are
A000079 and
A007283. -
N. J. A. Sloane, Jul 14 2022
-
[1] cat [(-1)^Floor(n/2)*2^Floor((n-1)/2): n in [1..50]]; // G. C. Greubel, Apr 19 2023
-
CoefficientList[Series[(1-x^3)/((1-x)(1+2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{0,-2},{1,1,-1},45] (* Harvey P. Dale, Apr 09 2018 *)
-
a(n)=if(n,(-1)^(n\2)<<((n-1)\2),1) \\ Charles R Greathouse IV, Jan 31 2012
-
def A117575(n): return 1 if (n==0) else (-1)^(n//2)*2^((n-1)//2)
[A117575(n) for n in range(51)] # G. C. Greubel, Apr 19 2023
Original entry on oeis.org
0, 1, 1, 1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152, 4194304
Offset: 0
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with
A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent:
A029744 (s(n));
A052955 (s(n)-1),
A027383 (s(n)-2),
A354788 (s(n)-3),
A347789 (s(n)-4),
A209721 (s(n)+1),
A209722 (s(n)+2),
A343177 (s(n)+3),
A209723 (s(n)+4);
A060482,
A136252 (minor differences from
A354788 at the start);
A354785 (3*s(n)),
A354789 (3*s(n)-7). The first differences of
A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences:
A016116,
A060546,
A117575,
A131572,
A152166,
A158780,
A163403,
A320770. The bisections of
A029744 are
A000079 and
A007283. -
N. J. A. Sloane, Jul 14 2022
-
[0,1] cat [2^Floor((n-2)/2): n in [2..50]]; // G. C. Greubel, Apr 19 2023
-
Table[(2^Floor[n/2] +Boole[n==1] -Boole[n==0])/2, {n,0,50}] (* or *) LinearRecurrence[{0,2}, {0,1,1,1}, 51] (* G. C. Greubel, Apr 19 2023 *)
-
a(n)=if(n>3,([0,1; 2,0]^n*[1;1])[1,1]/2,n>0) \\ Charles R Greathouse IV, Oct 18 2022
-
def A158780(n): return (2^(n//2) + int(n==1) - int(n==0))/2
[A158780(n) for n in range(51)] # G. C. Greubel, Apr 19 2023
A354789
a(2*n) = 9*2^n - 7, a(2*n+1) = 3*2^(n+2) - 7.
Original entry on oeis.org
2, 5, 11, 17, 29, 41, 65, 89, 137, 185, 281, 377, 569, 761, 1145, 1529, 2297, 3065, 4601, 6137, 9209, 12281, 18425, 24569, 36857, 49145, 73721, 98297, 147449, 196601, 294905, 393209, 589817, 786425, 1179641, 1572857, 2359289, 3145721, 4718585, 6291449, 9437177, 12582905, 18874361, 25165817, 37748729, 50331641, 75497465
Offset: 0
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with
A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent:
A029744 (s(n));
A052955 (s(n)-1),
A027383 (s(n)-2),
A354788 (s(n)-3),
A347789 (s(n)-4),
A209721 (s(n)+1),
A209722 (s(n)+2),
A343177 (s(n)+3),
A209723 (s(n)+4);
A060482,
A136252 (minor differences from
A354788 at the start);
A354785 (3*s(n)),
A354789 (3*s(n)-7). The first differences of
A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences:
A016116,
A060546,
A117575,
A131572,
A152166,
A158780,
A163403,
A320770. The bisections of
A029744 are
A000079 and
A007283.
-
LinearRecurrence[{1,2,-2},{2,5,11},100] (* Paolo Xausa, Oct 17 2023 *)
CoefficientList[Series[(2+3x+2x^2)/((1-x)(1-2x^2)),{x,0,50}],x] (* Harvey P. Dale, Jun 07 2024 *)
A131572
a(0) = 0 and a(1) = 1, continued such that absolute values of 2nd differences equal the original sequence.
Original entry on oeis.org
0, 1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576
Offset: 0
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with
A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent:
A029744 (s(n));
A052955 (s(n)-1),
A027383 (s(n)-2),
A354788 (s(n)-3),
A347789 (s(n)-4),
A209721 (s(n)+1),
A209722 (s(n)+2),
A343177 (s(n)+3),
A209723 (s(n)+4);
A060482,
A136252 (minor differences from
A354788 at the start);
A354785 (3*s(n)),
A354789 (3*s(n)-7). The first differences of
A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences:
A016116,
A060546,
A117575,
A131572,
A152166,
A158780,
A163403,
A320770. The bisections of
A029744 are
A000079 and
A007283. -
N. J. A. Sloane, Jul 14 2022
A354788
a(2*k) = 3*2^k - 3, a(2*k+1) = 2^(k+2) - 3.
Original entry on oeis.org
0, 1, 3, 5, 9, 13, 21, 29, 45, 61, 93, 125, 189, 253, 381, 509, 765, 1021, 1533, 2045, 3069, 4093, 6141, 8189, 12285, 16381, 24573, 32765, 49149, 65533, 98301, 131069, 196605, 262141, 393213, 524285, 786429, 1048573, 1572861, 2097149, 3145725, 4194301, 6291453, 8388605, 12582909, 16777213, 25165821, 33554429, 50331645
Offset: 0
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with
A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent:
A029744 (s(n));
A052955 (s(n)-1),
A027383 (s(n)-2),
A354788 (s(n)-3),
A347789 (s(n)-4),
A209721 (s(n)+1),
A209722 (s(n)+2),
A343177 (s(n)+3),
A209723 (s(n)+4);
A060482,
A136252 (minor differences from
A354788 at the start);
A354785 (3*s(n)),
A354789 (3*s(n)-7). The first differences of
A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences:
A016116,
A060546,
A117575,
A131572,
A152166,
A158780,
A163403,
A320770. The bisections of
A029744 are
A000079 and
A007283. -
N. J. A. Sloane, Jul 14 2022
-
f1:=proc(n) if (n mod 2) = 1 then 2^((n+3)/2)-3 else 3*2^(n/2)-3; fi; end;
[seq(f1(n),n=0..45)];
-
LinearRecurrence[{1,2,-2},{0,1,3},100] (* Paolo Xausa, Oct 17 2023 *)
Previous
Showing 61-70 of 246 results.
Next
Comments