A052539
a(n) = 4^n + 1.
Original entry on oeis.org
2, 5, 17, 65, 257, 1025, 4097, 16385, 65537, 262145, 1048577, 4194305, 16777217, 67108865, 268435457, 1073741825, 4294967297, 17179869185, 68719476737, 274877906945, 1099511627777, 4398046511105, 17592186044417
Offset: 0
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
- Vincenzo Librandi, Table of n, a(n) for n = 0..175
- Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
- Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 470.
- Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
- Amelia Carolina Sparavigna, Some Groupoids and their Representations by Means of Integer Sequences, International Journal of Sciences 8(10) (2019).
- Eric Weisstein's World of Mathematics, Edge Cover Number.
- Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph.
- Wikipedia, Lucas sequence: Specific names.
- Index entries for linear recurrences with constant coefficients, signature (5,-4).
Other powers:
A000051,
A034472,
A034474,
A062394,
A034491,
A062395,
A062396,
A062397,
A007689,
A063376,
A063481,
A074600-
A074624,
A034524,
A178248,
A228081.
-
List([0..30], n-> 4^n+1); # G. C. Greubel, May 09 2019
-
[4^n+1: n in [0..30] ]; // Vincenzo Librandi, Apr 30 2011
-
spec := [S,{S=Union(Sequence(Union(Z,Z,Z,Z)),Sequence(Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..30);
A052539:=n->4^n + 1; seq(A052539(n), n=0..30); # Wesley Ivan Hurt, Jun 12 2014
-
Table[4^n + 1, {n, 0, 30}]
(* From Eric W. Weisstein, Sep 20 2017 *)
4^Range[0, 30] + 1
LinearRecurrence[{5, -4}, {2, 5}, 30]
CoefficientList[Series[(2-5x)/(1-5x+4x^2), {x, 0, 30}], x] (* End *)
-
a(n)=4^n+1 \\ Charles R Greathouse IV, Nov 20 2011
-
[4^n+1 for n in (0..30)] # G. C. Greubel, May 09 2019
A024036
a(n) = 4^n - 1.
Original entry on oeis.org
0, 3, 15, 63, 255, 1023, 4095, 16383, 65535, 262143, 1048575, 4194303, 16777215, 67108863, 268435455, 1073741823, 4294967295, 17179869183, 68719476735, 274877906943, 1099511627775, 4398046511103, 17592186044415, 70368744177663, 281474976710655
Offset: 0
G.f. = 3*x + 15*x^2 + 63*x^3 + 255*x^4 + 1023*x^5 + 4095*x^6 + ...
- Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
- Felix Fröhlich, Table of n, a(n) for n = 0..99
- Alexander V. Kitaev, Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 15 (2019), 046, 53 pages; arXiv preprint, arXiv:1809.00122 [math.CA], 2018-2019.
- Eric Weisstein's World of Mathematics, Cyclomatic Number.
- Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph.
- Index entries for linear recurrences with constant coefficients, signature (5,-4).
Cf.
A000051,
A000120,
A000225,
A000302,
A002001,
A002063,
A002193,
A002450,
A005057,
A010503,
A010532,
A010541,
A010767,
A015521,
A020988,
A027637 (partial products),
A078904 (partial sums),
A079978,
A080674,
A164346 (first differences),
A178789,
A179857,
A248721.
-
a024036 = (subtract 1) . a000302
a024036_list = iterate ((+ 3) . (* 4)) 0
-- Reinhard Zumkeller, Oct 03 2012
-
A024036:=n->4^n-1; seq(A024036(n), n=0..30); # Wesley Ivan Hurt, Apr 04 2014
-
Array[4^# - 1 &, 50, 0] (* Vladimir Joseph Stephan Orlovsky, Nov 03 2009 *)
(* Start from Eric W. Weisstein, Sep 19 2017 *)
Table[4^n - 1, {n, 0, 20}]
4^Range[0, 20] - 1
LinearRecurrence[{5, -4}, {0, 3}, 20]
CoefficientList[Series[3 x/(1 - 5 x + 4 x^2), {x, 0, 20}], x]
(* End *)
-
for(n=0, 100, print1(4^n-1, ", ")) \\ Felix Fröhlich, Jul 04 2014
-
[gaussian_binomial(2*n,1, 2) for n in range(21)] # Zerinvary Lajos, May 28 2009
-
[stirling_number2(2*n+1, 2) for n in range(21)] # Zerinvary Lajos, Nov 26 2009
A084221
a(n+2) = 4*a(n), with a(0)=1, a(1)=3.
Original entry on oeis.org
1, 3, 4, 12, 16, 48, 64, 192, 256, 768, 1024, 3072, 4096, 12288, 16384, 49152, 65536, 196608, 262144, 786432, 1048576, 3145728, 4194304, 12582912, 16777216, 50331648, 67108864, 201326592, 268435456, 805306368, 1073741824, 3221225472, 4294967296, 12884901888
Offset: 0
Binary...............Decimal
1..........................1
11.........................3
100........................4
1100......................12
10000.....................16
110000....................48
1000000...................64
11000000.................192
100000000................256
1100000000...............768
10000000000.............1024
110000000000............3072, etc. - _Philippe Deléham_, Mar 21 2014
-
[(5*2^n-(-2)^n)/4: n in [0..40]]; // Vincenzo Librandi, Aug 13 2011
-
CoefficientList[Series[(-3*x - 1)/(4*x^2 - 1), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 10 2011 *)
-
a(n)=([0,1; 4,0]^n*[1;3])[1,1] \\ Charles R Greathouse IV, Oct 03 2016
Original entry on oeis.org
3, 21, 147, 1029, 7203, 50421, 352947, 2470629, 17294403, 121060821, 847425747, 5931980229, 41523861603, 290667031221, 2034669218547, 14242684529829, 99698791708803, 697891541961621, 4885240793731347, 34196685556119429, 239376798892836003, 1675637592249852021
Offset: 0
Original entry on oeis.org
3, 18, 108, 648, 3888, 23328, 139968, 839808, 5038848, 30233088, 181398528, 1088391168, 6530347008, 39182082048, 235092492288, 1410554953728, 8463329722368, 50779978334208, 304679870005248, 1828079220031488, 10968475320188928, 65810851921133568, 394865111526801408
Offset: 0
A178789
a(n) = 4^(n-1) + 2: Number of acute angles after n iterations of the Koch snowflake construction.
Original entry on oeis.org
3, 6, 18, 66, 258, 1026, 4098, 16386, 65538, 262146, 1048578, 4194306, 16777218, 67108866, 268435458, 1073741826, 4294967298, 17179869186, 68719476738, 274877906946, 1099511627778, 4398046511106, 17592186044418, 70368744177666
Offset: 1
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
- Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
- Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 113.
- Larry Riddle, Koch Curve.
- Index entries for linear recurrences with constant coefficients, signature (5,-4).
-
[2^(2*(n-1)) + 2: n in [1..30]]; // Vincenzo Librandi, Feb 02 2013
-
A178789:=n->2+4^(n-1); seq(A178789(n), n=1..30); # Wesley Ivan Hurt, Dec 17 2013
-
a=b=3;lst={a};Do[a=a+b;b*=4;AppendTo[lst,a],{n,40}];lst
Flatten[Table[2^(2*(n-1)) + 2, {n, 1, 50}]](* or *) CoefficientList[Series[(3 - 9*x)/(1 - 5*x + 4*x^2),{x, 0, 100}], x] (* Vincenzo Librandi, Feb 02 2013 *)
-
A178789=n->4^(n-1)+2 \\ - M. F. Hasler, Dec 17 2013
A279125
Lexicographically earliest sequence such that, for any distinct i and j, a(i)=a(j) implies (i AND j)=0 (where AND stands for the bitwise AND operator).
Original entry on oeis.org
0, 0, 1, 0, 2, 3, 4, 0, 3, 2, 5, 1, 6, 7, 8, 0, 7, 6, 9, 5, 10, 11, 12, 4, 13, 14, 15, 16, 17, 18, 19, 0, 11, 10, 16, 9, 14, 13, 20, 12, 21, 22, 23, 24, 25, 26, 27, 1, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 0, 18, 17, 24, 15, 22, 21, 35, 9
Offset: 1
-
with(Bits):
n:= 100:
l:= []:
g:=[seq(0, i = 0..n-1)]:
for i from 1 to n by 1
do
a:= 0;
while (And(g[a + 1], i)) > 0
do
a++;
end do:
g[a + 1] += i;
l:= [op(l), a];
end do:
print(l); # Reza K Ghazi, Dec 29 2021
-
n = 100;
l = {};
g = ConstantArray[0, n];
For[i = 0, i < n, i++; a = 0; While[BitAnd[g[[a + 1]], i] > 0, a++];
g[[a + 1]] += i;
l = Append[l, a]];
l (* Reza K Ghazi, Dec 29 2021 *)
-
g = vector(72); for (n=1, #g, a = 0; while (bitand(g[a+1],n)>0, a++); g[a+1] += n; print1 (a", "))
-
n = 100
g = n * [0]
for i in range(1, n + 1):
a = 0
while g[a] & i:
a += 1
g[a] += i
print(a, end=', ') # Reza K Ghazi, Dec 29 2021
A340666
A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 8, 3, 4, 3, 0, 1, 16, 3, 16, 5, 3, 0, 1, 32, 3, 64, 9, 6, 7, 0, 1, 64, 3, 256, 17, 12, 7, 1, 0, 1, 128, 3, 1024, 33, 24, 7, 8, 3, 0, 1, 256, 3, 4096, 65, 48, 7, 64, 9, 3, 0, 1, 512, 3, 16384, 129, 96, 7, 512, 33, 10, 7
Offset: 0
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, ...
3, 3, 3, 3, 3, 3, 3, 3, 3, ...
1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, ...
3, 5, 9, 17, 33, 65, 129, 257, 513, ...
3, 6, 12, 24, 48, 96, 192, 384, 768, ...
7, 7, 7, 7, 7, 7, 7, 7, 7, ...
1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, ...
...
Rows n=0..17, 19 give:
A000004,
A000012,
A000079,
A010701,
A000302,
A000051(k+1),
A007283,
A010727,
A001018,
A087289,
A007582(k+1),
A062709(k+2),
A164346,
A181565(k+1),
A005009,
A181404(k+3),
A001025,
A199493,
A253208(k+1).
-
A:= (n, k)-> Bits[Join](subs(0=[0$k][], Bits[Split](n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n<2, n,
`if`(irem(n, 2, 'r')=1, A(r, k)*2+1, A(r, k)*2^k))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
-
A[n_, k_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence @@ Table[0, {k}], 2];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 02 2021 *)
A178872
Partial sums of round(4^n/7).
Original entry on oeis.org
0, 1, 3, 12, 49, 195, 780, 3121, 12483, 49932, 199729, 798915, 3195660, 12782641, 51130563, 204522252, 818089009, 3272356035, 13089424140, 52357696561, 209430786243, 837723144972, 3350892579889, 13403570319555, 53614281278220, 214457125112881
Offset: 0
-
[Floor((4*4^n+5)/21): n in [0..30]]; // Vincenzo Librandi, May 01 2011
-
A178872 := proc(n) add( round(4^i/7),i=0..n) ; end proc:
-
Join[{a = b = 0}, Table[c = 4^n - a - b; a = b; b = c, {n, 0, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
Accumulate[Round[4^Range[0,30]/7]] (* or *) LinearRecurrence[{3,3,4},{0,1,3},30] (* Harvey P. Dale, Feb 18 2023 *)
-
a(n) = (4^(n+1)+5)\21; \\ Altug Alkan, Oct 05 2017
A280345
a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].
Original entry on oeis.org
3, 7, 12, 25, 48, 97, 192, 385, 768, 1537, 3072, 6145, 12288, 24577, 49152, 98305, 196608, 393217, 786432, 1572865, 3145728, 6291457, 12582912, 25165825, 50331648, 100663297, 201326592, 402653185, 805306368, 1610612737, 3221225472, 6442450945, 12884901888
Offset: 0
a(0) = 3, a(1) = 2*3 + 1 = 7, a(2) = 2*7 - 2 = 12, a(3) = 2*12 + 1 = 25.
-
a[0] = 3; a[n_] := a[n] = 2 a[n - 1] + 1 + (-3) Boole[EvenQ@ n]; Table[a@ n, {n, 0, 32}] (* or *)
CoefficientList[Series[(3 + x - 5 x^2)/((1 - x) (1 + x) (1 - 2 x)), {x, 0, 32}], x] (* Michael De Vlieger, Jan 01 2017 *)
-
Vec((3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Jan 01 2017
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