A052531 -id:A052531 - cp's OEIS frontend

A052539 a(n) = 4^n + 1.

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

The sequence is a Lucas sequence V(P,Q) with P = 5 and Q = 4, so if n is a prime number, then V_n(5,4) - 5 is divisible by n. The smallest pseudoprime q which divides V_q(5,4) - 5 is 15.
Also the edge cover number of the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Sep 20 2017
First bisection of A000051, A049332, A052531 and A014551. - Klaus Purath, Sep 23 2020

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).

Cf. A164346 (first differences).
Other powers: A000051, A034472, A034474, A062394, A034491, A062395, A062396, A062397, A007689, A063376, A063481, A074600-A074624, A034524, A178248, A228081.
Cf. A019434, A274903.

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

a(n) = 4^n + 1.
a(n) = 4*a(n-1) - 3 = 5*a(n-1) - 4*a(n-2).
G.f.: (2 - 5*x)/((1 - 4*x)*(1 - x)).
E.g.f.: exp(x) + exp(4*x). - Mohammad K. Azarian, Jan 02 2009
From Klaus Purath, Sep 23 2020: (Start)
a(n) = 3*4^(n-1) + a(n-1).
a(n) = (a(n-1)^2 + 9*4^(n-2))/a(n-2).
a(n) = A178675(n) - 3. (End)

A352633 Lexicographically earliest sequence of distinct positive integers such for any n > 0, a(n) and a(n+1) are coprime and have no common 1-bits in their binary expansions.

Original entry on oeis.org

1, 2, 5, 8, 3, 4, 9, 16, 7, 24, 35, 12, 17, 6, 25, 32, 11, 20, 33, 10, 21, 34, 13, 18, 37, 26, 69, 40, 19, 36, 65, 14, 81, 38, 73, 22, 41, 64, 15, 112, 129, 28, 67, 44, 83, 128, 23, 72, 49, 66, 29, 96, 31, 160, 27, 68, 43, 80, 39, 88, 131, 48, 71, 56, 135, 104

Offset: 1

Rémy Sigrist, May 07 2022

This sequence combines features of A000027 (where two consecutive terms are coprime) and of A109812 (where two consecutive terms have no common 1-bits in their binary expansions).
For any n > 0, n and a(n) have the same parity.
The sequence is well defined:
- after an odd term v: we can extend the sequence with a power of 2 greater than any previous term,
- after an even term v < 2^k: we can extend the sequence with a prime number of the form 1 + t*2^k (Dirichlet's theorem on arithmetic progressions guarantees us that there is an infinity of such prime numbers).
This sequence is a permutation of the positive integers (with inverse A353604):
- the sequence is clearly unbounded,
- so we have even terms of infinitely many different binary lengths,
- the first even term with binary length w > 1 is necessarily 2^(w-1),
- so we have infinitely many powers of 2 in the sequence,
- so eventually all odd numbers will appear in the sequence,
- and all prime numbers will appear in the sequence,
- and eventually any even number v < 2^k must appear in the sequence (for instance after a prime number of the form 1 + t*2^k).

			The first terms, alongside their binary expansion and distinct prime factors, are:
  n   a(n)  bin(a(n))  dpf(a(n))
  --  ----  ---------  ----------
   1     1          1  None
   2     2         10  2
   3     5        101      5
   4     8       1000  2
   5     3         11    3
   6     4        100  2
   7     9       1001    3
   8    16      10000  2
   9     7        111        7
  10    24      11000  2 3
  11    35     100011      5 7
  12    12       1100  2 3
  13    17      10001          17
  14     6        110  2 3

Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A052531, A109812, inverse (A353604).

{ s=0; v=1; for (n=1, 66, print1 (v", "); s+=2^v; for (w=1, oo, if (!bittest(s, w) && bitand(v,w)==0 && gcd(v,w)==1, v=w; break))) }

A052929 Expansion of g.f. (2-3x-x^2)/((1-x^2)(1-3*x)).

Original entry on oeis.org

2, 3, 10, 27, 82, 243, 730, 2187, 6562, 19683, 59050, 177147, 531442, 1594323, 4782970, 14348907, 43046722, 129140163, 387420490, 1162261467, 3486784402, 10460353203, 31381059610, 94143178827, 282429536482, 847288609443, 2541865828330, 7625597484987, 22876792454962

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 915.
Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Cf. A052531: 2^n + (1+(-1)^n)/2.

List([0..30], n-> 3^n + (1+(-1)^n)/2 ); # G. C. Greubel, Oct 17 2019

[&+[(-1)^k+2^k*Binomial(n,k): k in [0..n]]: n in [0..30]]; // Bruno Berselli, Aug 27 2013

spec:= [S, {S=Union(Sequence(Prod(Z,Z)), Sequence(Union(Z,Z,Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(3^n + (1+(-1)^n)/2, n=0..30); # G. C. Greubel, Oct 17 2019

Table[3^n + (1+(-1)^n)/2, {n, 0, 30}] (* Bruno Berselli, Aug 27 2013 *)
LinearRecurrence[{3, 1, -3}, {2, 3, 10}, 40] (* Vincenzo Librandi, Mar 09 2018 *)
Table[3^n + Fibonacci[n+1,0], {n,0,30}] (* G. C. Greubel, Oct 17 2019 *)

x='x+O('x^30); Vec((2-3*x-x^2)/((1-x^2)*(1-3*x))) \\ Altug Alkan, Mar 09 2018

[3^n + (1+(-1)^n)/2 for n in (0..30)] # G. C. Greubel, Oct 17 2019

G.f.: (2-3*x-x^2)/((1-x^2)*(1-3*x)).
a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3), a(0)=2, a(1)=3, a(2)=10.
a(n) = 3^n + Sum_{alpha=RootOf(-1+z^2)} alpha^(-n)/2.
a(n) = 2*A033113(n+1) - 3*A033113(n) - A033113(n-1). - R. J. Mathar, Nov 28 2011
From Bruno Berselli, Aug 27 2013: (Start)
a(n) = 3^n + (1 + (-1)^n)/2.
a(n) = Sum_{k=0..n} (-1)^k + 2^k*binomial(n,k). (End)
E.g.f.: exp(3*x) + cosh(x). - Elmo R. Oliveira, Mar 16 2025

More terms from James Sellers, Jun 05 2000

A052647 E.g.f. (2-2x-x^2)/((1-2x)(1-x^2)).

Original entry on oeis.org

2, 2, 10, 48, 408, 3840, 46800, 645120, 10362240, 185794560, 3719520000, 81749606400, 1962469555200, 51011754393600, 1428416301312000, 42849873690624000, 1371216880889856000, 46620662575398912000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 593

spec := [S,{S=Union(Sequence(Prod(Z,Z)),Sequence(Union(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

E.g.f.: -(-2+x^2+2*x)/(-1+2*x)/(-1+x^2)
Recurrence: {a(1)=2, a(2)=10, a(0)=2, (12+2*n^3+12*n^2+22*n)*a(n)+(-n^2-5*n-6)*a(n+1)+(-2*n-6)*a(n+2)+a(n+3)=0}
(2^n+Sum(1/2*_alpha^(-n), _alpha=RootOf(-1+_Z^2)))*n!
n!*[2^n+(n mod 2)].
a(n) = n!*A052531(n). - R. J. Mathar, Nov 27 2011

cp's OEIS Frontend

A052539 a(n) = 4^n + 1.

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A352633 Lexicographically earliest sequence of distinct positive integers such for any n > 0, a(n) and a(n+1) are coprime and have no common 1-bits in their binary expansions.

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A052929 Expansion of g.f. (2-3*x-x^2)/((1-x^2)*(1-3*x)).

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A052647 E.g.f. (2-2x-x^2)/((1-2x)(1-x^2)).

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Maple

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A052929 Expansion of g.f. (2-3x-x^2)/((1-x^2)(1-3*x)).