A257273 -id:A257273 - cp's OEIS frontend

A057732 Numbers k such that 2^k + 3 is prime.

1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 28, 30, 55, 67, 84, 228, 390, 784, 1110, 1704, 2008, 2139, 2191, 2367, 2370, 4002, 4060, 4062, 4552, 5547, 8739, 17187, 17220, 17934, 20724, 22732, 25927, 31854, 33028, 35754, 38244, 39796, 40347, 55456, 58312, 122550, 205962, 235326, 363120, 479844, 685578, 742452, 1213815, 1434400, 1594947, 1875552, 1940812, 2205444

Offset: 1

G. L. Honaker, Jr., Oct 29 2000

Some of the larger entries may only correspond to probable primes.

A number k is in this sequence iff A062709(k) is in A057733; this is the case iff A257273(k) is in A125246. - M. F. Hasler, Apr 27 2015

			For k = 6, 2^6 + 3 = 67 is prime.
For k = 28, 2^28 + 3 = 268435459 is prime.

Mike Oakes, posting to primenumbers(AT)yahoogroups.com on Jul 08 2001

Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+3, PRP Top Records.

Cf. A050414, A057733, A062709, A094076, A125246, A257273.

Cf. A019434 (primes 2^k+1), this sequence (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196(2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23).

[n: n in [0..1000] | IsPrime(2^n+3)]; // Vincenzo Librandi, Apr 27 2015

Select[Range[10000], PrimeQ[2^# + 3] &] (* Vincenzo Librandi, Apr 27 2015 *)

for(n=1, 2200, if(isprime(2^n+3), print1(n, ", ")));

for (n=1, 2, if (isprime(2^n+3), print1(n, ", "))); for(n=3, 100000, N=2^n+3 ; S=(N-5)/2 ; x=S ; for(j=1, n-1, x=Mod(x^2-2, N)) ; if(x==S , print1(n, ", "))) \\ produces terms corresponding to probable primes, see formula; Tony Reix, Aug 27 2015

Here is an LLT-like algorithm, using a cycle of the digraph x^2-2 modulo N, that finds terms of this sequence generating a PRP (PRobable Prime) of A057733 numbers: N=2^k+3; S0=(N-5)/2; s(0)=S0; s(i+1)=s(i)^2-2 modulo N; if s(k-1) == S0 then N is prime. - Tony Reix, Aug 27 2015

More terms from Jason Earls, Jul 18 2001 and Mike Oakes, Jul 28 2001
a(47)-a(50) from Donovan Johnson 2006, verified by Paul Bourdelais, Mar 22 2012
a(51) is a probable prime based on trial factoring to 1E9 and PRP testing base 3,5,7 (PFGW v3.3.1). Discovered by Paul Bourdelais, Apr 09 2012
a(52)-a(54) from Paul Bourdelais, Jun 18 2019
a(55) from Paul Bourdelais, Jul 16 2019
a(56) from Paul Bourdelais, Apr 22 2020
a(57) from Paul Bourdelais, Jun 12 2020
a(58) from Paul Bourdelais, Aug 04 2020

A257272 a(n) = 2^(n-1)*(2^n+7).

Original entry on oeis.org

4, 9, 22, 60, 184, 624, 2272, 8640, 33664, 132864, 527872, 2104320, 8402944, 33583104, 134275072, 536985600, 2147713024, 8590393344, 34360655872, 137440788480, 549759483904, 2199030595584, 8796107702272, 35184401448960, 140737547075584, 562950070861824, 2251800048566272, 9007199724503040

Offset: 0

M. F. Hasler, Apr 27 2015

For n in A057195, a(n) is of deficiency 8, i.e., in A125247.

Also, the third column (k=2) of the table given in A181444.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Cf. A000079, A007582, A028403, A256873, A257273, A256871.

Cf. A168415, A057195, A104066, A125247.

[2^(n-1)*(2^n+7): n in [0..25]]; // Vincenzo Librandi, Apr 27 2015

Table[2^(n - 1) (2^n + 7), {n, 0, 30}] (* Bruno Berselli, Apr 27 2015 *)
CoefficientList[Series[(4 - 15 x)/((1 - 4 x) (1 - 2 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 27 2015 *)

PARI
```
a(n)=2^(n-1)*(2^n+7)
```

Vec((4-15*x)/((1-4*x)*(1-2*x)) + O(x^100)) \\ Colin Barker, Apr 27 2015

a(n) = 2^(n-1)*A168415(n).
n in A057195 <=> A168415(n) in A104066 <=> a(n) in A125247.
G.f.: (4-15*x)/((1-4*x)*(1-2*x)). - Vincenzo Librandi, Apr 27 2015

A370933 Number of pairs of antipodal vertices in the level n>1 Sierpiński triangle graph.

Original entry on oeis.org

6, 15, 42, 132, 456, 1680, 6432, 25152, 99456, 395520, 1577472, 6300672, 25184256, 100700160, 402726912, 1610760192, 6442745856, 25770393600, 103080394752, 412319219712, 1649272160256, 6597079203840, 26388297940992, 105553154015232, 422212540563456, 1688850011258880, 6755399743045632

Offset: 2

Allan Bickle, Aug 07 2024

A level 1 Sierpiński triangle graph is a triangle. Level n+1 is formed from three copies of level n by identifying pairs of corner vertices of each pair of triangles.

Antipodal vertices are a pair of vertices at maximum distance in a graph. The diameter of the level n Sierpiński triangle graph is 2^(n-1).

			3 example graphs:                        o
                                        / \
                                       o---o
                                      / \ / \
                        o            o---o---o
                       / \          / \     / \
             o        o---o        o---o   o---o
            / \      / \ / \      / \ / \ / \ / \
           o---o    o---o---o    o---o---o---o---o
Graph:      S_1        S_2              S_3
For S_2, there are 3 pairs of corners and 3 pairs of a corner and a middle vertex, so a(2) = 6.

Paolo Xausa, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (6,-8).
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph

Cf. A007283, A029858, A067771, A193277, A233774, A233775, A246959, A298202 (Sierpiński triangle graphs).

Cf. A375256 (antipodal pairs in Hanoi graphs).

A370933[n_] := 3*2^(n - 3)*(2^(n - 2) + 3);
Array[A370933, 30, 2] (* or *)
LinearRecurrence[{6, -8}, {6, 15}, 30] (* Paolo Xausa, Sep 23 2024 *)

a(n) = 3*2^(n-3)*(2^(n-2)+3); \\ Michel Marcus, Aug 08 2024

a(n) = 3*2^(n-3)*(2^(n-2)+3).
a(n) = 3*A257273(n-2).
a(n) = A375256(n-1) + 3.

More terms from Michel Marcus, Aug 08 2024

A146769 Coefficients of polynomial P(n) by rows, with P(n) = (x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1) for n > 0 and P(0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 12, 18, 12, 1, 1, 25, 50, 50, 25, 1, 1, 54, 135, 180, 135, 54, 1, 1, 119, 357, 595, 595, 357, 119, 1, 1, 264, 924, 1848, 2310, 1848, 924, 264, 1, 1, 585, 2340, 5460, 8190, 8190, 5460, 2340, 585, 1, 1, 1290, 5805, 15480, 27090, 32508

Offset: 0

Roger L. Bagula, Nov 02 2008

Original name: A new symmetrical polynomial form to give a triangle sequence: p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]].
Row sums are:{1, 2, 5, 14, 44, 152, 560, 2144, 8384, 33152, 131840}.
Row sums are 1 and (6*2^k + 4^k)/8 for k >= 1 (see A257273). - Robert Israel, Apr 29 2015

			1;
1,    1;
1,    3,    1;
1,    6,    6,     1;
1,   12,   18,    12,     1;
1,   25,   50,    50,    25,     1;
1,   54,  135,   180,   135,    54,     1;
1,  119,  357,   595,   595,   357,   119,     1;
1,  264,  924,  1848,  2310,  1848,   924,   264,    1;
1,  585, 2340,  5460,  8190,  8190,  5460,  2340,  585,    1;
1, 1290, 5805, 15480, 27090, 32508, 27090, 15480, 5805, 1290, 1;
...

Cf. A257273 (row sums).

/* As triangle: */ [1]; for n in [1..10] do; R := PolynomialAlgebra(RationalField(), n); Coefficients((x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1)); end for; // Bruno Berselli, Apr 30 2015

p[x_, n_] = If[ n == 0, 1, (x + 1)^n + 2^(n - 4)*Sum[Binomial[n, m]*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]]; Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}]; Flatten[%]

G.f.: y/(4*(2*y-1)) - 1/(x*y+y-1) - 1/(8*(2*x*y+2*y-1)) + 1/(8*(2*x*y-1)). - Robert Israel, Apr 29 2015

New name from Charles R Greathouse IV, Apr 29 2015

A290075 Number of monomials in c(n) where c(1) = x, c(2) = y, c(n+2) = c(n+1) + c(n)^2.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 24, 44, 80, 152, 288, 560, 1088, 2144, 4224, 8384, 16640, 33152, 66048, 131840, 263168, 525824, 1050624, 2100224, 4198400, 8394752, 16785408, 33566720, 67125248, 134242304, 268468224, 536920064, 1073807360, 2147581952, 4295098368

Offset: 1

Michael Somos, Jul 19 2017

			G.f. = x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 14*x^7 + 24*x^8 + 44*x^9 + ...
c(3) = x^2 + y so a(3) = 2, c(4) = x^2 + (y + y^2) so a(4) = 3, c(5) = x^4 + x^2(2*y) + (y + 2*y^2) so a(5) = 5.

Colin Barker, Table of n, a(n) for n = 1..1000
Kai Williams, Number of monomials in a_n = a_{n-1} + (a_{n-2})^2 with a_1 = a, a_2 = b, Math StackExchange question 2363374, Jul 18 2017.
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A088966, A103484, A211525, A257273.

nn:=36; nn:=10; Rest[CoefficientList[Series[(x - x^2 - 2*x^3 + x^4 - x^5) / ((1 - 2*x) * (1 - 2*x^2)),{x, 0, nn}], x]] (* Georg Fischer, May 10 2020 *)

{a(n) = if( n<3, n>0, my(e=n%2, m=2^((n+e)/2-2)); m * (m+2+e) / (1+e))};

Vec(x*(1 - x - 2*x^2 + x^3 - x^4) / ((1 - 2*x)*(1 - 2*x^2)) + O(x^50)) \\ Colin Barker, Jul 22 2017

G.f.: (x - x^2 - 2*x^3 + x^4 - x^5) / ((1 - 2*x) * (1 - 2*x^2)).
0 = 4*a(n) - 2*a(n+1) - 2*a(n+2) + 1*a(n+3) for n>=3.
A088966(n) = a(2*n+2). A257273(n) = a(2*n+3). A211525(n) = a(n+8).
From Colin Barker, Jul 22 2017: (Start)
a(n) = 2^(n/2-1) + 2^(n-4) for n>2 and even.
a(n) = 3*2^((n-5)/2) + 2^(n-4) for n>2 and odd.
(End)
Given the sequence c(n, x, y), then the coefficients of: (1) c(n+2, sqrt(t), 0), (2) c(n+1, 0, t), and (3) c(n, t, t), each form the triangular sequence A103484. - Michael Somos, Jul 24 2017

cp's OEIS Frontend

A057732 Numbers k such that 2^k + 3 is prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

A257272 a(n) = 2^(n-1)*(2^n+7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A370933 Number of pairs of antipodal vertices in the level n>1 Sierpiński triangle graph.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A146769 Coefficients of polynomial P(n) by rows, with P(n) = (x+1)^n + 2^(n-3)*((x+1)^n - x^n - 1) for n > 0 and P(0) = 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A290075 Number of monomials in c(n) where c(1) = x, c(2) = y, c(n+2) = c(n+1) + c(n)^2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula