A154117 -id:A154117 - cp's OEIS frontend

A173769 Primes in A154117.

2, 7, 17, 37, 157, 317, 1277, 2557, 20477, 655357, 5242877, 671088637, 2684354557, 5368709117, 343597383677, 23058430092136939517, 23611832414348226068477, 48357032784585166988247037, 830767497365572420564879412675215357

Offset: 1

Vincenzo Librandi, Mar 05 2010

Apart from the first term, the same as A172156. - R. J. Mathar, Mar 17 2010

Primes of the form 5*2^n-3.

Vincenzo Librandi, Table of n, a(n) for n = 1..38

[ a: n in [0..150] | IsPrime(a) where a is 5*2^n-3 ];

 Select[Table[5*2^n-3,{n,0,300}],PrimeQ] (* Vincenzo Librandi, Jul 26 2012 *)

Edited by N. J. A. Sloane, Mar 17 2010

A154251 Expansion of (1-x+7x^2)/((1-x)(1-2x)).

Original entry on oeis.org

1, 2, 11, 29, 65, 137, 281, 569, 1145, 2297, 4601, 9209, 18425, 36857, 73721, 147449, 294905, 589817, 1179641, 2359289, 4718585, 9437177, 18874361, 37748729, 75497465, 150994937, 301989881, 603979769, 1207959545, 2415919097

Offset: 0

Philippe Deléham, Jan 05 2009

Binomial transform of 1,1,8,1,8,1,8,1,8,1,8,1,8,1,8,...

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

Cf.: A094373, A000079, A083329, A095121, A154117, A131128, A154118, A131130

Join[{1},LinearRecurrence[{3,-2},{2,11}, 25]] (* or *) Join[{1},Table[9*2^(n-1) - 7, {n,1,25}]] (* G. C. Greubel, Sep 08 2016 *)

Vec((1-x+7*x^2)/((1-x)*(1-2*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 3*a(n-1) - 2*a(n-2), n>2, with a(0)=1, a(1)=2, a(2)=11.
a(n) = 9*2^(n-1) - 7, n>0, with a(0)=1.
a(n) = 2*a(n-1) + 7, n>1, with a(0)=1, a(1)=2.
From G. C. Greubel, Sep 08 2016: (Start)
a(n) = 9*2^(n-1) - 7 for n >= 1.
E.g.f.: (1/2)*(9*exp(2*x) - 14*exp(x) + 7). (End)

A172156 Primes in the chain of repeated application of x->2*x+3, starting at x=2.

Original entry on oeis.org

7, 17, 37, 157, 317, 1277, 2557, 20477, 655357, 5242877, 671088637, 2684354557, 5368709117, 343597383677, 23058430092136939517, 23611832414348226068477, 48357032784585166988247037, 830767497365572420564879412675215357

Offset: 1

Vincenzo Librandi, Jan 27 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..37

Cf. A154117. [Klaus Brockhaus, Feb 28 2010]

x:=2; a:=[n eq 1 select 2*x+3 else 2*Self(n-1)+3: n in [1..120]]; [a[i]: i in [1..#a] | IsPrime(a[i])]; // Bruno Berselli, May 14 2013

Rest[Select[NestList[2#+3&,2,500],PrimeQ]] (* Harvey P. Dale, Nov 17 2012 *)

A154252 Expansion of (1-x+8x^2)/((1-x)(1-2x)) .

Original entry on oeis.org

1, 2, 12, 32, 72, 152, 312, 632, 1272, 2552, 5112, 10232, 20472, 40952, 81912, 163832, 327672, 655352, 1310712, 2621432, 5242872, 10485752, 20971512, 41943032, 83886072, 167772152, 335544312, 671088632, 1342177272, 2684354552, 5368709112, 10737418232

Offset: 0

Philippe Deléham, Jan 05 2009

Binomial transform of 1,1,9,1,9,1,9,1,9,1,9,1,9,1,9,...

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

Cf.: A094373, A000079, A083329, A095121, A154117, A131128, A154118, A131130, A154251

Join[{1},LinearRecurrence[{3,-2},{2,12},40]]  (* Harvey P. Dale, Dec 30 2014 *)
Join[{1},Table[5*2^n - 8, {n,1,25}]] (* G. C. Greubel, Sep 08 2016 *)

Vec((1-x+8*x^2)/((1-x)*(1-2*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 3*a(n-1) - 2*a(n-2), n>2, with a(0)=1, a(1)=2, a(2)=12.
a(n) = 2*a(n-1) + 8, n>1, with a(0)=1, a(1)=2.
a(n) = 10*2^(n-1) - 8, n>=1, with a(0)=1.
E.g.f.: 5*exp(2*x) - 8*exp(x) + 4. - G. C. Greubel, Sep 08 2016

Two terms corrected by Johannes W. Meijer, May 26 2011

A321237 Start with a square of dimension 1 X 1, and repeatedly append along the squares of the previous step squares with half their side length that do not overlap with any prior square; a(n) gives the number of squares appended at n-th step.

Original entry on oeis.org

1, 8, 28, 68, 148, 308, 628, 1268, 2548, 5108, 10228, 20468, 40948, 81908, 163828, 327668, 655348, 1310708, 2621428, 5242868, 10485748, 20971508, 41943028, 83886068, 167772148, 335544308, 671088628, 1342177268, 2684354548, 5368709108, 10737418228, 21474836468

Offset: 1

Rémy Sigrist, Nov 01 2018

The following diagram depicts the first three steps of the construction:
                       +----+----+----+----+
                       |  3 |  3 |  3 |  3 |
                  +----+----+----+----+----+----+
                  |  3 |         |         |  3 |
             +----+----+    2    |    2    +----+----+
             |  3 |  3 |         |         |  3 |  3 |
        +----+----+----+---------+---------+----+----+----+
        |  3 |         |                   |         |  3 |
        +----+    2    |                   |    2    +----+
        |  3 |         |                   |         |  3 |
        +----+---------+         1         +---------+----+
        |  3 |         |                   |         |  3 |
        +----+    2    |                   |    2    +----+
        |  3 |         |                   |         |  3 |
        +----+----+----+---------+---------+----+----+----+
             |  3 |  3 |         |         |  3 |  3 |
             +----+----+    2    |    2    +----+----+
                  |  3 |         |         |  3 |
                  +----+----+----+----+----+----+
                       |  3 |  3 |  3 |  3 |
                       +----+----+----+----+
A square of step n+1 touches one or two squares of step n.
The limiting construction is an octagon (truncated square); its area is 7 times the area of the initial square.
See A321257 for a similar sequence.

Colin Barker, Table of n, a(n) for n = 1..1000
Rémy Sigrist, Illustration of the construction after 7 steps
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A154117, A321257.

a(n) = if (n==1, return (1), return (4*( 2^(n-1) + 3 * floor( (2^(n-2)-1) ) )))

Vec(x*(1 + 2*x)*(1 + 3*x) / ((1 - x)*(1 - 2*x)) + O(x^40)) \\ Colin Barker, Nov 02 2018

a(n) = 4 * (2^(n-1) + 3 * (2^(n-2)-1)) for any n > 1.
a(n) = 4 * A154117(n-1) for any n > 1.
Sum_{n > 0} a(n) / 4^(n-1) = 7.
From Colin Barker, Nov 02 2018: (Start)
G.f.: x*(1 + 2*x)*(1 + 3*x) / ((1 - x)*(1 - 2*x)).
a(n) = 5*2^n - 12 for n>1.
a(n) = 3*a(n-1) - 2*a(n-2) for n>3.
(End)

A078113 Let u(1)=u(2)=1, u(3)=n, u(k) = (1/2)abs(2u(k-1) -u(k-2)-u(k-3)); sequence gives values of n such that Sum_{k>=1} u(k) is an integer.

Original entry on oeis.org

2, 6, 7, 15, 17, 33, 37, 69, 77, 141, 157, 285, 317, 573, 637, 1149, 1277, 2301, 2557, 4605, 5117, 9213, 10237, 18429, 20477, 36861, 40957, 73725, 81917, 147453, 163837, 294909, 327677, 589821, 655357, 1179645, 1310717, 2359293, 2621437, 4718589, 5242877

Offset: 1

Benoit Cloitre, Dec 04 2002

u(3)=7, Sum_{k>=1} u(k) = 28 is an integer, hence 7 is in the sequence.

Cf. A154117, A133257.

A078113(maxn, maxk) = {
  u=vector(maxk);
  u[1]=1; u[2]=1;
  for(n=1, maxn,
    u[3]=n;
    for(k=4, maxk, u[k]=abs(2*u[k-1]-u[k-2]-u[k-3])/2);
    s=sum(i=1, maxk, u[i]);
    if(ceil(s)-s < 1E-11, print1(n, ", ")) \\ Arbitrary 1E-11
  )
}
A078113(1000000, 200) \\ Colin Barker, Aug 14 2013

Conjecture: a(n) = -3+2^(1/2*(-5+n))*(10-10*(-1)^n+9*sqrt(2)+9*(-1)^n*sqrt(2)). a(n) = a(n-1)+2*a(n-2)-2*a(n-3). G.f.: x*(3*x^2-4*x-2) / ((x-1)*(2*x^2-1)). - Colin Barker, Aug 14 2013

Conjecture: a(n) = 2*a(n-2) + 3, n odd>2 = A154117((n+1)/2). - Bill McEachen, Jun 21 2025

a(11)-a(33) from Colin Barker, Aug 14 2013

a(34)-a(41) from Bill McEachen, Jun 21 2025

A377613 a(n) is the number of iterations of x -> 2*x + 3 until (# composites reached) = (# primes reached), starting with prime(n).

Original entry on oeis.org

19, 1, 15, 15, 1, 13, 13, 15, 1, 3, 1, 1, 1, 7, 27, 3, 1, 1, 25, 1, 3, 1, 1, 5, 23, 1, 1, 1, 1, 7, 3, 1, 23, 3, 1, 1, 9, 1, 17, 5, 1, 1, 1, 3, 19, 7, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 3, 1, 21, 1, 3, 1, 19, 1, 1, 1, 1, 3, 1, 3, 3, 1, 1, 1, 3, 3, 1, 17, 1, 3, 1

Offset: 1

Clark Kimberling, Nov 13 2024

nonn

For a guide to related sequences, see A377609.

			Starting with prime(1) = 2, we have 2*2+3 = 7, then 2*7+3 = 17, etc., resulting in a chain 2, 7, 17, 37, 77, 157, 317, 637, 1277, 2557, 5117, 10237, 20477, 40957, 81917, 163837, 327677, 655357, 1310717, 2621437 having 10 primes and 10 composites. Since every initial subchain has fewer composites than primes, a(1) = 20-1 = 19. (For more terms from the mapping x -> 2*x+3, see A154117.)

Cf. A377609, A154117.

chain[{start_, u_, v_}] := NestWhile[Append[#, u*Last[#] + v] &, {start}, !
     Count[#, ?PrimeQ] == Count[#, ?(! PrimeQ[#] &)] &];
chain[{Prime[1], 2, 3}]
Map[Length[chain[{Prime[#], 2, 3}]] &, Range[100]] - 1
(* Peter J. C. Moses Oct 31 2024 *)

A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 0, 7, 9, 0, 0, 0, 0, 15, 17, 0, 0, 0, 0, 0, 31, 33, 0, 0, 0, 0, 0, 0, 63, 65, 0, 0, 0, 0, 0, 0, 0, 127, 129, 0, 0, 0, 0, 0, 0, 0, 0, 255, 257, 0, 0, 0, 0, 0, 0, 0, 0, 0, 511, 513, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1023, 1025, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2047

Offset: 0

Philippe Deléham, Jan 07 2009

Column sums give A003945.

			Triangle begins:
1;
0, 2;
0, 1, 3;
0, 0, 3, 5;
0, 0, 0, 7, 9;
0, 0, 0, 0, 15, 17; ...

Cf. A000225, A094373

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A040000(n), A094373(n), A000079(n), A083329(n), A095121(n), A154117(n), A131128(n), A154118(n), A131130(n), A154251(n), A154252(n) for x = -1,0,1,2,3,4,5,6,7,8,9 respectively.
G.f.: (1-x*y+x^2*y-x^2*y^2)/(1-3*x*y+2*x^2*y^2). - Philippe Deléham, Nov 02 2013
T(n,k) = 3*T(n-1,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(2,0) = 0, T(2,1) = 1, T(2,2) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 02 2013

cp's OEIS Frontend

A173769 Primes in A154117.

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A154251 Expansion of (1-x+7x^2)/((1-x)(1-2x)).

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A172156 Primes in the chain of repeated application of x->2*x+3, starting at x=2.

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A154252 Expansion of (1-x+8x^2)/((1-x)(1-2x)) .

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A321237 Start with a square of dimension 1 X 1, and repeatedly append along the squares of the previous step squares with half their side length that do not overlap with any prior square; a(n) gives the number of squares appended at n-th step.

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A078113 Let u(1)=u(2)=1, u(3)=n, u(k) = (1/2)*abs(2*u(k-1) -u(k-2)-u(k-3)); sequence gives values of n such that Sum_{k>=1} u(k) is an integer.

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A377613 a(n) is the number of iterations of x -> 2*x + 3 until (# composites reached) = (# primes reached), starting with prime(n).

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A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

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A078113 Let u(1)=u(2)=1, u(3)=n, u(k) = (1/2)abs(2u(k-1) -u(k-2)-u(k-3)); sequence gives values of n such that Sum_{k>=1} u(k) is an integer.