A154251 -id:A154251 - cp's OEIS frontend

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

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

A377615 a(n) is the number of iterations of x -> 2*x + 7 until (# composites reached) = (# primes reached), starting with prime(n).

Original entry on oeis.org

23, 7, 9, 1, 21, 1, 7, 1, 21, 1, 1, 1, 3, 1, 3, 19, 1, 1, 1, 5, 1, 1, 7, 1, 1, 1, 1, 1, 1, 17, 1, 9, 17, 1, 1, 1, 1, 1, 1, 5, 1, 1, 3, 1, 15, 1, 1, 1, 9, 1, 1, 1, 1, 3, 17, 1, 1, 1, 1, 15, 1, 11, 1, 1, 1, 5, 1, 1, 11, 1, 1, 1, 1, 1, 1, 23, 1, 1, 11, 1, 1, 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+7 = 11, then 2*11+7 = 29, etc., resulting in a chain 2, 11, 29, 65, 137, 281, 569, 1145, 2297, 4601, 9209, 18425, 36857, 73721, 147449, 294905, 589817, 1179641, 2359289, 4718585, 9437177, 18874361, 37748729, 75497465 having 24 primes and 24 composites. Since every initial subchain has fewer composites than primes, a(1) =   24-1 = 23. (For more terms from the mapping x -> 2x+7, see A154251.)

Cf. A377609, A154251.

chain[{start_, u_, v_}] := NestWhile[Append[#, u*Last[#] + v] &, {start}, !
     Count[#, ?PrimeQ] == Count[#, ?(! PrimeQ[#] &)] &];
chain[{Prime[1], 2, 7}]
Map[Length[chain[{Prime[#], 2, 7}]] &, 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

A370882 Square array T(n,k) = 9*2^k - n read by ascending antidiagonals.

Original entry on oeis.org

9, 8, 18, 7, 17, 36, 6, 16, 35, 72, 5, 15, 34, 71, 144, 4, 14, 33, 70, 143, 288, 3, 13, 32, 69, 142, 287, 576, 2, 12, 31, 68, 141, 286, 575, 1152, 1, 11, 30, 67, 140, 285, 574, 1151, 2304, 0, 10, 29, 66, 139, 284, 573, 1150, 2303, 4608, -1, 9, 28, 65, 138, 283, 572, 1149, 2302, 4607, 9216

Offset: 0

Paul Curtz, Mar 05 2024

Just after A367559 and A368826.

			Table begins:
       k=0  1  2  3   4   5
  n=0:   9 18 36 72 144 288 ...
  n=1:   8 17 35 71 143 287 ...
  n=2:   7 16 34 70 142 286 ...
  n=3:   6 15 33 69 141 285 ...
  n=4:   5 14 32 68 140 284 ...
  n=5:   4 13 31 67 139 283 ...
Every line has the signature (3,-2). For n=1: 3*17 - 2*8 = 35.
Main diagonal's difference table:
  9   17   34   69   140   283   570  1145  ...  =  b(n)
  8   17   35   71   143   287   575  1151  ...  =  A052996(n+2)
  9   18   36   72   144   288   576  1152  ...  =  A005010(n)
  ...
b(n+1) - 2*b(n) = A023443(n).

Cf. A000225, A033484, A048491, A005010, A052996, A053209, A083329, A154251, A176449, A304383, A367559, A368826.

Cf. A023443.

T[n_, k_] := 9*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 06 2024 *)

T(0,k) = 9*2^k     =   A005010(k);
T(1,k) = 9*2^k - 1 =   A052996(k+2);
T(2,k) = 9*2^k - 2 =   A176449(k);
T(3,k) = 9*2^k - 3 = 3*A083329(k);
T(4,k) = 9*2^k - 4 =   A053209(k);
T(5,k) = 9*2^k - 5 =   A304383(k+3);
T(6,k) = 9*2^k - 6 = 3*A033484(k);
T(7,k) = 9*2^k - 7 =   A154251(k+1);
T(8,k) = 9*2^k - 8 =   A048491(k);
T(9,k) = 9*2^k - 9 = 3*A000225(k).
G.f.: (9 - 9*y + x*(11*y - 10))/((1 - x)^2*(1 - y)*(1 - 2*y)). - Stefano Spezia, Mar 17 2024

cp's OEIS Frontend

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

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A377615 a(n) is the number of iterations of x -> 2*x + 7 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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A370882 Square array T(n,k) = 9*2^k - n read by ascending antidiagonals.

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