A129929 -id:A129929 - cp's OEIS frontend

A138003 Binomial transform of 1, 1, 0, -1, -1 (periodically continued).

1, 2, 3, 3, 0, -8, -21, -34, -34, 0, 89, 233, 377, 377, 0, -987, -2584, -4181, -4181, 0, 10946, 28657, 46368, 46368, 0, -121393, -317811, -514229, -514229, 0, 1346269, 3524578, 5702887, 5702887, 0, -14930352, -39088169, -63245986, -63245986

Offset: 0

Paul Curtz, May 01 2008

Shares many elements with A103311, as already indicated by the similarity of the two generating functions. First differences are essentially in A105371. - R. J. Mathar, May 02 2008

The longer of the two recurrences ensures that the sequence (like A133476) equals its 5th differences. - R. J. Mathar, May 02 2008

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,2,-1).

Cf. A129929.

LinearRecurrence[{3,-4,2,-1},{1,2,3,3},50] (* Paolo Xausa, Dec 05 2023 *)

a=[1,2,3,3];for(i=1,99,a=concat(a,3*a[#a]-4*a[#a-1]+2*a[#a-2]-a[#a-3]));a \\ Charles R Greathouse IV, Jun 02 2011

From R. J. Mathar, May 02 2008: (Start)
O.g.f.: (x^2-x+1)/(x^4-2*x^3+4*x^2-3*x+1).
a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+2a(n-5).
a(n) = 3a(n-1)-4a(n-2)+2a(n-3)-a(n-4). (End)

Edited by R. J. Mathar, May 02 2008

A361376 Rewrite A129912(n), a product of distinct primorials P(i) = A002110(i) instead as a sum of powers 2^(i-1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 16, 11, 17, 12, 13, 18, 19, 32, 14, 33, 20, 15, 21, 34, 35, 22, 24, 64, 23, 36, 25, 65, 37, 26, 66, 38, 27, 67, 40, 128, 39, 41, 28, 68, 129, 29, 69, 42, 130, 48, 43, 30, 70, 72, 131, 49, 31, 71, 44, 73, 256, 132, 45, 50, 257, 133, 74, 51, 46, 80, 75, 258, 134, 136

Offset: 1

Michael De Vlieger, Jun 08 2023

nonn

Permutation of nonnegative numbers.

			a(1) = 0 by convention.
a(8) = 8 comes before a(9) = 7, since we interpret 8 = 2^3 instead as P(4) = 210, while for a(9), 7 = 2^2 + 2^1 + 2^0 becomes P(3)*P(2)*P(1) = 30*6*2 = 360. Because 210 < 360, 8 appears before 7 in this sequence.
Table relating a(n), n=1..19 with the set S(n) of indices of distinct primorial factors of A129912(n):
   n A129912(n)  S(n)   a(n)  A272011(a(n))
  -----------------------------------------
   1         1            0
   2         2   1        1   0
   3         6   2        2   1
   4        12   2,1      3   1,0
   5        30   3        4   2
   6        60   3,1      5   2,0
   7       180   3,2      6   2,1
   8       210   4        8   3
   9       360   3,2,1    7   2,1,0
  10       420   4,1      9   3,0
  11      1260   4,2     10   3,1
  12      2310   5       16   4
  13      2520   4,2,1   11   3,1,0
  14      4620   5,1     17   4,0
  15      6300   4,3     12   3,2
  16     12600   4,3,1   13   3,2,0
  17     13860   5,2     18   4,1
  18     27720   5,2,1   19   4,1,0
  19     30030   6       32   5
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..15303 (a(15303) = 2^29.)
Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^6.
Michael De Vlieger, Plot terms S(n) = A272011(a(n)) at (x,y) = (n,S(n,k)) for n = 1..2^11.

Cf. A002110, A129912, A272011, A283477.

a6939[n_] := Product[Prime[n + 1 - i]^i, {i, n}];
g[m_] := Block[{f, j = 1},
  f[n_, i_, e_] :=
   If[n < m, Block[{p = Prime[i + 1]}, If[e == 1, Sow@ n];
     f[n p^e, i + 1, e];
     If[e > 1, f[n p^(e - 1), i + 1, e - 1]]]];
  Sort@ Reap[While[a6939[j] < m, f[2^j, 1, j]; j++]][[-1, 1]] ];
Map[Total@
     Map[2^(# - 1) &,
      Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} ] &[
FactorInteger[#][[All, -1]]] &, g[2^31]] (* Michael De Vlieger, Jun 08 2023, after Giovanni Resta at A129929 *)

Let S(n) be the set of indices of primorials P(i), reverse sorted, such that A129912(n) = Product_{k=1..m} S(n,k), where m = | S(n) |. Then a(n) = Sum_{k=1..m} 2^(S(n,k)-1).

cp's OEIS Frontend

A138003 Binomial transform of 1, 1, 0, -1, -1 (periodically continued).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions