cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A130624 Binomial transform of A101000.

Original entry on oeis.org

0, 1, 5, 12, 23, 43, 84, 169, 341, 684, 1367, 2731, 5460, 10921, 21845, 43692, 87383, 174763, 349524, 699049, 1398101, 2796204, 5592407, 11184811, 22369620, 44739241, 89478485, 178956972, 357913943, 715827883, 1431655764, 2863311529, 5726623061, 11453246124
Offset: 0

Views

Author

Paul Curtz, Jun 18 2007

Keywords

Links

Crossrefs

Cf. A101000, A119910, A130625 (first differences), A130626 (second differences).

Programs

  • Magma
    m:=32; S:=[[0, 1, 3][(n-1) mod 3 +1]: n in [1..m]]; [&+[Binomial(i-1, k-1)*S[k]: k in [1..i]]: i in [1..m]]; /* Klaus Brockhaus, Jun 21 2007 */
  • Magma
    I:=[0,1,5]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+2*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Nov 15 2018
  • Mathematica
    LinearRecurrence[{3,-3,2},{0,1,5},40] (* Harvey P. Dale, Mar 05 2013 *)
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(2^n) + a[n-1] - a[n-2]}, a, {n, 50}] (* Vincenzo Librandi, Nov 15 2018 *)
  • PARI
    {m=32; v=concat([0, 1, 5], vector(m-3)); for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} /* Klaus Brockhaus, Jun 21 2007 */

Formula

a(0)=0, a(1)=1, a(2)=5; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(0)=0; a(n+1) = 2*a(n) + A119910(n).
G.f.: x*(1 + 2*x)/((1 - 2*x)*(1 - x + x^2)).
a(n) = 2^n + a(n-1) - a(n-2). - Jon Maiga, Nov 14 2018

Extensions

Edited and extended by Klaus Brockhaus, Jun 21 2007

A130625 First differences of A130624.

Original entry on oeis.org

1, 4, 7, 11, 20, 41, 85, 172, 343, 683, 1364, 2729, 5461, 10924, 21847, 43691, 87380, 174761, 349525, 699052, 1398103, 2796203, 5592404, 11184809, 22369621, 44739244, 89478487, 178956971, 357913940, 715827881, 1431655765, 2863311532
Offset: 0

Views

Author

Paul Curtz, Jun 18 2007

Keywords

Comments

a(n) = A130624(n+1) - A130624(n).

Links

Crossrefs

Cf. A130624, A130626 (second differences).

Programs

  • Magma
    m:=33; S:=[ [0, 1, 3][ (n-1) mod 3 +1 ]: n in [1..m] ]; T:=[ &+[ Binomial(i-1, k-1)*S[k]: k in [1..i] ]: i in [1..m] ]; [ T[n+1]-T[n]: n in[1..m-1] ]; /* Klaus Brockhaus, Jun 21 2007 */
  • Mathematica
    LinearRecurrence[{3,-3,2},{1,4,7},40] (* Harvey P. Dale, Apr 27 2015 *)

Formula

G.f.: (1-x)*(1+2*x)/((1-2*x)*(1-x+x^2)).
a(n) = 3a(n-1) - 3a(n-2) + 2a(n-3). Sequence is identical to its third differences. Binomial transform of 1, 3, 0. - Paul Curtz, Nov 23 2007

Extensions

Edited and extended by Klaus Brockhaus, Jun 21 2007
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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