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 1-3 of 3 results.

A052913 a(n+2) = 5*a(n+1) - 2*a(n), with a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 18, 82, 374, 1706, 7782, 35498, 161926, 738634, 3369318, 15369322, 70107974, 319801226, 1458790182, 6654348458, 30354161926, 138462112714, 631602239718, 2881086973162, 13142230386374, 59948977985546, 273460429154982, 1247404189803818, 5690100090709126
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

Main diagonal of the array: m(1,j)=3^(j-1), m(i,1)=1; m(i,j) = m(i-1,j) + m(i,j-1): 1 3 9 27 81 ... / 1 4 13 40 ... / 1 5 18 58 ... / 1 6 24 82 ... - Benoit Cloitre, Aug 05 2002
a(n) is also the number of 3 X n matrices of integers for which the upper-left hand corner is a 1, the rows and columns are weakly increasing, and two adjacent entries differ by at most 1. - Richard Stanley, Jun 06 2010
a(n) is the number of compositions of n when there are 4 types of 1 and 2 types of other natural numbers. - Milan Janjic, Aug 13 2010
If a Stern's sequence based enumeration system of positive irreducible fractions is considered (for example, A007305/A047679, or A162909/A162910, or A071766/A229742, or A245325/A245326, ...), and if it is organized by blocks or levels (n) with 2^n terms (n>=0), and the products numerator*denominator, term by term, are summed at each level n, then the resulting sequence of integers is a(n). - Yosu Yurramendi, May 23 2015
Number of 1’s in the substitution system {0 -> 110, 1 -> 11110} at step n from initial string "1" (1 -> 11110 -> 11110111101111011110110 -> ...) . - Ilya Gutkovskiy, Apr 10 2017

Links

Crossrefs

Cf. A007482 (inverse binomial transform).

Programs

  • GAP
    a:=[1,4];; for n in [3..30] do a[n]:=5*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Oct 16 2019
  • Magma
    I:=[1,4]; [n le 2 select I[n] else 5*Self(n-1)-2*Self(n-2): n in [1..35]]; // Vincenzo Librandi, May 24 2015
  • Magma
    R:=PowerSeriesRing(Integers(), 25); Coefficients(R!((1-x)/(1-5*x+2*x^2))); // Marius A. Burtea, Oct 16 2019
  • Maple
    spec := [S,{S=Sequence(Union(Prod(Sequence(Z),Union(Z,Z)),Z,Z))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(coeff(series((1-x)/(1-5*x+2*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 16 2019
  • Mathematica
    Transpose[NestList[{Last[#],5Last[#]-2First[#]}&, {1,4},20]][[1]] (* Harvey P. Dale, Mar 12 2011 *)
LinearRecurrence[{5, -2}, {1, 4}, 25] (* Jean-François Alcover, Jan 08 2019 *)
  • PARI
    Vec((1-x)/(1-5*x+2*x^2) + O(x^30)) \\ Michel Marcus, Mar 05 2015
  • Sage
    def A052913_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-5*x+2*x^2)).list()
A052913_list(30) # G. C. Greubel, Oct 16 2019

Formula

G.f.: (1-x)/(1-5*x+2*x^2).
a(n) = Sum_{alpha=RootOf(1 - 5*z + 2*z^2)} (1/17)*(3+alpha)*alpha^(-1-n).
a(n) = ((17+3*sqrt(17))/34)*((5+sqrt(17))/2)^n + ((17-3*sqrt(17))/34)*((5-sqrt(17))/2)^n. - N. J. A. Sloane, Jun 03 2002
a(n) = A107839(n) - A107839(n-1). - R. J. Mathar, May 21 2015
a(n) = 2*A020698(n-1), n>1. - R. J. Mathar, Nov 23 2015
E.g.f.: (1/17)*exp(5*x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2)). - Stefano Spezia, Oct 16 2019
a(n) = 3*A107839(n-1) + (-1)^n*A152594(n) with A107839(-1) = 0. - Klaus Purath, Jul 29 2020

Extensions

Typo in definition corrected by Bruno Berselli, Jun 07 2010

A152620 a(n)=-8*a(n-1)-6*a(n-2), n>1 ; a(0)=1, a(1)=-2 .

Original entry on oeis.org

1, -2, 10, -68, 484, -3464, 24808, -177680, 1272592, -9114656, 65281696, -467565632, 3348834880, -23985285248, 171789272704, -1230402470144, 8812484124928, -63117458178560, 452064760678912, -3237813336359936
Offset: 0

Views

Author

Philippe Deléham, Dec 10 2008

Keywords

Links

Crossrefs

Cf. A152594

Programs

  • Mathematica
    LinearRecurrence[{-8,-6},{1,-2},30] (* Harvey P. Dale, Apr 03 2015 *)

Formula

G.f.: (1+6*x)/(1+8*x+6*x^2). a(n)=Sum_{k, 0<=k<=n}A147703(n,k)*(-3)^(n-k).

A199479 Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,0,0,...) DELTA (1,1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 5, 9, 5, 1, 7, 20, 27, 13, 1, 9, 35, 73, 80, 34, 1, 11, 54, 151, 252, 234, 89, 1, 13, 77, 269, 597, 837, 677, 233, 1, 15, 104, 435, 1199, 2225, 2702, 1941, 610, 1, 17, 135, 657, 2158, 4956, 7943, 8533, 5523, 1597
Offset: 0

Views

Author

Philippe Deléham, Nov 06 2011

Keywords

Comments

Mirror image of triangle in A147703.

Examples

			Triangle begins:
  1;
  1,  1;
  1,  3,  2;
  1,  5,  9,  5;
  1,  7, 20, 27, 13;
  1,  9, 35, 73, 80, 34;

Crossrefs

Cf. A001519, A059502, A000012, A005408, A014107.

Formula

Sum_{k=0..n} T(n,k)*x^k = A152620(n), A152594(n), A000007(n), A000012(n), A006012(n), A152596(n), A152599(n) for x=-3,-2,-1,0,1,2,3 respectively.
T(n,n) = A001519(n).
G.f.: (1-2y*x)/(1-(1+3y)*x+y*(1+y)*x^2).
Showing 1-3 of 3 results.

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

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