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.

A016161 Expansion of g.f. 1/((1-5*x)*(1-7*x)).

Original entry on oeis.org

1, 12, 109, 888, 6841, 51012, 372709, 2687088, 19200241, 136354812, 964249309, 6798573288, 47834153641, 336059778612, 2358521965909, 16540171339488, 115933787267041, 812299450322412, 5689910849522509
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also, this is the number of incongruent integer-edged Heron triangles whose circumdiameter is the product of n distinct primes each of shape 4k + 1. Cf. A003462, A109021. - R. K. Guy, Jan 31 2007

Links

Crossrefs

Cf. A003462, A080962, A081200, A109021.
Cf. A016162, A016163, A016164, A016165, A016166.

Programs

  • Magma
    [n le 2 select 12^(n-1) else 12*Self(n-1) -35*Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 09 2024
  • Mathematica
    CoefficientList[Series[1/((1-5x)(1-7x)),{x,0,30}],x] (* or *) LinearRecurrence[ {12,-35},{1,12},30] (* Harvey P. Dale, Nov 16 2021 *)
  • PARI
    Vec(1/((1-5*x)*(1-7*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012
  • SageMath
    A016161=BinaryRecurrenceSequence(12,-35,1,12)
[A016161(n) for n in range(31)] # G. C. Greubel, Nov 09 2024

Formula

From R. J. Mathar, Sep 18 2008: (Start)
a(n) = (7^(n+1) - 5^(n+1))/2 = A081200(n+1).
Binomial transform of A080962. (End)
a(n) = 7*a(n-1) + 5^n. - Vincenzo Librandi, Feb 09 2011
E.g.f.: exp(5*x)*(7*exp(2*x) - 5)/2. - Stefano Spezia, Oct 25 2023

A080961 Fourth binomial transform of A010686 (period 2: repeat 1,5).

Original entry on oeis.org

1, 9, 57, 321, 1713, 8889, 45417, 230001, 1158753, 5820009, 29178777, 146130081, 731358993, 3658920729, 18300980937, 91524036561, 457677578433, 2288560079049, 11443316955897, 57218134461441, 286095321353073, 1430490553902969, 7152494610927657, 35762598578876721
Offset: 0

Views

Author

Paul Barry, Mar 03 2003

Keywords

Examples

			G.f. = 1 + 9*x + 57*x^2 + 321*x^3 + 1713*x^4 + 8889*x^5 + 45417*x^6 + 230001*x^7 + ...

Links

Crossrefs

Cf. A003948, A005059, A010686, A080960, A080962.

Programs

  • Magma
    binomtf:=func< V | [ &+[ Binomial(i-1, k-1)*V[k]: k in [1..i] ]: i in [1..#V] ] >;
binomtf(binomtf(binomtf(binomtf(&cat[ [1, 5]: n in [1..11] ])))); // Klaus Brockhaus, Nov 26 2009
  • Magma
    [3*5^n - 2*3^n: n in [0..30]]; // Vincenzo Librandi, Dec 07 2012
  • Maple
    A080961:=n->3*5^n-2*3^n: seq(A080961(n), n=0..30); # Wesley Ivan Hurt, Dec 08 2016
  • Mathematica
    CoefficientList[Series[(1 + x)/((1 - 3*x) * (1 - 5*x)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 07 2012 *)

Formula

a(n) = 5*a(n-1) + 4*3^(n-1).
a(n) = 3*5^n - 2*3^n.
G.f.: (1+x)/((1-3*x)*(1-5*x)). - Klaus Brockhaus, Nov 26 2009
From Mario C. Enriquez, Dec 08 2016: (Start)
a(n) = A005059(n+1) + A005059(n) = (5^(n+1)+5^n-3^(n+1)-3^n)/2.
a(n) = Sum_{k=0..n} A003948(n-k)*3^k = Sum_{k=0..n} (3^k * ceiling(Sum_{v=0..n-k} (5^v - 5^(v-2)))). (End)
a(n) = 8*a(n-1) - 15*a(n-2) for n > 1. - Wesley Ivan Hurt, Dec 08 2016
E.g.f.: exp(3*x)*(3*exp(2*x) - 2). - Stefano Spezia, Jul 23 2024

Extensions

Definition corrected, edited by Klaus Brockhaus, Nov 26 2009

A081033 6th binomial transform of the periodic sequence (1,7,1,1,7,1...).

Original entry on oeis.org

1, 13, 121, 997, 7729, 57853, 423721, 3059797, 21887329, 155555053, 1100604121, 7762822597, 54632726929, 383893932253, 2694581744521, 18898693305397, 132473958606529, 928233237589453, 6502210299844921, 45538360282508197, 318882962895526129, 2232752944858526653
Offset: 0

Views

Author

Paul Barry, Mar 03 2003

Keywords

Links

Crossrefs

Cf. A080134, A080962.

Programs

  • Magma
    [4*7^n-3*5^n: n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
  • Mathematica
    CoefficientList[Series[(1 + x) / ((1 - 5 x) (1 - 7 x)),{x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)
LinearRecurrence[{12,-35},{1,13},30] (* Harvey P. Dale, Aug 18 2015 *)

Formula

a(n) = 7*a(n-1) + 6*5^(n-1).
a(n) = 4*7^n - 3*5^n.
G.f.: (1+x)/((1-5*x)*(1-7*x)). - Vincenzo Librandi, Aug 06 2013
E.g.f.: exp(5*x)*(4*exp(2*x) - 3). - Stefano Spezia, Jul 23 2024
Showing 1-3 of 3 results.

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

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