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.

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A204450 2*A203579 - A204449. Difference between the exponential convolution of A000032 (Lucas) with itself and the corresponding exponential half-convolution.

Original entry on oeis.org

0, 2, 6, 17, 30, 177, 417, 1857, 4302, 19457, 47731, 203777, 509769, 2134017, 5462701, 22347777, 58104062, 234029057, 616919457, 2450784257, 6533317815, 25664946177, 69085604341, 268766806017, 729558799305, 2814562533377
Offset: 0

Views

Author

Wolfdieter Lang, Jan 16 2012

Keywords

Comments

See A204449 for the exponential half-convolution of A000032 (Lucas). The present sequence gives the numbers to be added to A204449 to obtain the corresponding (full) exponential convolution A203579.

Examples

			With A000032 = {2, 1, 3, 4, 7, 11, ...}
a(4) = 4*4*1 + 1*7*2 = 30.
a(5) = 10*4*3 + 5*7*1 + 1*11*2 = 177.

Crossrefs

Cf. A000032, A203579, A204449.

Formula

a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=floor(n/2)+1..n), n>=0,
with the Lucas numbers L(n)=A000032(n). For n=0 this is 0.
E.g.f.: exp(x)*(cosh(sqrt(5)*x)+1) - (BesselI(0,2*phi*x) + BesselI(0,2*(phi-1)*x) + 2*BesselI(0,2*I*x))/2. Compare this with the e.g.f. of A204449, where phi and BesselI are explained.
Bisection: a(2*k) = (2^(2*k)-binomial(2*k,k))*L(2*k)/2 + 1 - ((-1)^k)*binomial(2*k,k), a(2*k+1) = 2^(2*k)*L(2*k+1)+ 1 = A204449(2*k+1), k>=0.

A014335 Exponential convolution of Fibonacci numbers with themselves (divided by 2).

Original entry on oeis.org

0, 0, 1, 3, 11, 35, 115, 371, 1203, 3891, 12595, 40755, 131891, 426803, 1381171, 4469555, 14463795, 46805811, 151466803, 490156851, 1586180915, 5132989235, 16610702131, 53753361203, 173949530931, 562912506675, 1821623137075, 5894896300851, 19076285150003
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

It can be noticed that A014335/A011782 is an "autosequence", that is a sequence which is identical to its inverse binomial transform, except for alternating signs. - Jean-François Alcover, Jun 15 2016

Links

Crossrefs

Cf. A000032, A000045, A014334, A081057, A203579.
Cf. (partial sums of) A063727.
Column k=2 of A346415.

Programs

  • Magma
    [(2^n*Lucas(n)-2)/10: n in [0..40]]; // G. C. Greubel, Jan 06 2023
  • Maple
    a[0]:=0:a[1]:=0:for n from 2 to 50 do a[n]:=2*a[n-1]+4*a[n-2]+1 od: seq(a[n], n=0..29); # Zerinvary Lajos, Dec 14 2008
# second Maple program:
a:= n-> (<<0|1|0>, <0|0|1>, <-4|2|3>>^n)[1,3]:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 04 2016
  • Mathematica
    LinearRecurrence[{3,2,-4}, {0,0,1}, 41] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
Table[(2^n LucasL[n] - 2)/10, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 06 2016 *)
  • SageMath
    [(2^n*lucas_number2(n,1,-1) -2)/10 for n in range(41)] # G. C. Greubel, Jan 06 2023

Formula

a(n) = A014334(n)/2.
G.f.: x^2/((1-x)*(1-2*x-4*x^2)). - Vladeta Jovovic, Mar 05 2003
E.g.f.: exp(x)*(cosh(sqrt(5)*x)-1)/5. - Vladeta Jovovic, Sep 01 2004
From Benoit Cloitre, Sep 25 2004: (Start)
a(n+1) = Sum_{i=0..n} A000045(i)*2^(i-1).
a(n) = (1/5)*(2^(n-1)*A000032(n) - 1). (End)
a(n) = 2*a(n-1) + 4*a(n-2) + 1, a(0)=0; a(1)=0. - Zerinvary Lajos, Dec 14 2008
G.f.: G(0)*x^2/(2*(1-x)^2), where G(k)= 1 + 1/(1 - x*(5*k-1)/(x*(5*k+4) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = (A203579(n) - 2)/5. - Vladimir Reshetnikov, Oct 06 2016

A204449 Exponential (or binomial) half-convolution of A000032 (Lucas) with itself.

Original entry on oeis.org

4, 2, 8, 17, 84, 177, 737, 1857, 7732, 19457, 78223, 203777, 809145, 2134017, 8349013, 22347777, 86533892, 234029057, 897748577, 2450784257, 9328491339, 25664946177, 97021416973, 268766806017, 1009936510009, 2814562533377
Offset: 0

Views

Author

Wolfdieter Lang, Jan 16 2012

Keywords

Comments

For the definition of the exponential (also known as binomial) half-convolution of a sequence with itself see A203576. There the rule for the e.g.f. is also found.
The other half of this exponential half-convolution is found under A204450.

Examples

			With A000032 = {2, 1, 3, 4, 7, 11,...}
  a(4) = 1*2*7 + 4*1*4 + 6*3*3 = 84,
  a(5) = 1*2*11 + 5*1*7 + 10*3*4 = 177.

Crossrefs

Cf. A000032, 2*A203579 (exponential convolution), A204450.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*LucasL[k]*LucasL[n-k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vaclav Kotesovec, Sep 25 2019 *)

Formula

a(n) = sum(binomial(n,k)*L(k)*L(n-k),k=0..floor(n/2)), n>=0, with L(n)=A000032(n).
E.g.f.: (l(x)^2 + L2(x^2))/2 with the e.g.f. l(x) of A000032, and the o.g.f. L2(x) of the sequence {(L(n)/n!)^2}.
l(x)^2 = 2*exp(x)*(cosh(sqrt(5)*x)+1) (see 2*A203579).
L2(x^2) = BesselI(0,2*phi*x) + BesselI(0,2*(phi-1)*x) + 2*BesselI(0,2*I*x), with the golden section phi:=(1+sqrt(5))/2, and for BesselI see Abramowitz-Stegun (reference and link given under A008277), p. 375, eq. 9.6.10.
BesselI(0,2*sqrt(x)) = hypergeom([],[1],x) is the e.g.f. of {1/n!}.
Bisection: a(2*k) = (2^(2*k)+binomial(2*k,k))*L(2*k)/2 +1 + ((-1)^k)*binomial(2*k,k), a(2*k+1) = 2^(2*k)*L(2*k+1)+1, k>=0. For (2^(2*k)+binomial(2*k,k))/2 see A032443(k).
Showing 1-3 of 3 results.

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