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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A122909 a(n) = F(n+1)*F(2n+2) + F(n)*F(2n).

Original entry on oeis.org

1, 4, 19, 79, 338, 1427, 6053, 25628, 108583, 459931, 1948354, 8253271, 34961561, 148099316, 627359147, 2657535383, 11257501522, 47687540107, 202007664157, 855718193164, 3624880442591, 15355239954179, 65045840274434
Offset: 0

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Author

Paul Barry, Sep 18 2006

Keywords

Comments

Let M be the matrix M(n,k)=F(k+1)*sum{j=0..n, (-1)^(n-j)C(n,j)C(j+1,k+1)}. a(n) gives the row sums of M^3.

Links

Crossrefs

Cf. A000045, A037451, A061084, A163063.

Programs

  • Mathematica
    Table[Fibonacci[n+1]Fibonacci[2n+2]+Fibonacci[n]Fibonacci[2n],{n,0,30}] (* or *) LinearRecurrence[{3,6,-3,-1},{1,4,19,79},30] (* Harvey P. Dale, Dec 11 2016 *)

Formula

G.f.: (1+x)*(1+x^2) / ( (x^2+4*x-1)*(x^2-x-1) ).
a(n) = (sqrt(5)+2)^n(sqrt(5)/5+3/5)-2^(-n-1)(sqrt(5)-1)^n(sqrt(5)/5+1/5)+ 2^(-n-1)(sqrt(5)+1)^n(sqrt(5)/5-1/5)(-1)^n+(sqrt(5)-2)^n(3/5-sqrt(5)/5)(-1)^n;
a(n) = (2*A163063(n) -A061084(n))/5. - R. J. Mathar, Jun 08 2016

A386014 Distinct values occurring in first differences of A030655(n), listed in order of first appearance.

Original entry on oeis.org

22, 832, 202, 89302, 2002, 8993002, 20002, 899930002, 200002, 89999300002, 2000002, 8999993000002, 20000002, 899999930000002, 200000002, 89999999300000002, 2000000002, 8999999993000000002, 20000000002, 899999999930000000002, 200000000002, 89999999999300000000002, 2000000000002
Offset: 1

Views

Author

Carin Maria Sakin, Jul 14 2025

Keywords

Comments

A030655(t) = 2*t-1||2*t = (2*t-1)*10^k + 2*t, where || indicates digit concatenation and 2*t has digit length k.
Difference A030655(t+1) - A030655(t) = 2*(10^k+1) when 2*t and 2*t+2 are both digit length k and this is the odd n case in the formulas below.
2*t and 2*t+2 differ in length only at t = j(k) = 5*10^(k-1)-1 with 2*t+2 then having length k+1, and this is the even n case in the formulas below.

Examples

			At t = j(3) = 499, difference A030655(t+1) - A030655(t) = 9991000 - 997998 = 8993002 which is term a(6).

Links

Crossrefs

Cf. A007376, A030655, A057147, A061084.

Programs

  • Maple
    a := proc(n)
    if type(n, even) then
        k := n/2;
        return 9*100^k - 7*10^k + 2;
    else
        k := (n + 1)/2;
        return 2*(10^k + 1);
    end if;
end proc:
seq(a(n), n=1..12);
  • Mathematica
    a[n_] := Which[
  EvenQ[n], Module[{k = n/2}, 9*100^k - 7*10^k + 2],
  OddQ[n], Module[{k = (n + 1)/2}, 2*(10^k + 1)]
];
Table[a[n], {n, 1, 12}]
  • PARI
    a(n) = if(n%2==1, 2*10^((n+1)/2)+2, 9*10^n - 7*10^(n/2) + 2) \\ David A. Corneth, Jul 17 2025
  • Python
    def a(n):
    k = (n + 1)//2
    if n % 2 == 0:
        return 9 * 100**k - 7 * 10**k + 2
    else:
        return 2 * (10**k + 1)

Formula

a(n) = 2*(10^k + 1) for odd n, where k = (n+1)/2.
a(n) = 9*100^k - 7*10^k + 2 for even n, where k = n/2.

Extensions

More terms from Michel Marcus, Jul 16 2025
More terms from David A. Corneth, Jul 17 2025
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