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.

A051656 Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2*i).

Original entry on oeis.org

0, 1, 3, 12, 21, 77, 168, 609, 987, 3572, 7755, 28059, 47376, 171409, 372099, 1346268, 2178309, 7881197, 17108664, 61899729, 104512485, 378129724, 820851717, 2969869413, 4809706272, 17401680769, 37775923491, 136674575148
Offset: 0

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Author

Antti Karttunen, Nov 30 1999

Keywords

Comments

Positions in the first column (A003622) of Wythoff array of the terms which have their Zeckendorf Expansion patterned as row[2n+1] in Pascal's Triangle computed modulo 2 (A047999)

References

  • Proof in preparation, to be published (see A048757).

Links

Crossrefs

Cf. A048757, A047999, A035513, A038183, A051256. First row of A050609, First column of A050610.
a(n) = A019586[A048757[n]]. A048757[n] = SS(Athis_sequence[n])+1, where SSx means the second Fibonacci Successor of x (= x's Z.E. shifted left twice).
Cf. A001906.

Programs

  • Haskell
    a051656 = sum . zipWith (*) a001906_list . a047999_row
-- Reinhard Zumkeller, Feb 27 2011
  • Mathematica
    Table[Sum[Mod[Binomial[n,i],2]Fibonacci[2i],{i,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 30 2011 *)
  • PARI
    a(n)=sum(i=0,n,if(!bitand(i,n-i),fibonacci(2*i))) \\ Charles R Greathouse IV, Jan 04 2013

Formula

a(n) = sum_{i=0..n} (C(2n, 2i) mod 2)*F(2*i) = FL(n)product_{i=0..inf} L(2^i)^bit(n, i) where L is n-th Lucas number (A000032) and FL is defined as in A048757: FL(n) = n-th fibonacci number if n even, n-th Lucas number if n odd.

A051257 Numbers formed from binomial coefficients (mod 2+k) interpreted as digits in factorial base.

Original entry on oeis.org

1, 3, 11, 43, 231, 1337, 9739, 76209, 706109, 6914977, 78150249, 920172983, 12216376453, 168531536319, 2571960399839, 40581616143967, 701349512411763, 12460393480873445, 240094506439569631, 4749510978132662277
Offset: 0

Views

Author

Antti Karttunen, Oct 24 1999

Keywords

Examples

			a(5) = (1 mod 2)1!+(5 mod 3)2!+(10 mod 4)3!+(10 mod 5)4!+(5 mod 6)5!+(1 mod 7)6! = 1*1+2*2+2*6+0*24+5*120+1*720 = 1337

Links

Crossrefs

Cf. A001317, A001339, A048757, A047999, A051256.

Programs

  • Maple
    a(n) := proc(n) local i; RETURN(add(((binomial(n,i)mod(i+2))*((i+1)!)),i=0..n)); end;
  • Mathematica
    a[n_] := Sum[(k+1)!*Mod[Binomial[n, k], 2+k], {k, 0, n}]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Sep 09 2013 *)

Formula

a(n) = Sum (k+1)!(C(n, k) mod (2+k)), k=0..n
Showing 1-2 of 2 results.

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