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.

A127969 Inverse of number triangle A127967.

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 1, 0, -2, 1, 2, 0, -3, 0, 1, -3, 0, 5, 0, -3, 1, -8, 0, 13, 0, -6, 0, 1, 17, 0, -28, 0, 14, 0, -4, 1, 56, 0, -92, 0, 45, 0, -10, 0, 1, -155, 0, 255, 0, -126, 0, 30, 0, -5, 1, -608
Offset: 0

Views

Author

Paul Barry, Feb 09 2007

Keywords

Comments

First column is (-1)^C(n+1,2)*A099960(n), signed interleaved Genocchi numbers of the first and second kind. Row sums are (1,0,0,0,...).

Examples

			Triangle begins
1,
-1, 1,
-1, 0, 1,
1, 0, -2, 1,
2, 0, -3, 0, 1,
-3, 0, 5, 0, -3, 1,
-8, 0, 13, 0, -6, 0, 1,
17, 0, -28, 0, 14, 0, -4, 1,
56, 0, -92, 0, 45, 0, -10, 0, 1,
-155, 0, 255, 0, -126, 0, 30, 0, -5, 1,
-608, 0, 1000, 0, -493, 0, 115, 0, -15, 0, 1,
2073, 0, -3410, 0, 1683, 0, -396, 0, 55, 0, -6, 1,
9440, 0, -15528, 0, 7662, 0, -1799, 0, 245, 0, -21, 0, 1

A127970 Number triangle A127967 modulo 2.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0
Offset: 0

Views

Author

Paul Barry, Feb 09 2007

Keywords

Comments

Row sums are A127971.

Examples

			Triangle begins
1,
1, 1,
1, 0, 1,
1, 0, 0, 1,
1, 0, 1, 0, 1,
1, 0, 0, 0, 1, 1,
1, 0, 1, 0, 0, 0, 1,
1, 0, 0, 0, 0, 0, 0, 1,
1, 0, 1, 0, 1, 0, 0, 0, 1,
1, 0, 0, 0, 1, 0, 0, 0, 1, 1,
1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1,
1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1,
1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1

A127968 a(n) = F(n+1) + (1-(-1)^n)/2, where F() = Fibonacci numbers A000045.

Original entry on oeis.org

1, 2, 2, 4, 5, 9, 13, 22, 34, 56, 89, 145, 233, 378, 610, 988, 1597, 2585, 4181, 6766, 10946, 17712, 28657, 46369, 75025, 121394, 196418, 317812, 514229, 832041, 1346269, 2178310, 3524578, 5702888, 9227465, 14930353, 24157817, 39088170, 63245986, 102334156
Offset: 0

Views

Author

Paul Barry, Feb 09 2007

Keywords

Comments

Row sums of A127967.
The sequence beginning 1,1,2,2,4,... with g.f. x/(1-x-x^2) + 1/(1-x^2) has general term a(n) = F(n) + (1+(-1)^n)/2.

Links

Programs

  • Magma
    I:=[1,2,2,4]; [n le 4 select I[n] else Self(n-1) +2*Self(n-2) - Self(n-3) -Self(n-4): n in [1..30]]; // G. C. Greubel, May 04 2018
  • Mathematica
    LinearRecurrence[{1,2,-1,-1},{1,2,2,4},40] (* Harvey P. Dale, Jun 19 2013 *)
  • PARI
    Vec((1+x-2*x^2-x^3)/((1-x)*(1+x)*(1-x-x^2)) + O(x^50)) \\ Colin Barker, Jul 12 2017

Formula

G.f.: 1 / (1 - x - x^2) + x / (1 - x^2).
G.f.: (1 + x - 2*x^2 - x^3) / ((1 - x)*(1 + x)*(1 - x - x^2)).
From Colin Barker, Jul 12 2017: (Start)
a(n) = (5 - 5*(-1)^n + 2^(-n)*sqrt(5)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))) / 10.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>3.
(End)
Showing 1-3 of 3 results.

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