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 1-5 of 5 results.

A117377 Riordan array (1-4x,x(1-x)).

Original entry on oeis.org

1, -4, 1, 0, -5, 1, 0, 4, -6, 1, 0, 0, 9, -7, 1, 0, 0, -4, 15, -8, 1, 0, 0, 0, -13, 22, -9, 1, 0, 0, 0, 4, -28, 30, -10, 1, 0, 0, 0, 0, 17, -50, 39, -11, 1, 0, 0, 0, 0, -4, 45, -80, 49, -12, 1, 0, 0, 0, 0, 0, -21, 95, -119, 60, -13, 1
Offset: 0

Views

Author

Paul Barry, Mar 10 2006

Keywords

Comments

Row sums are A117378. Diagonal sums are A117379. Inverse is A117380.

Examples

			Triangle begins
1,
-4, 1,
0, -5, 1,
0, 4, -6, 1,
0, 0, 9, -7, 1,
0, 0, -4, 15, -8, 1,
0, 0, 0, -13, 22, -9, 1,
0, 0, 0, 4, -28, 30, -10, 1

Formula

Number triangle T(n,k)=(-1)^(n-k)(C(k,n-k)+4*C(k, n-k-1))

A155751 A variation on 10^n mod 17.

Original entry on oeis.org

1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5, 1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5, 1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5, 1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5, 1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5
Offset: 0

Views

Author

Ferruccio Guidi (fguidi(AT)cs.unibo.it), Jan 26 2009, Feb 08 2009

Keywords

Comments

This is 10^n mod 17, using values -8,-7,...,7,8 (instead of 0..16). - Don Reble, Sep 02 2017.
This sequence can be employed in a test for divisibility by 17 and works like A033940 works for 7.
The use of negative coefficients ensures the termination of the test because the modulus of the intermediate sum at each step of the test decreases strictly.
The test is successful if the final sum is 0.
The negative coefficients have the form (10^n mod 17) - 17 when 10^n mod 17 > 8.
Example: 9996 is divisible by 17 since |6*1 + 9*(-7) + 9*(-2) + 9*(-3)| = 102 and 2*1 + 0*(-7) + 1*(-2) = 0.

Crossrefs

Cf. A033940, A119910, A117378.

Formula

a(n)= -a(n-8). G.f.:(1-7x-2x^2-3x^3+4x^4+6x^5-8x^6+5x^7)/(1+x^8). [From R. J. Mathar, Feb 13 2009]

A174559 Triangle T(n,k)of the coefficients [x^(n-k)] of the polynomials q(0,x)=-1, q(1,x)=3*x, q(n,x)=x*q(n-1,x)-q(n-2,x) in row n,column k. A companion to A193002(n).

Original entry on oeis.org

-1, 3, 0, 3, 0, 1, 3, 0, -2, 0, 3, 0, -5, 0, -1, 3, 0, -8, 0, 1, 0, 3, 0, -11, 0, 6, 0, 1, 3, 0, -14, 0, 14, 0, 0, 0, 3, 0, -17, 0, 25, 0, -6, 0, -1, 3, 0, -20, 0, 39, 0, -20, 0, -1, 0, 3, 0, -23, 0, 56, 0, -45, 0, 5, 0, 1
Offset: 0

Views

Author

Paul Curtz, Aug 20 2011

Keywords

Comments

a(n)=
-1, :-1,
3, 0, :3*x,
3, 0, 1, :3*x^2+1,
3, 0, -2, 0, :3*x^3-2*x,
3, 0, -5, 0, -1,
3, 0, -8, 0, 1, 0
3, 0, -11, 0, 6, 0, 1.
Row sum=period 6:repeat -1, 3, 4, 1, -3, 4=-A117378(n)=A117378(n+3).

Crossrefs

Cf. A192011.

Formula

a(n) + A193002(n)=4*A192174(n).
a(n) - A193002(n)=2*A053119(n), Chebyshev's S(n,x).

A144471 Inverse binomial transform of A020806.

Original entry on oeis.org

1, 3, -5, 13, -30, 61, -119, 234, -467, 937, -1878, 3757, -7511, 15018, -30035, 60073, -120150, 240301, -480599, 961194, -1922387, 3844777, -7689558, 15379117, -30758231, 61516458, -123032915, 246065833, -492131670, 984263341, -1968526679, 3937053354, -7874106707
Offset: 0

Views

Author

Paul Curtz, Oct 10 2008

Keywords

Links

Crossrefs

Cf. A141425, A141532.

Programs

  • Maple
    Digits := 200 ; read("transforms") ; read("transforms3") ; x := 1/7 ; L := CONSTTOLIST(x) ; BINOMIALi(L) ; # R. J. Mathar, Sep 07 2009
  • Mathematica
    LinearRecurrence[{-3,-3,-2},{1,3,-5,13},40] (* Harvey P. Dale, Nov 11 2017 *)

Formula

|a(n+1)| - 2*|a(n)| = -A117378(n-1) = A117378(n+2), n>0.
a(n) = -3*a(n-1) - 3*a(n-2) - 2*a(n-3), n > 3.
G.f.: (6*x+7*x^2+9*x^3+1) / ((2*x+1) * (1+x+x^2)). - R. J. Mathar, Sep 07 2009

Extensions

Edited and extended by R. J. Mathar, Sep 07 2009

A155754 A variation on 10^n mod 19.

Original entry on oeis.org

1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2, 1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2, 1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2, 1, -9, 5, -7, 6, 3, -8, -4, -2, -1, 9, -5, 7, -6, -3, 8, 4, 2
Offset: 0

Views

Author

Ferruccio Guidi (fguidi(AT)cs.unibo.it), Jan 26 2009

Keywords

Comments

This sequence can be employed in a test for divisibility by 19 and works like A033940 works for 7.
The use of negative coefficients ensures the termination of the test because the modulus of the intermediate sum at each step of the test decreases strictly.
The test is successful if the final sum is 0.
The negative coefficients have the form (10^n mod 19) - 19 when 10^n mod 19 > 9.
Example: 8284 is divisible by 19 since |4*1 + 8*(-9) + 2*5 + 8*(-7)| = 114 and 4*1 + 1*(-9) + 1*5 = 0.

Links

Crossrefs

Cf. A033940, A119910, A117378.

Formula

a(n) = -a(n-9). G.f.: (-2*x^8-4*x^7-8*x^6+3*x^5+6*x^4-7*x^3+5*x^2-9*x+1) / (x^9+1). [Colin Barker, Feb 14 2013]
Showing 1-5 of 5 results.

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