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 1-4 of 4 results.

A140420 Binomial transform of 0, 1, 1, 7, 7, 31, 31, ..., zero followed by duplicated A083420.

Original entry on oeis.org

0, 1, 3, 13, 45, 151, 483, 1513, 4665, 14251, 43263, 130813, 394485, 1187551, 3570843, 10728913, 32219505, 96724051, 290303223, 871171813, 2614039725, 7843167751, 23531600403, 70598995513, 211805375145, 635432902651
Offset: 0

Views

Author

Paul Curtz, Jun 18 2008, corrected Jun 23 2008

Keywords

Links

Formula

a(n+1)-3a(n) = A099430(n).
O.g.f.: (3x^2-2x+1)x/((2x-1)(1+x)(3x-1)(1-x)). - R. J. Mathar, Jul 10 2008
a(n)+a(n+1)=A126644(n). - R. J. Mathar, Jul 10 2008

Extensions

More terms from R. J. Mathar, Jul 10 2008

A254522 Numerators of (2^n - 1 + (-1)^n)/(2*n), n > 0.

Original entry on oeis.org

0, 1, 1, 2, 3, 16, 9, 16, 85, 256, 93, 512, 315, 4096, 5461, 2048, 3855, 65536, 13797, 131072, 349525, 1048576, 182361, 1048576, 3355443, 16777216, 22369621, 33554432, 9256395, 268435456, 34636833, 67108864, 1431655765, 4294967296, 17179869183, 8589934592, 1857283155, 68719476736, 91625968981
Offset: 1

Views

Author

Paul Curtz, Jan 31 2015

Keywords

Comments

An autosequence of the first kind is a sequence which main diagonal is A000004.
Difference table of a(n)/A093803(n):
0, 1, 1, 2, 3, 16/3, ...
1, 0, 1, 1, 7/3, 11/3, ...
-1, 1, 0, 4/3, 4/3, 10/3, ...
2, -1, 4/3, 0, 2, 2, ...
-3, 7/3, -4/3, 2, 0, 16/5, ...
16/3, -11/3, 10/3, -2, 16/5, 0, ...
etc.
This is an autosequence of the first kind.
Its first (or second) upper diagonal is A075101(n)/(2*A000265(n)).
From Robert Israel, Apr 03 2017: (Start)
If p is a prime == 5 (mod 8), then a(5*p) = (2^(5*p-1)-1)/5 and a(5*p+3) = 2^(5*p) = 10*a(5*p)+2. This explains pairs such as
a(25) = 3355443
a(28) = 33554432
and
a(65) = 3689348814741910323
a(68) = 36893488147419103232. (End)

Links

Crossrefs

Cf. A000004, A075101, A093803, A099430.

Programs

  • Maple
    seq(numer((2^n-1+(-1)^n)/(2*n)), n=1..50); # Robert Israel, Feb 01 2015
  • Mathematica
    Table[Numerator[(2^n - 1 + (-1)^n)/(2*n)], {n, 39}] (* Michael De Vlieger, Feb 01 2015 *)

Extensions

a(25) corrected by Robert Israel, Apr 03 2017

A290968 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) + a(n-5), with a(0)=a(1)=a(2)=1, a(3)=-1 and a(4)=1.

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 5, 5, 9, 11, 21, 33, 57, 89, 145, 231, 377, 609, 989, 1597, 2585, 4179, 6765, 10945, 17713, 28657, 46369, 75023, 121393, 196417, 317813, 514229, 832041, 1346267, 2178309, 3524577, 5702889, 9227465, 14930353, 24157815
Offset: 0

Views

Author

Jean-François Alcover and Paul Curtz, Aug 16 2017

Keywords

Comments

The array of successive differences begins:
1, 1, 1, -1, 1, 1, 5, 5, 9, 11, 21, 33, 57, ...
0, 0, -2, 2, 0, 4, 0, 4, 2, 10, 12, 24, 32, ...
0, -2, 4, -2, 4, -4, 4, -2, 8, 2, 12, 8, 24, ...
-2, 6, -6, 6, -8, 8, -6, 10, -6, 10, -4, 16, 6, ...
8, -12, 12, -14, 16, -14, 16, -16, 16, -14, 20, -10, 24, ...
...
First row is a(n) = 2*A141325(n) - A141325(n+1).
Main diagonal is A099430(n).
The first upper subdiagonal, 1, -2, -2, -8, -14, ..., has -3*A078008(n) as first differences.
The second upper subdiagonal is A000079(n) = 2^n.
a(n) is related to Fibonacci numbers a(n) = A000045(n-2) + period 6: repeat [2, 0, 1, -2, 0, -1].

Links

Crossrefs

Cf. A000045, A078008, A099430, A131531, A141325.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^2-2*x^3+x^4)/((1+x^3)*(1-x-x^2)) )); // G. C. Greubel, Jun 11 2019
  • Mathematica
    LinearRecurrence[{1,1,-1,1,1}, {1,1,1,-1,1}, 40]
  • PARI
    my(x='x+O('x^40)); Vec((1-x^2-2*x^3+x^4)/((1+x^3)*(1-x-x^2))) \\ G. C. Greubel, Jun 11 2019
  • Sage
    ((1-x^2-2*x^3+x^4)/((1+x^3)*(1-x-x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 11 2019

Formula

G.f.: (1-x^2-2*x^3+x^4)/((1+x)*(1-x+x^2)*(1-x-x^2)).
a(n) ~ phi^(n-2)/sqrt(5), where phi is the golden ratio.
a(n) = (1/2 + sqrt(5)/2)^n*(3*sqrt(5)/10-1/2) - (-1/2 + sqrt(5)/2)^n*(3*sqrt(5)/10 + 1/2)*(-1)^n + 2*sqrt(3)*sin(Pi*(n/3 + 1/3))/3 + (-1)^n. - Eric Simon Jacob, Jul 11 2024

A099431 Expansion of x(1-2x+3x^2)/(1-x-2x)^2;.

Original entry on oeis.org

0, 1, 0, 6, 8, 30, 60, 154, 336, 774, 1700, 3762, 8184, 17758, 38220, 81930, 174752, 371382, 786420, 1660258, 3495240, 7340046, 15379100, 32156346, 67108848, 139810150, 290805060, 603979794, 1252698776, 2594876094, 5368709100
Offset: 0

Views

Author

Paul Barry, Oct 15 2004

Keywords

Comments

Convolution of A078008 and A099430(n-1)+3*0^n/2.

Links

Formula

a(n)=sum{k=0..n, (2^(n-k)+2(-1)^(n-k))/3*(2^(k-1)-(-1)^k-1+3*0^k/20)}; a(n)=sum{k=0..n+1, A078008(n-k)binomial(n-k+1, k)binomial(1, (k+1)/2)(1-(-1)^k)/2}.
Showing 1-4 of 4 results.

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