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.

A111711 Leading column of triangle mentioned in A111710.

Original entry on oeis.org

1, 2, 5, 8, 14, 19, 28, 35, 47, 56, 71, 82, 100, 113, 134, 149, 173, 190, 217, 236, 266, 287, 320, 343, 379, 404, 443, 470, 512, 541, 586, 617, 665, 698, 749, 784, 838, 875, 932, 971, 1031, 1072, 1135, 1178, 1244, 1289, 1358, 1405, 1477, 1526, 1601, 1652
Offset: 1

Views

Author

Amarnath Murthy, Aug 24 2005

Keywords

Comments

Also partial sums of A257143. - Reinhard Zumkeller, Apr 17 2015

Links

Crossrefs

Cf. A111710, A111712, A080512, A257143.

Programs

  • Haskell
    a111711 n = a111711_list !! (n-1)
a111711_list = 1 : zipWith (+) a111711_list a080512_list
-- Reinhard Zumkeller, Apr 17 2015
  • Mathematica
    LinearRecurrence[{1,2,-2,-1,1},{1,2,5,8,14},60] (* Harvey P. Dale, Jun 21 2023 *)

Formula

a(1)=1, a(2n) = a(2n-1)+2n-1, a(2n+1)=a(2n)+3n; a(n) = A111710(n-1)+1. - Franklin T. Adams-Watters, May 01 2006
From Chai Wah Wu, Mar 05 2021: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5.
G.f.: x*(-x^4 - x^3 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2). (End)
a(n) = (10*n*(n-1) + (-1)^n*(1-2*n)+15)/16. - Eric Simon Jacob, Jun 11 2022

Extensions

More terms from Franklin T. Adams-Watters, May 01 2006

A257145 a(n) = 5 * floor( (n+2) / 5) - n with a(0) = 1.

Original entry on oeis.org

1, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0, -1, -2, 2, 1, 0
Offset: 0

Views

Author

Michael Somos, Apr 16 2015

Keywords

Comments

Cycle period is 5, {0, -1, -2, 2, 1} after the first five terms. - Robert G. Wilson v, Aug 02 2018

Examples

			G.f. = 1 - x - 2*x^2 + 2*x^3 + x^4 - x^6 - 2*x^7 + 2*x^8 + x^9 - x^11 + ...

Links

Crossrefs

Cf. A257143, A253262, A117444.

Programs

  • Haskell
    a257145 0 = 1
a257145 n = div (n + 2) 5 * 5 - n  -- Reinhard Zumkeller, Apr 17 2015
  • Magma
    m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)*(1-x^2)^2/(1-x^5))); // G. C. Greubel, Aug 02 2018
  • Mathematica
    a[ n_] := If[ n==0, 1, -Mod[ n, 5, -2]];
a[ n_] := If[ n==0, 1, Sign[n] SeriesCoefficient[ (1 - x) * (1 - x^2)^2 / (1 - x^5), {x, 0, Abs@n}]];
CoefficientList[Series[(1-x)*(1-x^2)^2/(1-x^5), {x,0,60}], x] (* G. C. Greubel, Aug 02 2018 *)
a[n_] := 5 Floor[(n + 2)/5] - n; Array[a, 77, 0] (* or *)
CoefficientList[ Series[(x - 1)^2 (x + 1)^2/(x^4 + x^3 + x^2 + x + 1), {x, 0, 76}], x] (* or *)
LinearRecurrence[{-1, -1, -1, -1}, {1, -1, -2, 2, 1, 0}, 76] (* Robert G. Wilson v, Aug 02 2018*)
  • PARI
    {a(n) = if( n==0, 1, (n+2) \ 5 * 5 - n)};
  • PARI
    {a(n) = if( n==0, 1, [0, -1, -2, 2, 1][n%5 + 1])};
  • PARI
    {a(n) = if( n==0, 1, sign(n) * polcoeff( (1 - x) * (1 - x^2)^2 / (1 - x^5) + x * O(x^abs(n)), abs(n)))};
  • PARI
    x='x+O('x^60); Vec((1-x)*(1-x^2)^2/(1-x^5)) \\ G. C. Greubel, Aug 02 2018

Formula

Euler transform of length 5 sequence [-1, -2, 0, 0, 1].
a(5*n) = 0 for all n in Z except n=0.
a(n) = -a(-n) for all n in Z except n=0.
a(n) = a(n+5) for all n in Z except n=-5 or n=0.
Convolution inverse is A257143.
G.f.: (1 - x) * (1 - x^2)^2 / (1 - x^5).
G.f.: (1 - 2*x^2 + x^4) / (1 + x + x^2 + x^3 + x^4).
a(n) = -A117444(n), n>0. - R. J. Mathar, Oct 05 2017

A280167 a(2*n) = 3*n if n>0, a(2*n + 1) = -(2*n + 1), a(0) = 1.

Original entry on oeis.org

1, -1, 3, -3, 6, -5, 9, -7, 12, -9, 15, -11, 18, -13, 21, -15, 24, -17, 27, -19, 30, -21, 33, -23, 36, -25, 39, -27, 42, -29, 45, -31, 48, -33, 51, -35, 54, -37, 57, -39, 60, -41, 63, -43, 66, -45, 69, -47, 72, -49, 75, -51, 78, -53, 81, -55, 84, -57, 87, -59
Offset: 0

Views

Author

Michael Somos, Dec 27 2016

Keywords

Examples

			G.f. = 1 - x + 3*x^2 - 3*x^3 + 6*x^4 - 5*x^5 + 9*x^6 - 7*x^7 + 12*x^8 + ...

Links

Crossrefs

Cf. A080512, A084964, A257143.

Programs

  • Magma
    I:=[-1,3,-3,6]; [1] cat [n le 4 select I[n] else 2*Self(n-2) - Self(n-4): n in [1..30]]; // G. C. Greubel, Aug 01 2018
  • Mathematica
    a[ n_] := Which[ n < 1, Boole[n == 0], OddQ[n], -n, True, 3 n/2];
a[ n_] := SeriesCoefficient[ (1 - x + x^2 - x^3 + x^4) / (1 - 2*x^2 + x^4), {x, 0, n}];
Join[{1}, LinearRecurrence[{0,2,0,-1}, {-1,3,-3,6}, 50]] (* G. C. Greubel, Aug 01 2018 *)
  • PARI
    {a(n) = if( n<1, n==0, n%2, -n, 3*n/2)};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( (1 - x) * (1 - x^10) / ((1 - x^2)^3 * (1 - x^5)) + x * O(x^n), n))};

Formula

b(n) = -a(n) for n > 0 is multiplicative with b(2^e) = -3 * 2^(e-1) if e > 0, b(p^e) = p^e for prime p > 2.
Euler transform of length 10 sequence [-1, 3, 0, 0, 1, 0, 0, 0, 0, -1].
G.f.: (1 - x + x^2 - x^3 + x^4) / (1 - 2*x^2 + x^4).
G.f.: (1 - x) * (1 - x^10) / ((1 - x^2)^3 * (1 - x^5)).
a(n) = (-1)^n * A257143(n). a(n) = (-1)^n * A080512(n) if n>0.
a(n) + a(n+1) = A084964(n-1) if n>0.
Showing 1-3 of 3 results.

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