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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.

User: Dragan Stevanovic

Dragan Stevanovic's wiki page.

Dragan Stevanovic has authored 2 sequences.

A030119 a(n) = a(n-1) + a(n-2) + n, a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 4, 8, 16, 29, 51, 87, 146, 242, 398, 651, 1061, 1725, 2800, 4540, 7356, 11913, 19287, 31219, 50526, 81766, 132314, 214103, 346441, 560569, 907036, 1467632, 2374696, 3842357, 6217083, 10059471, 16276586, 26336090, 42612710, 68948835, 111561581
Offset: 0

Views

Author

Dragan Stevanovic

Keywords

Links

Crossrefs

Cf. A000032, A000045.

Programs

  • GAP
    F:=Fibonacci;; List([0..40], n-> F(n+3)+2*F(n+1)-n-3); # G. C. Greubel, Jul 24 2019
  • Magma
    [Lucas(n+2) + Fibonacci(n+1) - (n+3) : n in [0..40]]; // Vincenzo Librandi, Nov 16 2011
  • Mathematica
    RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1]+a[n-2]+n},a,{n,40}] (* or *) LinearRecurrence[{3,-2,-1,1},{1,1,4,8},40] (* Harvey P. Dale, Nov 06 2011 *)
  • PARI
    Vec((1-2*x+3*x^2-x^3)/((1-x-x^2)*(1-x)^2) + O(x^40)) \\ Colin Barker, Mar 11 2017
  • PARI
    vector(40, n, n--; f=fibonacci; f(n+3)+2*f(n+1)-n-3) \\ G. C. Greubel, Jul 24 2019
  • Sage
    f=fibonacci; [f(n+3)+2*f(n+1)-n-3 for n in (0..40)] # G. C. Greubel, Jul 24 2019

Formula

Periodic mod 6.
G.f.: (1 - 2*x + 3*x^2 - x^3) / ((1 - x - x^2)*(1-x)^2).
a(n) = Lucas(n+2) + Fibonacci(n+1) - (n+3).
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4); a(0)=1, a(1)=1, a(2)=4, a(3)=8. - Harvey P. Dale, Nov 06 2011
a(n) = -3 + (2^(-n)*((1-sqrt(5))^n*(-3+2*sqrt(5)) + (1+sqrt(5))^n*(3+2*sqrt(5)))) / sqrt(5) - n. - Colin Barker, Mar 11 2017

Extensions

Description corrected and sequence extended by Erich Friedman

A030118 a(0) = 1, a(1) = 1, a(n) = a(n-1) - a(n-2) + n.

Original entry on oeis.org

1, 1, 2, 4, 6, 7, 7, 7, 8, 10, 12, 13, 13, 13, 14, 16, 18, 19, 19, 19, 20, 22, 24, 25, 25, 25, 26, 28, 30, 31, 31, 31, 32, 34, 36, 37, 37, 37, 38, 40, 42, 43, 43, 43, 44, 46, 48, 49, 49, 49, 50, 52, 54, 55, 55, 55, 56, 58, 60, 61, 61, 61, 62, 64, 66, 67, 67, 67, 68, 70, 72, 73
Offset: 0

Views

Author

Dragan Stevanovic

Keywords

Comments

Contains all positive integers except for 3 mod 6 and 5 mod 6 (A047270). - Jon Perry, Nov 02 2014

Links

Crossrefs

Cf. A010892, A047270.

Programs

  • GAP
    Concatenation([1], List([1..80], n-> n+1 - ((-1)^Int((n-1)/3) + (-1)^Int(n/3))/2 )); # G. C. Greubel, Jul 24 2019
  • Magma
    [1] cat [n le 2 select (n) else n + Self(n-1)-Self(n-2): n in [1..80]]; // Vincenzo Librandi, Nov 02 2014
  • Maple
    A:= gfun:-rectoproc({a(n)=a(n-1)-a(n-2)+n , a(0)=1,a(1)=1},a(n),remember):
seq(A(n),n=0..80); # Robert Israel, Nov 02 2014
  • Mathematica
    Table[n+1 -((-1)^Floor[(n-1)/3] +(-1)^Floor[n/3])/2, {n, 0, 80}] (* G. C. Greubel, Jul 24 2019 *)
nxt[{n_,a_,b_}]:={n+1,b,b-a+n+1}; NestList[nxt,{1,1,1},80][[;;,2]] (* or *) LinearRecurrence[{3,-4,3,-1},{1,1,2,4},80] (* Harvey P. Dale, May 16 2025 *)
  • PARI
    vector(80, n, n--; n+1 - ((-1)^floor((n-1)/3) + (-1)^floor(n/3))/2) \\ G. C. Greubel, Jul 24 2019
  • Sage
    [lucas_number1(n+1,2,1)-lucas_number1(n,1,1) for n in range(0, 80)] # Zerinvary Lajos, Nov 10 2009

Formula

a(
G.f.: (1-2*x+3*x^2-x^3)/((1-x+x^2)*(1-x)^2). - Robert Israel, Nov 02 2014
a(n) = n iff n is either 1 or 2 mod 6. - Jon Perry, Nov 02 2014
a(n) = n + 1 - ((-1)^floor((n-1)/3) + (-1)^floor(n/3))/2) = n + 1 - A010892(n+5). - G. C. Greubel, Jul 24 2019
For k >= 1, a(6*k-1) = a(6*k) = a(6*k+1) = 6*k+1; a(6*k+2) = 6*k+2; a(6*k+3) = 6*k+4; a(6*k+4) = 6*k+6. - Bernard Schott, Jul 24 2019
a(n) = 3*a(n-1) - 4*a(n-2) + 3*a(n-3) - a(n-4) for n > 3. - Chai Wah Wu, Jun 30 2020

Extensions

More terms from Erich Friedman

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