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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: Mehmet A. Ates

Mehmet A. Ates's wiki page.

Mehmet A. Ates has authored 4 sequences.

A375079 a(n) = a(n-1) + a(n-2) + ... + a(n-k) where k = (a(n-1) mod (n-1)) + 1 for n >= 3, with a(1) = 1 and a(2) = 2.

Original entry on oeis.org

1, 2, 2, 5, 7, 14, 26, 56, 56, 138, 306, 612, 612, 1224, 3004, 5758, 11822, 23476, 45284, 91792, 184140, 368224, 735948, 1472492, 2944996, 5889992, 11411652, 23191624, 46290860, 92672900, 185346856, 370693871, 741375929, 1479818680, 2962582344, 5925164688
Offset: 1

Views

Author

Mehmet A. Ates, Jul 29 2024

Keywords

Comments

It appears that the ratio a(n+1)/a(n) -> 2.

Examples

			For n = 7, we add up the previous a(7-1) mod (7-1) + 1 = 3 terms to get a(7) = a(6) + a(5) + a(4) = 14 + 7 + 5 = 26.

Programs

  • Mathematica
    Modanacci={1,2};Do[AppendTo[Modanacci,Sum[Modanacci[[-i]],{i,Mod[Modanacci[[-1]],Length[Modanacci]]+1}]],100]

Formula

a(n) = Sum_{i=1 .. (a(n-1) mod (n-1)) + 1} a(n-i).

A332616 a(n) = value of the cubic form A^3 + B^3 + C^3 - 3ABC evaluated at row n of the table in A331195.

Original entry on oeis.org

0, 1, 2, 0, 8, 9, 4, 16, 5, 0, 27, 28, 20, 35, 18, 7, 54, 28, 8, 0, 64, 65, 54, 72, 49, 32, 91, 56, 27, 10, 128, 81, 40, 11, 0, 125, 126, 112, 133, 104, 81, 152, 108, 70, 44, 189, 130, 77, 36, 13, 250, 176, 108, 52, 14, 0, 216, 217, 200, 224, 189, 160, 243
Offset: 0

Views

Author

Mehmet A. Ates, Jun 08 2020

Keywords

Comments

No term in the sequence is congruent to 3 or 6 (mod 9).

Examples

			For n=3, a(n) = f[1,1,0] = 1^3 + 1^3 + 0^3 - 3*1*1*0 = 2.

Links

Crossrefs

Cf. A056556, A056557, A056558.
Cf. A330013, A331195.
Cf. A074232 (in ascending order, strictly positive & without duplicates).

Programs

  • Mathematica
    SeqSize = 30;
ListSize = 120;
F3List = List[];
f3[a_, b_, c_] := a^3 + b^3 + c^3 - 3*a*b*c
For[i = 0, i <= SeqSize, i++, For[j = 0, j <= i, j++, For[k = 0, k <= j, k++, AppendTo[F3List, f3[i, j, k]]]]]
ListPlot[F3List, PlotLabel -> "a(n)"]
Print["First ", ListSize, " elements of a(n): ", Take[F3List, ListSize]]

Formula

a(n) = A056556(n)^3 + A056557(n)^3 + A056558(n)^3 - 3*A056556(n)*A056557(n)*A056558(n).

Extensions

Edited by N. J. A. Sloane, Aug 06 2020

A330709 Two-column table read by rows: pairs (i,j) in order sorted from the left.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 0, 2, 1, 2, 2, 3, 0, 3, 1, 3, 2, 3, 3, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, 5, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 6, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6, 7, 0, 7, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 7, 7, 8, 0, 8, 1, 8, 2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 8, 8, 9, 0, 9, 1, 9, 2, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 9, 9, 10, 0
Offset: 0

Views

Author

Mehmet A. Ates, Jun 08 2020

Keywords

Links

Crossrefs

a(2n)= A003056(n),
a(2n+1)= A002262(n).
Cf. A331195.

Programs

  • Mathematica
    TwoDVectors = List[];
SeqSize = 20;
For[i = 0, i <= SeqSize, i++,
  For[j = 0, j <= i, j++,
    AppendTo[TwoDVectors, {i, j}]
  ]
];
Flatten[TwoDVectors]

A331195 Three-column table read by rows: triples (i,j,k) in order sorted from the left.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 2, 0, 0, 2, 1, 0, 2, 1, 1, 2, 2, 0, 2, 2, 1, 2, 2, 2, 3, 0, 0, 3, 1, 0, 3, 1, 1, 3, 2, 0, 3, 2, 1, 3, 2, 2, 3, 3, 0, 3, 3, 1, 3, 3, 2, 3, 3, 3, 4, 0, 0, 4, 1, 0, 4, 1, 1, 4, 2, 0, 4, 2, 1, 4, 2, 2, 4, 3, 0, 4, 3, 1, 4, 3, 2, 4, 3, 3, 4, 4, 0, 4, 4, 1, 4, 4, 2, 4, 4, 3, 4, 4, 4, 5, 0, 0
Offset: 0

Views

Author

Mehmet A. Ates, Jun 08 2020

Keywords

Examples

			For n=[0,1,2] to n=[12,13,14], a[n,n+1,n+2] counts up as such: [0,0,0], [1,0,0], [1,1,0], [1,1,1], [2,0,0], etc.

Links

Crossrefs

Cf. A056556, A056557, A056558, A330709.
See A372667 for the norms of these triples.

Programs

  • Mathematica
    ThreeDVectors = List[];
SeqSize = 10;
For[i = 0, i <= SeqSize, i++,
  For[j = 0, j <= i, j++,
    For[k = 0, k <= j, k++,
      AppendTo[ThreeDVectors, {i, j, k}]
    ]
  ]
];
Flatten[ThreeDVectors]
  • Python
    from math import comb, isqrt
from sympy import integer_nthroot
def A331195(n): return (m:=integer_nthroot((n<<1)+6,3)[0])-(n<3*comb(m+2,3)) if not (a:=n%3) else (k:=isqrt(r:=(b:=n//3)+1-comb((m:=integer_nthroot((n<<1)-1,3)[0])-(b=comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2) # Chai Wah Wu, Nov 23 2024

Formula

a(3*n) = A056556(n).
a(3*n+1) = A056557(n).
a(3*n+2) = A056558(n).

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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