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.

A119457 Triangle read by rows: T(n, 1) = n, T(n, 2) = 2*(n-1) for n>1 and T(n, k) = T(n-1, k-1) + T(n-2, k-2) for 2 < k <= n.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 5, 5, 8, 9, 10, 8, 6, 10, 12, 15, 16, 13, 7, 12, 15, 20, 24, 26, 21, 8, 14, 18, 25, 32, 39, 42, 34, 9, 16, 21, 30, 40, 52, 63, 68, 55, 10, 18, 24, 35, 48, 65, 84, 102, 110, 89, 11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144, 12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233
Offset: 1

Views

Author

Reinhard Zumkeller, May 20 2006

Keywords

Examples

			Triangle begins as:
   1;
   2,  2;
   3,  4,  3;
   4,  6,  6,  5;
   5,  8,  9, 10,  8;
   6, 10, 12, 15, 16, 13;
   7, 12, 15, 20, 24, 26,  21;
   8, 14, 18, 25, 32, 39,  42,  34;
   9, 16, 21, 30, 40, 52,  63,  68,  55;
  10, 18, 24, 35, 48, 65,  84, 102, 110,  89;
  11, 20, 27, 40, 56, 78, 105, 136, 165, 178, 144;
  12, 22, 30, 45, 64, 91, 126, 170, 220, 267, 288, 233;

Links

Crossrefs

Main diagonal: A023607(n).
Sums: A001891 (row), A355020 (signed row).
Columns: A000027(n) (k=1), A005843(n-1) (k=2), A008585(n-2) (k=3), A008587(n-3) (k=4), A008590(n-4) (k=5), A008595(n-5) (k=6), A008603(n-6) (k=7).
Diagonals: A000045(n+1) (k=n), A006355(n+1) (k=n-1), A022086(n-1) (k=n-2), A022087(n-2) (k=n-3), A022088(n-3) (k=n-4), A022089(n-4) (k=n-5), A022090(n-5) (k=n-6).
Cf. A023652, A112469.

Programs

  • Magma
    A119457:= func< n,k | (n-k+1)*Fibonacci(k+1) >;
[A119457(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 16 2025
  • Mathematica
    (* First program *)
T[n_, 1] := n;
T[n_ /; n > 1, 2] := 2 n - 2;
T[n_, k_] /; 2 < k <= n := T[n, k] = T[n - 1, k - 1] + T[n - 2, k - 2];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)
(* Second program *)
A119457[n_,k_]:= (n-k+1)*Fibonacci[k+1];
Table[A119457[n,k], {n,13}, {k,n}]//Flatten (* G. C. Greubel, Apr 16 2025 *)
  • SageMath
    def A119457(n,k): return (n-k+1)*fibonacci(k+1)
print(flatten([[A119457(n,k) for k in range(1,n+1)] for n in range(1,13)])) # G. C. Greubel, Apr 16 2025

Formula

T(n, k) = (n-k+1)*T(k,k) for 1 <= k < n, with T(n, n) = A000045(n+1).
From G. C. Greubel, Apr 15 2025: (Start)
T(n, k) = (n-k+1)*Fibonacci(k+1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A023652(floor((n+1)/2)) + (1+(-1)^n)*A001891(floor(n/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n)*A112469(floor((n-1)/2)) + (1+(-1)^n)*A355020(floor((n-2)/2)). (End)

A355021 a(n) = (-1)^n * A000032(n) - 1.

Original entry on oeis.org

1, -2, 2, -5, 6, -12, 17, -30, 46, -77, 122, -200, 321, -522, 842, -1365, 2206, -3572, 5777, -9350, 15126, -24477, 39602, -64080, 103681, -167762, 271442, -439205, 710646, -1149852, 1860497, -3010350, 4870846, -7881197, 12752042, -20633240, 33385281
Offset: 0

Views

Author

Clark Kimberling, Jun 21 2022

Keywords

Comments

There are the partial sums of L(1) - L(2) + L(3) - L(4) + L(5) - ... .
Closely related (Fibonacci, A000045) partial sums of F(1) - F(2) + F(3) - F(4) + F(5) - ... are given by A355020.
Apart from signs, same as A098600 and A181716.

Examples

			a(0) = 1;
a(1) = 1 - 3 = -2;
a(2) = 1 - 3 + 4 = 2;
a(3) = 1 - 3 + 4 - 7 = -5.

Links

Crossrefs

Cf. A000032, A000045, A098600, A181716, A355018, A355019, A355020.

Programs

  • Magma
    [Lucas(-n) -1: n in [0..50]]; // G. C. Greubel, Mar 17 2024
  • Mathematica
    f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
f1 = Table[(-1)^n f[n] + 1, {n, 0, 40}]   (* A355020 *)
g1 = Table[(-1)^n g[n] - 1, {n, 0, 40}]   (* this sequence *)
LucasL[-Range[0, 50]] - 1 (* G. C. Greubel, Mar 17 2024 *)
LinearRecurrence[{0,2,-1},{1,-2,2},40] (* Harvey P. Dale, Sep 06 2024 *)
  • SageMath
    [lucas_number2(-n,1,-1) -1 for n in range(51)] # G. C. Greubel, Mar 17 2024

Formula

a(n) = 2*a(n-2) - a(n-3) for n >= 3. [Corrected by Georg Fischer, Sep 30 2022]
G.f.: (1 - 2*x)/(1 - 2*x^2 + x^3).

A355018 Partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ..., where F = A000045 and L = A000032.

Original entry on oeis.org

1, 0, 1, -2, 0, -4, -1, -8, -3, -14, -6, -24, -11, -40, -19, -66, -32, -108, -53, -176, -87, -286, -142, -464, -231, -752, -375, -1218, -608, -1972, -985, -3192, -1595, -5166, -2582, -8360, -4179, -13528, -6763, -21890, -10944, -35420, -17709, -57312, -28655
Offset: 0

Views

Author

Clark Kimberling, Jun 16 2022

Keywords

Comments

The closely related partial sums of L(1) - F(1) + L(2) - F(2) + L(3) - F(3) + .... are given by A355019.

Examples

			a(0) = 1
a(1) = 1 - 1 = 0
a(2) = 1 - 1 + 1 = 1
a(3) = 1 - 1 + 1 - 3 = -2.

Links

Crossrefs

Cf. A000032, A000045, A355019, A355020, A355021.

Programs

  • Magma
    F:=Fibonacci; [2 - (((n+1) mod 2)*F(Floor((n+2)/2)) + 2*(n mod 2)*F(Floor((n+3)/2))) : n in [0..60]]; // G. C. Greubel, Mar 17 2024
  • Mathematica
    f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
f1[n_] := If[OddQ[n], 2 - 2 f[(n + 3)/2], 2 - f[(n + 2)/2]]
f2 = Table[f1[n], {n, 0, 20}]  (* this sequence *)
g1[n_] := If[OddQ[n], -2 + 2 f[(n + 3)/2], -2 + f[(n + 8)/2]]
g2 = Table[g1[n], {n, 0, 20}]  (* A355019 *)
LinearRecurrence[{1,1,-1,1,-1}, {1,0,1,-2,0}, 61] (* G. C. Greubel, Mar 17 2024 *)
  • SageMath
    f=fibonacci; [2 - (((n+1)%2)*f(((n+2)//2)) +2*(n%2)*f((n+3)//2)) for n in range(61)] # G. C. Greubel, Mar 17 2024

Formula

a(n) = 2 - 2*F((n+3)/2) if n is odd, a(n) = 2 - F((n+2)/2) if n is even, where F = A000045 (Fibonacci numbers).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n >= 5.
G.f.: (1 - x - 2*x^3)/((1 - x)*(1 - x^2 - x^4)).
From G. C. Greubel, Mar 17 2024: (Start)
a(n) = (1/2)*Sum_{j=0..n} ( (1+(-1)^j)*Fibonacci(floor((j+3)/2)) - (1 - (-1)^j)*Lucas(floor((j+1)/2)) ).
a(n) = 2 - (1/2)*( (1+(-1)^n)*Fibonacci(floor((n+2)/2)) + 2*(1-(-1)^n)* Fibonacci(floor((n+3)/2)) ). (End)

A355019 Partial sums of L(1) - F(1) + L(2) - F(2) + L(3) - F(3) + ..., where L = A000032 and F = A000045.

Original entry on oeis.org

1, 0, 3, 2, 6, 4, 11, 8, 19, 14, 32, 24, 53, 40, 87, 66, 142, 108, 231, 176, 375, 286, 608, 464, 985, 752, 1595, 1218, 2582, 1972, 4179, 3192, 6763, 5166, 10944, 8360, 17709, 13528, 28655, 21890, 46366, 35420, 75023, 57312, 121391, 92734, 196416, 150048
Offset: 0

Views

Author

Clark Kimberling, Jun 16 2022

Keywords

Comments

The closely related partial sums of F(1) - L(1) + F(2) - L(2) + F(3) - L(3) + ... are given by A355018.

Examples

			a(0) = 1
a(1) = 1 - 1 = 0
a(2) = 1 - 1 + 3 = 3
a(3) = 1 - 1 + 3  - 1 = 2.

Links

Crossrefs

Cf. A000032, A000045, A355018, A355020, A355021.

Programs

  • Magma
    F:=Fibonacci; [(((n+1) mod 2)*F(Floor(n/2)+4) + 2*(n mod 2)*F(Floor((n+3)/2))) - 2: n in [0..60]]; // G. C. Greubel, Mar 17 2024
  • Mathematica
    f[n_] := Fibonacci[n]; g[n_] := LucasL[n];
f1[n_] := If[OddQ[n], 2 - 2 f[(n + 3)/2], 2 - f[(n + 2)/2]]
f2 = Table[f1[n], {n, 0, 20}]  (* A355018 *)
g1[n_] := If[OddQ[n], -2 + 2 f[(n + 3)/2], -2 + f[(n + 8)/2]]
g2 = Table[g1[n], {n, 0, 20}]  (* this sequence *)
LinearRecurrence[{1,1,-1,1,-1}, {1,0,3,2,6}, 61] (* G. C. Greubel, Mar 17 2024 *)
  • SageMath
    f=fibonacci; [(((n+1)%2)*f((n//2)+4) +2*(n%2)*f((n+3)//2)) -2 for n in range(61)] # G. C. Greubel, Mar 17 2024

Formula

a(n) = -2 + 2 F((n+3)/2) if n is odd, a(n) = - 2 + F((n+8)/2) if n is even, where F = A000045 (Fibonacci numbers).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) for n >= 5.
G.f.: (1 - x + 2*x^2)/((1 - x)*(1 - x^2 - x^4)).
From G. C. Greubel, Mar 17 2024: (Start)
a(n) = (1/2)*Sum_{j=0..n} ( (1+(-1)^j)*Lucas(floor(j/2) +1) - (1-(-1)^j) *Fibonacci(floor((j+1)/2)) ).
a(n) = (1/2)*( (1+(-1)^n)*Fibonacci(floor(n/2) +4) + 2*(1-(-1)^n)* Fibonacci(floor((n+3)/2)) ) - 2. (End)

A128618 Triangle read by rows: A128174 * A127647 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 2, 0, 5, 0, 1, 0, 3, 0, 8, 1, 0, 2, 0, 5, 0, 13, 0, 1, 0, 3, 0, 8, 0, 21, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55, 0, 144, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 233
Offset: 1

Views

Author

Gary W. Adamson, Mar 14 2007

Keywords

Comments

This triangle is different from A128619, which is A128619 = A127647 * A128174.

Examples

			First few rows of the triangle are:
  1;
  0, 1;
  1, 0, 2;
  0, 1, 0, 3;
  1, 0, 2, 0, 5;
  0, 1, 0, 3, 0, 8;
  1, 0, 2, 0, 5, 0, 13;
  0, 1, 0, 3, 0, 8,  0, 21;
  1, 0, 2, 0, 5, 0, 13,  0, 34;
  0, 1, 0, 3, 0, 8,  0, 21,  0, 55;
  1, 0, 2, 0, 5, 0, 13,  0, 34,  0, 89;
  ...

Links

Crossrefs

Cf. A000045, A052952, A127647, A128174, A128619, A355020.

Programs

  • Magma
    [((n+k+1) mod 2)*Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 17 2024
  • Mathematica
    Table[Fibonacci[k]*Mod[n-k+1,2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 17 2024 *)
  • SageMath
    flatten([[((n-k+1)%2)*fibonacci(k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 17 2024

Formula

By columns, Fibonacci(k) interspersed with alternate zeros in every column, k=1,2,3,...
Sum_{k=1..n} T(n, k) = A052952(n-1) (row sums).
From G. C. Greubel, Mar 17 2024: (Start)
T(n, k) = (1/2)*(1 + (-1)^(n+k))*Fibonacci(k).
T(n, n) = A000045(n).
T(2*n-1, n) = (1/2)*(1-(-1)^n)*A000045(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A052952(n-1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1 - (-1)^n)*(Fibonacci((n+ 5)/2) - 1).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n) * A355020(floor((n-1)/2)). (End)

Extensions

a(6) corrected and more terms from Georg Fischer, May 30 2023
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