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.

A338225 a(n) = F(n+3) * F(n+1) + (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

Original entry on oeis.org

2, 11, 23, 66, 167, 443, 1154, 3027, 7919, 20738, 54287, 142131, 372098, 974171, 2550407, 6677058, 17480759, 45765227, 119814914, 313679523, 821223647, 2149991426, 5628750623, 14736260451, 38580030722, 101003831723, 264431464439, 692290561602, 1812440220359, 4745030099483
Offset: 1

Views

Author

Burak Muslu, Jan 30 2021

Keywords

Comments

Twice the difference between the areas of consecutive deltoids with cross lengths F(n+3) and F(n).
Also it is the difference between the areas of consecutive rectangles with side lengths F(n+3) and F(n).
Also first differences of A226205 for n > 2.

Examples

			For n = 2, a(2) = F(2+3) * F(2+1) + (-1)^2 = 5 * 2 + 1 = 11.

References

  • Burak Muslu, Sayılar ve Bağlantılar, Luna, 2021, p. 50.

Links

Crossrefs

Cf. A000045, A007598, A226205, A341208.

Programs

  • Mathematica
    a[n_] := Fibonacci[n + 3] * Fibonacci[n + 1] + (-1)^n; Array[a, 30] (* Amiram Eldar, Jan 30 2021 *)
  • PARI
    a(n) = fibonacci(n+3)*fibonacci(n+1) + (-1)^n; \\ Michel Marcus, Mar 25 2021

Formula

a(n) = F(n+3) * F(n+1) + (-1)^n, for n > 0.
G.f.: x*(2 + 7*x - 3*x^2)/(1 - 2*x - 2*x^2 + x^3). - Stefano Spezia, Jan 30 2021
a(n) = F(n+2)^2 + 2*(-1)^n = A007598(n+2) + 2*(-1)^n. - Amiram Eldar, Jan 11 2022

A341928 a(n) = F(n+4) * F(n+2) + 7 * (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

Original entry on oeis.org

3, 31, 58, 175, 435, 1162, 3019, 7927, 20730, 54295, 142123, 372106, 974163, 2550415, 6677050, 17480767, 45765219, 119814922, 313679515, 821223655, 2149991418, 5628750631, 14736260443, 38580030730, 101003831715, 264431464447, 692290561594, 1812440220367
Offset: 1

Views

Author

Burak Muslu, Feb 23 2021

Keywords

Comments

First differences of A341208.
Second differences of A338225.
Third differences of A226205 n > 2.
Third differences between the areas of consecutive rectangles with side lengths F(n+3) and F(n).
Twice the third differences between the areas of consecutive deltoids with cross lengths F(n+3) and F(n).
Twice the third differences between the areas of consecutive triangles with the height and base length are F(n+3) and F(n).

Examples

			For n = 2, a(2) = F(2+4) * F(2+2) + 7 * (-1)^2 = 8 * 3 + 7 = 31.

References

  • Burak Muslu, Sayılar ve Bağlantılar, Luna, 2021, p. 51 (in Turkish).

Links

Crossrefs

Cf. A000045, A341208, A338225, A226205.

Programs

  • Mathematica
    Table[Fibonacci[n + 4] * Fibonacci[n + 2] + 7 * (-1)^n, {n, 1, 28}] (* Amiram Eldar, Feb 23 2021 *)
  • PARI
    a(n) = fibonacci(n+4)*fibonacci(n+2) + 7*(-1)^n; \\ Michel Marcus, Feb 23 2021

Formula

a(n) = F(n+4) * F(n+2) + 7 * (-1)^n.
G.f.: x*(3 + 25*x - 10*x^2)/(1 - 2*x - 2*x^2 + x^3).

A343008 a(n) = F(n+5) * F(n+2) - 12 * (-1)^n where F(n) = A000045(n) are the Fibonacci numbers.

Original entry on oeis.org

28, 27, 117, 260, 727, 1857, 4908, 12803, 33565, 87828, 229983, 602057, 1576252, 4126635, 10803717, 28284452, 74049703, 193864593, 507544140, 1328767763, 3478759213, 9107509812, 23843770287, 62423800985, 163427632732, 427859097147, 1120149658773
Offset: 1

Views

Author

Burak Muslu, Apr 02 2021

Keywords

Comments

First differences of A341928.
Second differences of A341208.
Third differences of A338225.
Fourth differences of A226205.
Fourth differences between the areas of consecutive rectangles with side lengths F(n+3) and F(n).
Twice the fourth differences between the areas of consecutive deltoids with cross lengths F(n+3) and F(n).
Twice the fourth differences between the areas of consecutive triangles with the height and base length are F(n+3) and F(n).

Examples

			For n = 2, a(2) = F(2+5) * F(2+2) - 12 * (-1)^2 = 13 * 3 - 12 = 27.

References

  • B. Muslu, Sayılar ve Bağlantılar, Luna, 2021, p. 52.

Links

Crossrefs

Cf. A000045, A226205, A338225, A341208, A341928.

Programs

  • Mathematica
    a[n_]:=Fibonacci[n+5]*Fibonacci[n+2]-12(-1)^n
Array[a,30] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

Formula

a(n) = F(n+5) * F(n+2) - 12 * (-1)^n.
G.f.: x*(28 - 29*x + 7*x^2)/(1 - 2*x - 2*x^2 + x^3).
Showing 1-3 of 3 results.

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