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 1-6 of 6 results.

A104803 a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2)), a(1)=1, a(2)=3.

Original entry on oeis.org

1, 3, 3, 4, 5, 6, 7, 9, 11, 14, 17, 22, 27, 34, 43, 54, 69, 87, 111, 141, 179, 227, 289, 367, 467, 593, 754, 959, 1219, 1551, 1972, 2508, 3190, 4057, 5160, 6563, 8348, 10618, 13506, 17180, 21853, 27797, 35358, 44976, 57210, 72772, 92567, 117747, 149776
Offset: 1

Views

Author

Zak Seidov, Mar 26 2005

Keywords

Links

Crossrefs

Cf. A104804, A104805, A104810.

Programs

  • Magma
    A104803:= func< n| n lt 3 select (2*n-1) else Floor(Sqrt(Self(n-1)^2 +Self(n-2)^2)) >;
[A104803(n): n in [1..60]]; // G. C. Greubel, Jun 27 2021
  • Mathematica
    a[n_]:= a[n]= If[n<3, 2*n-1, Floor[Sqrt[a[n-1]^2 +a[n-2]^2]]]; Table[a[n], {n, 50}] (* Robert G. Wilson v, Mar 28 2005 *)
nxt[{a_,b_}]:={b,Floor[Sqrt[a^2+b^2]]}; Transpose[NestList[nxt,{1,3}, 50]] [[1]] (* Harvey P. Dale, Oct 29 2012 *)
  • Sage
    @CachedFunction
def a(n): return (2*n-1) if (n<3) else floor(sqrt(a(n-1)^2 + a(n-2)^2))
[a(n) for n in (1..60)] # G. C. Greubel, Jun 27 2021

Formula

a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2)) with a(1)=1, a(2)=3.

Extensions

More terms from Robert G. Wilson v, Mar 28 2005

A104805 "Ceiling of hypotenuses": a(n) = ceiling(sqrt(a(n-1)^2 + a(n-2)^2)), a(1)=1, a(2)=3.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 12, 15, 20, 25, 33, 42, 54, 69, 88, 112, 143, 182, 232, 295, 376, 478, 609, 775, 986, 1255, 1597, 2032, 2585, 3289, 4184, 5322, 6770, 8612, 10955, 13935, 17726, 22548, 28682, 36484, 46409, 59033, 75092, 95519, 121502, 154553, 196595, 250073
Offset: 1

Views

Author

Zak Seidov, Mar 26 2005

Keywords

Crossrefs

Cf. A104803, A104804.

Programs

  • Mathematica
    a[n_] := a[n] = Ceiling[ Sqrt[ a[n - 1]^2 + a[n - 2]^2]]; a[1] = 1; a[2] = 3; Table[ a[n], {n, 48}] (* Robert G. Wilson v, Mar 28 2005 *)

Extensions

More terms from Robert G. Wilson v, Mar 28 2005

A104806 "Round of hypotenuse", see comments.

Original entry on oeis.org

1, 3, 3, 4, 5, 7, 9, 11, 14, 18, 23, 29, 37, 47, 60, 77, 97, 124, 158, 201, 255, 324, 413, 525, 668, 849, 1081, 1374, 1748, 2224, 2829, 3598
Offset: 1

Views

Author

Zak Seidov, Mar 26 2005

Keywords

Comments

Start with a=1, b=3; repeating procedure "c=sqrt[a^2+b^2], a=b,b=c" gives: s = 1,3, Sqrt[10], Sqrt[19], Sqrt[29], 4*Sqrt[3], Sqrt[77],5*Sqrt[5], Sqrt[202],Sqrt[327], 23, 2*Sqrt[214],Sqrt[1385], 3*Sqrt[249],... Then A104806(n)=round[s(n)].

Crossrefs

Cf. A104803, A104804, A104805.

A104863 a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2)), a(1)=10, a(2)=30.

Original entry on oeis.org

10, 30, 31, 43, 53, 68, 86, 109, 138, 175, 222, 282, 358, 455, 578, 735, 935, 1189, 1512, 1923, 2446, 3111, 3957, 5033, 6402, 8143, 10358, 13175, 16759, 21317, 27116, 34491, 43873, 55807, 70987, 90297, 114859, 146103, 185845, 236398, 300703, 382500, 486547, 618897
Offset: 1

Views

Author

Zak Seidov, Mar 28 2005

Keywords

Links

Crossrefs

Cf. A104803, A104804, A104805, A104806, A104864.

Programs

  • Magma
    A104863:= func< n| n lt 3 select 10*(2*n-1) else Floor(Sqrt(Self(n-1)^2 +Self(n-2)^2)) >;
[A104863(n): n in [1..60]]; // G. C. Greubel, Jun 27 2021
  • Mathematica
    nxt[{a_,b_}]:={b,Floor[Sqrt[a^2+b^2]]}; Transpose[NestList[nxt,{10,30},60]][[1]] (* Harvey P. Dale, Jun 18 2013 *)
  • Sage
    @CachedFunction
def a(n): return 10*(2*n-1) if (n<3) else floor(sqrt(a(n-1)^2 + a(n-2)^2))
[a(n) for n in (1..60)] # G. C. Greubel, Jun 27 2021

Formula

For n>=17, a(n) = a(n-2) + a(n-4) + 1 (conjectured). If true then for m>5, a(2*m+1) = 4*F(m) + 25*F(m+1) + 1 and a(2*m+2) = 8*F(m) + 30*F(m+1) + 1 with F(n) = A000045(n). - Ralf Stephan, Nov 15 2010

A104864 "Floor of hypotenuse": a(n)=A104863(n)-10*A104803(n).

Original entry on oeis.org

0, 0, 1, 3, 3, 8, 16, 19, 28, 35, 52, 62, 88, 115, 148, 195, 245, 319, 402, 513, 656, 841, 1067, 1363, 1732, 2213, 2818, 3585, 4569, 5807, 7396, 9411
Offset: 0

Views

Author

Zak Seidov, Mar 28 2005

Keywords

Crossrefs

Cf. A104803, A104804, A104805, A104806, A104863.

A128424 a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2 + a(n-1)*a(n-2))), a(1)=1, a(2)=3.

Original entry on oeis.org

1, 3, 3, 5, 7, 10, 14, 20, 29, 42, 61, 89, 130, 190, 278, 407, 596, 873, 1279, 1874, 2746, 4024, 5897, 8642, 12665, 18561, 27202, 39866, 58426, 85627, 125492, 183917, 269543, 395034, 578950, 848492, 1243525, 1822474, 2670965, 3914489, 5736962
Offset: 1

Views

Author

Zak Seidov, May 04 2007

Keywords

Comments

For a triangle with sides a(n-1) and a(n-2) and a 120-degree angle between them, a(n) is the floor of the value of the third side.
a(n) = A020711(n-4) for 4 <= n <= 41. - Georg Fischer, Nov 02 2018

Crossrefs

Cf. A020711, A104803, A104804, A104805.

Programs

  • Mathematica
    a[1]=1;a[2]=3;a[n_]:=a[n]=Floor[Sqrt[a[n-1]^2+a[n-2]^2+a[n-1]*a[n-2]]] Table[a[n],{n,45}]
RecurrenceTable[{a[1]==1,a[2]==3,a[n]==Floor[Sqrt[a[n-1]^2+a[n-2]^2+ a[n-1]*a[n-2]]]},a,{n,50}] (* Harvey P. Dale, Oct 01 2018 *)

Formula

Conjectures from Colin Barker, Nov 03 2018: (Start)
G.f.: x*(1 + x - 2*x^2 + x^3 - 2*x^4 + x^5 - x^6) / ((1 - x)*(1 - x - x^3)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) for n>7.
(End)
Showing 1-6 of 6 results.

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