A104803 -id:A104803 - cp's OEIS frontend

A104910 a(n) = A104908(n) - 100*A104803(n).

0, 0, 16, 35, 37, 91, 175, 214, 316, 401, 590, 713, 1005, 1313, 1694, 2224, 2798, 3635, 4590, 5858, 7486, 9591, 12175, 15548, 19760, 25238, 32134, 40885, 52093, 66222, 84326, 107306

Offset: 1

Zak Seidov, Mar 29 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A104803, A104908, A104909.

A104803[n_]:= A104803[n]= If[n<3, (2*n-1), Floor[Sqrt[A104803[n-1]^2 +A104803[n- 2]^2]]];
A104908[n_]:= A104908[n]= If[n<3, 100*(2*n-1), Floor[Sqrt[A104908[n-1]^2 +A104908[n -2]^2]]];
A104910[n_]:= A104908[n] - 100*A104803[n];
Table[A104910[n], {n, 60}] (* G. C. Greubel, Jun 27 2021 *)

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

Zak Seidov, Mar 28 2005

nonn

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

A104804 "Rounded hypotenuses": a(n) = round(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, 8, 10, 13, 16, 21, 26, 33, 42, 53, 68, 86, 110, 140, 178, 226, 288, 366, 466, 593, 754, 959, 1220, 1552, 1974, 2511, 3194, 4063, 5168, 6574, 8362, 10637, 13530, 17211, 21892, 27847, 35422, 45057, 57314, 72904, 92736, 117962, 150050

Offset: 1

Zak Seidov, Mar 26 2005

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..2000

Cf. A104803, A104805.

a[n_] := a[n] = Round[ 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 *)

from gmpy2 import isqrt_rem
A104804_list = [1,3]
for _ in range(1000):
    i, j = isqrt_rem(A104804_list[-1]**2+A104804_list[-2]**2)
    A104804_list.append(int(i+ int(4*(j-i) >= 1))) # Chai Wah Wu, Aug 16 2016

a(n) = A063827(n) for n > 2. - Georg Fischer, Oct 07 2018

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

Zak Seidov, Mar 26 2005

nonn

Cf. A104803, A104804.

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 *)

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

Zak Seidov, Mar 26 2005

nonn

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)].

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

Zak Seidov, Mar 28 2005

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

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

nxt[{a_,b_}]:={b,Floor[Sqrt[a^2+b^2]]}; Transpose[NestList[nxt,{10,30},60]][[1]] (* Harvey P. Dale, Jun 18 2013 *)

@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

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

A104908 a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2)), a(1)=100, a(2)=300.

Original entry on oeis.org

100, 300, 316, 435, 537, 691, 875, 1114, 1416, 1801, 2290, 2913, 3705, 4713, 5994, 7624, 9698, 12335, 15690, 19958, 25386, 32291, 41075, 52248, 66460, 84538, 107534, 136785, 173993, 221322, 281526, 358106

Offset: 1

Zak Seidov, Mar 29 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A104803, A104909, A104910.

A104908:= func< n| n lt 3 select 100*(2*n-1) else Floor(Sqrt(Self(n-1)^2 +Self(n-2)^2)) >;
[A104908(n): n in [1..40]]; // G. C. Greubel, Jun 27 2021

a[n_]:= a[n]= If[n<3, 100*(2*n-1), Floor[Sqrt[a[n-1]^2 + a[n-2]^2]] ];
Table[a[n], {n, 40}] (* G. C. Greubel, Jun 27 2021 *)

@CachedFunction
def a(n): return 100*(2*n-1) if (n<3) else floor(sqrt(a(n-1)^2 + a(n-2)^2))
[a(n) for n in (1..40)] # G. C. Greubel, Jun 27 2021

a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2)) with a(1) = 100, a(2) = 300.

A104909 a(n) = A104908(n) - 10*A104863(n).

Original entry on oeis.org

0, 0, 6, 5, 7, 11, 15, 24, 36, 51, 70, 93, 125, 163, 214, 274, 348, 445, 570, 728, 926, 1181, 1505, 1918, 2440, 3108, 3954, 5035, 6403, 8152, 10366, 13196, 16788, 21357, 27172, 34562, 43969, 55928, 71150, 90509, 115128, 146446, 186287, 236964, 301424, 383419

Offset: 1

Zak Seidov, Mar 29 2005

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A104803, A104908, A104910.

A104863[n_]:= A104863[n]= If[n<3, 10*(2*n-1), Floor[Sqrt[A104863[n-1]^2 + A104863[n - 2]^2]]];
A104908[n_]:= A104908[n]= If[n<3, 100*(2*n-1), Floor[Sqrt[A104908[n-1]^2 + A104908[n-2]^2]]];
A104909[n_]:= A104908[n] -10*A104863[n];
Table[A104909[n], {n, 60}] (* G. C. Greubel, Jun 27 2021 *)

Terms a(33) onward added by G. C. Greubel, Jun 27 2021

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

Zak Seidov, May 04 2007

nonn

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

Cf. A020711, A104803, A104804, A104805.

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 *)

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)

cp's OEIS Frontend

A104910 a(n) = A104908(n) - 100*A104803(n).

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

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A104804 "Rounded hypotenuses": a(n) = round(sqrt(a(n-1)^2 + a(n-2)^2)), a(1)=1, a(2)=3.

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

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A104806 "Round of hypotenuse", see comments.

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

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A104908 a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2)), a(1)=100, a(2)=300.

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Sage

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A104909 a(n) = A104908(n) - 10*A104863(n).

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Mathematica

Extensions

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.

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