A006336 -id:A006336 - cp's OEIS frontend

A197878 a(n) = floor(2(1 + sqrt(2))n).

4, 9, 14, 19, 24, 28, 33, 38, 43, 48, 53, 57, 62, 67, 72, 77, 82, 86, 91, 96, 101, 106, 111, 115, 120, 125, 130, 135, 140, 144, 149, 154, 159, 164, 168, 173, 178, 183, 188, 193, 197, 202, 207, 212, 217, 222, 226, 231, 236, 241, 246, 251, 255, 260, 265, 270

Offset: 1

Zak Seidov, Oct 18 2011

First differences are 4 and 5. Also, there is no immediate pattern in parity of a(n).
Are similar sequences well defined (in terms of rounding problems)? See also A086843, A086844, A196468.
Answer: I would not call the sequences A086843, A086844, A196468 'similar' to (a(n)). The first differences d =5,5,5,5,4,5,5,5,5,4,... are a Sturmian sequence (d(n)) with slope alpha = 2 + sqrt(8) and intercept 0. We give d offset 0 by setting d(0):=4. By Hofstadter's Fundamental Theorem of eta-sequences, the chunks 45555 and 455555 occur following a Sturmian sequence with density beta = (sqrt(8) - 2)/(3 - sqrt(8)). Since beta = 2 + sqrt(8) = alpha, the sequence (d(n)) is fixed point of the substitution 4->45555, 5->455555. See A197879 for a complete description of the parity pattern of (a(n)). - Michel Dekking, Jan 24 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012; J. Int. Seq. 16 (2013) #13.1.8
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Cf. A001030. - Michel Dekking, Jan 24 2017
Cf. A086843, A086844, A196468.
A bisection of A003151.

[Floor(2*(1 + Sqrt(2))*n): n in [1..100]]; // G. C. Greubel, Aug 18 2018

Mathematica
```
Table[Floor[((2+Sqrt[8]))*n], {n,100}]
```

a(n)=2*n+sqrtint(8*n^2) \\ Charles R Greathouse IV, Oct 25 2011

a(n) = A003151(2n). - R. J. Mathar, Oct 20 2011

A249039 a(1)=1, a(2)=2; thereafter a(n) = a(n-1) + a(n-1-(number of even terms so far)) + a(n-1-(number of odd terms so far)).

Original entry on oeis.org

1, 2, 4, 7, 11, 17, 26, 37, 52, 70, 92, 120, 157, 200, 254, 323, 401, 490, 597, 719, 859, 1021, 1211, 1438, 1687, 1979, 2325, 2740, 3183, 3704, 4262, 4863, 5553, 6350, 7201, 8174, 9216, 10336, 11545, 12894, 14350, 15928, 17646, 19526, 21596, 23893, 26352, 29060, 32060, 35406, 39167

Offset: 1

N. J. A. Sloane, Oct 26 2014

nonn

Suggested by A006336, A007604 and A249036-A249038.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006336, A007604, A249036, A249037, A249038.

A249040 and A249041 give numbers of even and odd terms so far.

import Data.List (genericIndex)
a249039 n = genericIndex a249039_list (n - 1)
a249039_list = 1 : 2 : f 2 2 1 1 where
   f x u v w = y : f (x + 1) y (v + 1 - mod y 2) (w + mod y 2)
               where y = u + a249039 (x - v) + a249039 (x - w)
-- Reinhard Zumkeller, Nov 11 2014

M:=100;
v[1]:=1; v[2]:=2; w[1]:=0; w[2]:=1; x[1]:=1; x[2]:=1;
for n from 3 to M do
v[n]:=v[n-1]+v[n-1-w[n-1]]+v[n-1-x[n-1]];
if v[n] mod 2 = 0 then w[n]:=w[n-1]+1; x[n]:=x[n-1];
else w[n]:=w[n-1]; x[n]:=x[n-1]+1; fi;
od:
[seq(v[n], n=1..M)]; # A249039
[seq(w[n], n=1..M)]; # A249040
[seq(x[n], n=1..M)]; # A249041

For n > 1: a(n+1) = a(n) + a(n - A249040(n)) + a(n - A249041(n)) by mutual recursion. - Reinhard Zumkeller, Nov 11 2014

A131882 a(0)=1; thereafter a(n)=a(n-1)+a([n/Phi]), where Phi=(1+sqrt(5))/2, the golden ratio.

Original entry on oeis.org

1, 2, 4, 6, 10, 16, 22, 32, 42, 58, 80, 102, 134, 176, 218, 276, 334, 414, 516, 618, 752, 886, 1062, 1280, 1498, 1774, 2108, 2442, 2856, 3270, 3786, 4404, 5022, 5774, 6660, 7546, 8608, 9670, 10950, 12448, 13946, 15720, 17494, 19602, 22044, 24486, 27342

Offset: 0

T. D. Noe, Jul 23 2007

nonn

Same recursion as A006336, but different initial condition.

T. D. Noe, Table of n, a(n) for n = 0..1000

a[0]=1; a[n_] := a[n] = a[n-1] + a[Floor[n/GoldenRatio]]; Table[a[n], {n,0,100}]

A060729 a(n+1) = a(n) + a(n minus the number of terms of the same parity as n so far).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 16, 19, 25, 31, 37, 49, 55, 74, 83, 108, 120, 145, 161, 198, 217, 266, 291, 346, 377, 451, 482, 565, 602, 710, 765, 885, 940, 1085, 1159, 1357, 1431, 1697, 1771, 2117, 2191, 2642, 2725, 3207, 3290, 3855, 3963, 4673, 4781

Offset: 1

Robert G. Wilson v, Apr 22 2001

nonn

			a(5) = a(4) + a(4 - the number of even terms so far) = a(4) + a(4-2) = 4 + 2 = 6.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A006336, A006604.

a[ 1 ] = 1; a[ 2 ] = 2; a[ n_ ] := a[ n ] = Block[ {e = 0}, Do[ If[ EvenQ[ a[ k ] ], e++ ], {k, 1, n - 1} ]; If[ OddQ[ n ], a[ n - 1 ] + a[ n - 1 - e ], a[ n - 1 ] + a[ e ] ] ]; Table[ a[ n ], {n, 1, 50} ]

A060714 a(n) = a(n-1) + a(n-1 minus the number of terms of the same parity as n so far).

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 11, 17, 19, 30, 33, 50, 55, 74, 80, 99, 108, 138, 155, 188, 207, 257, 276, 331, 361, 441, 471, 579, 609, 764, 797, 985, 1018, 1225, 1275, 1551, 1601, 1962, 2017, 2458, 2532, 2973, 3053, 3632, 3731, 4340, 4448, 5057, 5195, 5992

Offset: 1

Robert G. Wilson v, Apr 21 2001

nonn

			a(5) = a(4) + a(4 - the number of odd terms so far) = a(4) + a(4-3) = 5 + 1 = 6.

Cf. A006336, A007604.

a[ 1 ] = 1; a[ 2 ] = 2; a[ n_ ] := a[ n ] = Block[ {e = 0}, Do[ If[ EvenQ[ a[ k ] ], e++ ], {k, 1, n - 1} ]; If[ EvenQ[ n ], a[ n - 1 ] + a[ n - 1 - e ], a[ n - 1 ] + a[ e ] ] ]; Table[ a[ n ], {n, 1, 15} ]

cp's OEIS Frontend

A197878 a(n) = floor(2*(1 + sqrt(2))*n).

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A249039 a(1)=1, a(2)=2; thereafter a(n) = a(n-1) + a(n-1-(number of even terms so far)) + a(n-1-(number of odd terms so far)).

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A131882 a(0)=1; thereafter a(n)=a(n-1)+a([n/Phi]), where Phi=(1+sqrt(5))/2, the golden ratio.

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A060729 a(n+1) = a(n) + a(n minus the number of terms of the same parity as n so far).

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Mathematica

A060714 a(n) = a(n-1) + a(n-1 minus the number of terms of the same parity as n so far).

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Mathematica

A197878 a(n) = floor(2(1 + sqrt(2))n).