A122237 -id:A122237 - cp's OEIS frontend

A122241 Iterates of A122237, starting from 4.

4, 22, 54, 169, 516, 1841, 6076, 19256, 66140, 252691, 888179, 2900616

Offset: 1

Antti Karttunen, Sep 14 2006

nonn

It might be more natural to use offset 0 for the initial value a(0) = 4 since that's not the result of an actual iteration of the map A122237. However, some other sequences (A122242, A179845, A179841, ...?) depend on this and have b-files that would require to be changed. - M. F. Hasler, Jul 18 2025

Cf. A122237 = A057548 o A082358, A122242 = A014486 o A122241.

(define (A122241 n) (if (= 1 n) 4 (A122237 (A122241 (- n 1)))))

a(n+1) = A122237(a(n)) for n >= 1; a(1) = 4.

A122244 Iterates of A122237, starting from 5.

Original entry on oeis.org

5, 21, 55, 183, 512, 1724, 6085, 20899, 66106, 231841, 888275, 3188220

Offset: 1

Antti Karttunen, Sep 14 2006

nonn

A122245.

(define (A122244 n) (if (= 1 n) 5 (A122237 (A122244 (- n 1)))))

A179756 Iterates of A122237 starting from the initial value 8.

Original entry on oeis.org

8, 20, 57, 167, 569, 1832, 5693, 19241, 72004, 251434, 817685, 2912603, 11464927, 41520649, 137255398, 502535651, 2038965137, 7539964485, 25351037019, 94403539934, 389919393256, 1461664899103, 4982806264910, 18769209002168

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

See the illustrations at A179757.

Antti Karttunen, Table of n, a(n) for n = 1..1024

Cf. A122237, A179757.

(define (A179756 n) (if (= 1 n) 8 (A122237 (A179756 (-1+ n)))))

a(1) = 8, a(n) = A122237(A179756(a(n-1))).

A179754 Iterates of A122237 starting from the initial value 6.

Original entry on oeis.org

6, 18, 48, 141, 431, 1427, 4692, 15417, 52833, 189576, 667458, 2303485, 8261483, 30854617, 112235989, 396544978, 1457813051, 5573491517, 20668205325, 74283571059, 277368667214, 1076038620556, 4041473193751, 14706699067814

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

See the illustrations at A179755.

Antti Karttunen, Table of n, a(n) for n = 1..1024

a(n+1) = A127307(A179756(n)).

(define (A179754 n) (if (= 1 n) 6 (A122237 (A179754 (-1+ n)))))

a(1) = 6, a(n) = A122237(A179754(a(n-1))).

A122242 a(n) = A014486(A122241(n)).

Original entry on oeis.org

42, 240, 916, 3748, 14960, 62104, 248176, 969304, 3876576, 15962544, 63772488, 248169896, 993554240, 4086635408, 16350541128, 63529835824, 254129143040, 1046249323840, 4184725760584, 16276030608712, 65054467548432, 267635134298624

Offset: 1

Antti Karttunen, Sep 14 2006

Question: to which Wolfram's class does this simple program belong, class 3 or class 4, or is such categorization at all applicable here?

Antti Karttunen, Table of n, a(n) for n = 1..1650
Antti Karttunen, Python program for computing this sequence and the associated images.
Antti Karttunen, Terms a(1)-a(768) drawn as binary strings, in Wolframesque fashion.
Antti Karttunen, Terms a(1)-a(256) drawn as binary strings, showing details better.
Antti Karttunen, The central parts of terms a(1) - a(20000) drawn in similar fashion. (Note the "foreshadowing ghost columns" present in the "central skyscraper" attractor that are not present at A122245. Compare also to illustrations in A376412)
Index entries for sequences related to binary expansion of n
Index entries for encodings of plane rooted trees
Index entries for sequences related to parenthesizing

Cf. A014486, A057548, A082358, A122237, A122241, A122243 (same sequence in binary).
Compare to similar Wolframesque plots given in A080070, A122229, A122232, A122235, A122239, A122245.
Cf. also A376402, A376412.

Python
```
# See the Links section
```

For n >= 1, a(1+n) = 2*a(n) XOR A376402(n), a(4+n) = 16*a(n) XOR A376412(n). - Antti Karttunen, Sep 23 2024

A106191 Expansion of sqrt(1-4x)/(1-x).

Original entry on oeis.org

1, -1, -3, -7, -17, -45, -129, -393, -1251, -4111, -13835, -47427, -164999, -581023, -2066823, -7415703, -26805393, -97520733, -356810313, -1312087713, -4846614093, -17974854933, -66907388973, -249872516253, -935991743553, -3515800038201, -13239692841105

Offset: 0

Paul Barry, Apr 24 2005

Row sums of number triangle A106190. Partial sums of A002420.

For n >= 1, the absolute values also give the iterates of A122237, starting from 0. (A122237(0), A122237(A122237(0)), A122237(A122237(A122237(0))), ...), this stems from the fact that the sequence gives the positions of terms with binary expansion 1(10){n-1}0 in A014486 (see A080675).

|a(n)| = A080300(A080675(n)) = A075161(A001348(n)) (for n >= 1) = A075163(A000244(A008578(n-2))) = A014137(n-1)+A014138(n-2) = 2*A014137(n-1)-1, for n >= 2 (because binomial(2n+2, n+1)/(2n+1) = 2*A000108(n)).

a(n) = Sum_{k=0..n} binomial(2k, k)/(1-2k).
G.f.: (2/(1-x))/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
D-finite with recurrence: a(0)=1, a(1)=-1; for n>1, a(n) = (1/n)*((5*n-6)*a(n-1) - (4*n-6)*a(n-2)). - Tani Akinari, Aug 25 2013

Barry's formula made more succinct, as well as comments regarding interpretation as absolute values added by Antti Karttunen, Sep 14 2006

A122227 a(n) = A057548(A057117(n)).

Original entry on oeis.org

1, 3, 7, 8, 17, 18, 21, 22, 20, 45, 46, 49, 50, 48, 58, 59, 63, 64, 62, 54, 55, 57, 61, 129, 130, 133, 134, 132, 142, 143, 147, 148, 146, 138, 139, 141, 145, 170, 171, 175, 176, 174, 189, 190, 195, 196, 194, 184, 185, 188, 193, 157, 158, 161, 162, 160, 169, 173

Offset: 0

Antti Karttunen, Sep 14 2006

nonn

Iterates: A122228, A122231, A122234, A122238. Cf. A080067, A122237.

A376402 Bitwise XOR (centrally aligned) of two consecutive terms of A122242.

Original entry on oeis.org

164, 628, 2444, 10040, 34424, 142400, 612536, 2536016, 8772720, 36320296, 156298040, 648930320, 2246427920, 9290072680, 40123676576, 166398412640, 574717970240, 2376856817864, 10244120543704, 42544644116352, 146496800436256, 607708669110320, 2625008220416552, 10882360875506928, 37586414897168848, 156056124134144296

Offset: 1

Antti Karttunen, Sep 22 2024

Antti Karttunen, Table of n, a(n) for n = 1..1650
Antti Karttunen, Python program for computing this sequence and the associated images.
Antti Karttunen, Terms a(1)-a(2048) drawn as binary strings, in Wolframesque fashion.
Index entries for sequences related to binary expansion of n

Cf. A122242 (A014486, A057548, A082358, A122237, A122241).

Cf. also A327973, A376405, A376412.

A376402(n) = bitxor(A122242(1+n), 2*A122242(n));

Python
```
# See the Links section
```

a(n) = A122242(1+n) XOR 2*A122242(n).

A376412 a(n) = A122242(4+n) XOR 16*A122242(n).

Original entry on oeis.org

14544, 64920, 258096, 925720, 3703264, 16562224, 66158664, 237396008, 948568384, 4239130768, 16949182920, 60767078320, 243080890624, 1085016114240, 4341071150792, 15535530051144, 62225888982288, 277421534227968, 1111070191401136, 3979658311943880, 15908408006551904, 71162952082313488, 284082756324759560, 1019946695587234480

Offset: 1

Antti Karttunen, Sep 22 2024

See comments in A376415.

Antti Karttunen, Table of n, a(n) for n = 1..1650
Antti Karttunen, Python program for computing this sequence and the associated images.
Antti Karttunen, Terms a(1)-a(768) drawn as binary strings, in Wolframesque fashion.
Antti Karttunen, The central parts of terms a(1) - a(20000) drawn in similar fashion. (the widening blank corridor in the middle corresponds with the "central crystallization" attractor present in A122242)
Index entries for sequences related to binary expansion of n

Cf. A122242 (A014486, A057548, A082358, A122237, A122241).

Cf. also A327973, A376402, A376415.

A376412(n) = bitxor(A122242(4+n), 16*A122242(n));

Python
```
# See the Links section
```

a(n) = A122242(4+n) XOR 16*A122242(n).

A376405 Bitwise XOR (centrally aligned) of two consecutive terms of A122245.

Original entry on oeis.org

176, 584, 2068, 9232, 38952, 135296, 567096, 2444568, 10136288, 34920688, 144732808, 625792608, 2598216176, 8989149792, 37119010736, 160422522664, 665656629200, 2297815400576, 9505629937992, 41066855413976, 169932530966160, 589165636912400, 2439104800321640, 10514745879265952, 43543845360254320, 149771860125187648

Offset: 1

Antti Karttunen, Sep 22 2024

Antti Karttunen, Table of n, a(n) for n = 1..1650
Antti Karttunen, Python program for computing this sequence and the associated images.
Antti Karttunen, Terms a(1)-a(2048) drawn as binary strings, in Wolframesque fashion.
Index entries for sequences related to binary expansion of n

Cf. A122245 (A014486, A057548, A082358, A122237, A122244).

Cf. also A327973, A376402, A376415.

A376405(n) = bitxor(A122245(1+n), 2*A122245(n));

Python
```
# See the Links section
```

a(n) = A122245(1+n) XOR 2*A122245(n).

cp's OEIS Frontend

A122241 Iterates of A122237, starting from 4.

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A122244 Iterates of A122237, starting from 5.

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A179756 Iterates of A122237 starting from the initial value 8.

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A179754 Iterates of A122237 starting from the initial value 6.

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A122242 a(n) = A014486(A122241(n)).

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A106191 Expansion of sqrt(1-4x)/(1-x).

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A122227 a(n) = A057548(A057117(n)).

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A376402 Bitwise XOR (centrally aligned) of two consecutive terms of A122242.

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PARI

Python

Formula

A376412 a(n) = A122242(4+n) XOR 16*A122242(n).

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PARI

Python