A085194 -id:A085194 - cp's OEIS frontend

A085195 Partial sums of A085194.

0, 1, 4, 5, 7, 16, 17, 20, 21, 23, 28, 29, 31, 35, 64, 65, 68, 69, 71, 80, 81, 84, 85, 87, 92, 93, 95, 99, 112, 113, 116, 117, 119, 124, 125, 127, 131, 140, 141, 143, 147, 155, 256, 257, 260, 261, 263, 272, 273, 276, 277, 279, 284, 285, 287, 291, 320, 321, 324, 325

Offset: 0

Antti Karttunen, Jun 14 2003

nonn

Same sequence in base 4: A085186. a(A000108(n+1)) = A000302(n). Cf. A085188.

a(0)=0, a(n) = a(n-1) + A085194(n-1).

A079319 a(0) = 1; for n >= 1, a(n) = 4*a(n-1) - (2^n-1).

Original entry on oeis.org

1, 3, 9, 29, 101, 373, 1429, 5589, 22101, 87893, 350549, 1400149, 5596501, 22377813, 89494869, 357946709, 1431721301, 5726754133, 22906754389, 91626493269, 366504924501, 1466017600853, 5864066209109, 23456256447829

Offset: 0

N. J. A. Sloane, Feb 12 2003

Paolo Xausa, Table of n, a(n) for n = 0..1000
David Singmaster, On the cellular automaton of Ulam and Warburton, M500 Magazine of the Open University, #195 (December 2003), pp. 2-7. Also cached copy, included with permission.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A079314, A079315, A079316, A079317, A079318.

Records in A085194.

A079319list[nmax_]:=LinearRecurrence[{7,-14,8},{1,3,9},nmax+1];A079319list[50] (* Paolo Xausa, Jul 30 2023 *)

PARI
```
a(n)=if(n<0,0,2^n+(4^n-1)/3)
```

A079319=lambda n: 2**n + 4**n//3 # M. F. Hasler, May 28 2024

a(n) = 2^n + (4^n-1)/3, n>=0.
a(n) = Sum_{i = 0..2^n - 1} A079314(i).
G.f.: (1-4x+2x^2)/((1-x)(1-2x)(1-4x)).

A085193 Repeating part of A085192.

Original entry on oeis.org

2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 58, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 26, 2, 6, 2, 4, 10, 2, 4, 8, 18, 2, 4, 8, 16, 202, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 58, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 26, 2, 6, 2, 4, 10, 2, 4, 8, 18, 2, 4, 8, 16, 74, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10

Offset: 0

Antti Karttunen, Jun 14 2003

nonn

Antti Karttunen, Notes concerning A080237-tree and related sequences.

Same sequence divided by 2: A085194. Cf. A085190.

a(n) = A085192(A081291(n+1)-1).

a(n) = 4*A085193(A085182(n+1)-1) + 2 - (2^A007001(n+1)) if A007001(n+2)==1, otherwise 2^A007001(n+1).

A085189 Terms of A085190 halved.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 2, 1, 3, 8, 1, 3, 12, 17, 1, 2, 1, 3, 5, 1, 2, 1, 3, 8, 1, 3, 12, 29, 1, 2, 1, 3, 8, 1, 3, 12, 44, 1, 3, 12, 60, 77, 1, 2, 1, 3, 5, 1, 2, 1, 3, 8, 1, 3, 12, 17, 1, 2, 1, 3, 5, 1, 2, 1, 3, 8, 1, 3, 12, 29, 1, 2, 1, 3, 8, 1, 3, 12, 44, 1, 3, 12, 60, 137, 1, 2, 1, 3, 5, 1, 2, 1, 3, 8, 1, 3

Offset: 0

Antti Karttunen, Jun 14 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1429

Partial sums: A085188. Cf. A085194.

a(n) = A085190(n)/2.

cp's OEIS Frontend

A085195 Partial sums of A085194.

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A079319 a(0) = 1; for n >= 1, a(n) = 4*a(n-1) - (2^n-1).

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A085193 Repeating part of A085192.

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Formula

A085189 Terms of A085190 halved.

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