A001592 -id:A001592 - cp's OEIS frontend

A111429 Tribonacci(hexanacci(n)).

0, 0, 0, 0, 0, 0, 0, 1, 2, 24, 3136, 53798080, 8607945812375585, 220376367601372354229355484029120, 78531983922879942351416192215114163135375656803468317366190276600

Offset: 0

Jonathan Vos Post, Nov 13 2005

Alois P. Heinz, Table of n, a(n) for n = 0..17

Cf. A000045, A000073, A001592, A007570.

a:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^((<<0|1|0|0|0|0>, <0|0|1|0|0|0>,
    <0|0|0|1|0|0>, <0|0|0|0|1|0>, <0|0|0|0|0|1>, <1|1|1|1|1|1>>^n)[1, 6]))[1, 3]:
seq(a(n), n=0..14);  # Alois P. Heinz, Nov 07 2018

a(n) = A000073(A001592(n)).

A143662 a(n) is the number of n-tosses having a run of 6 or more heads for a fair coin (i.e., probability is a(n)/2^n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 8, 20, 48, 112, 256, 575, 1275, 2798, 6088, 13152, 28240, 60320, 128257, 271623, 573216, 1205880, 2529680, 5293264, 11050496, 23021311, 47868151, 99357390, 205897508, 426042552, 880346272, 1816750912, 3744698241, 7709963787, 15857441096, 32582726508, 66886894112

Offset: 0

DoZerg (daidodo(AT)gmail.com), Aug 28 2008

nonn

a(n-1) is the number of compositions of n with at least one part >=7. - Joerg Arndt, Aug 06 2012

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1,-1,-1,-1,-2).

LinearRecurrence[{3,-1,-1,-1,-1,-1,-2},{0,0,0,0,0,0,1},40] (* Harvey P. Dale, Dec 06 2018 *)

N=66;  x='x+O('x^N);
gf = (1-x)/(1-2*x); /* A011782(n): compositions of n */
gf -= 1/(1 - (x+x^2+x^3+x^4+x^5+x^6)); /* A001592(n+5): compositions of n into parts <=6 */
v143662=Vec(gf + 'a0);  v143662[1]=0; /* kludge to get all terms */
v143662 /* show terms */
/* Joerg Arndt, Aug 06 2012 */

G.f.: x^6/ ( (2*x-1)*(x^6+x^5+x^4+x^3+x^2+x-1) ). - Joerg Arndt, Aug 06 2012

a(n) = 2^n - A001592(n+6). - R. J. Mathar, Aug 06 2012

A181190 Maximal length of chain-addition sequence mod 10 with window of size n.

Original entry on oeis.org

1, 60, 124, 1560, 4686, 1456, 18744, 585936, 4882810, 212784

Offset: 1

Alexander Dashevsky (atanvarnoalda(AT)gmail.com), Oct 10 2010

Chain addition mod 10 with window n: take an n-digit 'seed'. Take the sum of its digits mod 10 and append to the seed. Repeat with the last n digits of the string, until the seed appears again.
This sequence shows the lengths of the longest sequences for different window sizes.
a(1)-a(10) all occur for seed 1 (among others). If this is always true, the sequence continues: 406224, 12695306, 4272460934, 380859180, 122070312496, 518798826, 3433227539058. - Lars Blomberg, Feb 12 2013
Comment from Michel Lagneau, Jan 20 2017, edited by N. J. A. Sloane, Jan 24 2017: (Start)
If seed 1 is always as good as or better than any other, as seems likely, then this sequence has the following alternative description.
Consider the n initial terms of an infinite sequence S(k, n) of decimal digits given by 0, 0,..., 0, 1. The succeeding terms are given by the final digits in the sum of the n immediately preceding terms. The sequence lists the period of each sequence corresponding to n = 2, 3, ...
a(2) = period of A000045 mod 10 (Fibonacci numbers mod 10) = A001175(10).
a(3) = period of A000073 mod 10 (tribonacci numbers mod 10) = A046738(10).
a(4) = period of A000078 mod 10 (tetranacci numbers mod 10) = A106295(10).
a(5) = period of A001591 mod 10 (pentanacci numbers mod 10) = A106303(10).
a(6) = period of A001592 mod 10 (hexanacci numbers mod 10).
a(7) = period of A122189 mod 10 (heptanacci numbers mod 10).
a(8) = period of A079262 mod 10 (octanacci numbers mod 10).
a(4) = 1560 because the four initial terms 0, 0, 0, 1 => S(k, 4) = 0, 0, 0, 1, 1, 2, 4, 8, 5, 9, 6, 8, 8, 1, 3, 0, 2, 6, 1, 9, 8, ... (tetranacci numbers mod 10). This sequence is periodic with period 1560:
S(1560 + 1, 4) = S(1, 4) = 0,
S(1560 + 2, 4) = S(2, 4) = 0,
S(1560 + 3, 4) = S(3, 4) = 0,
S(1560 + 4, 4) = S(4, 4) = 1.
(End)

			For n=2, the longest sequence begins with '01' (among others):
01123583145943707741561785381909987527965167303369549325729101.
It is 60 digits long (not counting the second '01' at the end).
For n=3, one of the longest sequences begins again with '001':
00112473441944756893025746770415061742394699425184352079627546556679289964992013
48570291225960516297849144970639807524172091001 (124 digits long without the second '001').

Cf. A000045, A000073, A000078, A001591, A001592, A003893, A079262, A122189

a(8)-a(10) from Lars Blomberg, Feb 12 2013

A349903 Array read by ascending antidiagonals. Inverse Euler transform of the right-shifted k-bonacci numbers.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, -1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, -1, 0, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 0, 1, 3, 4, 0, 0, 0, 0, 0, 0, 1, 2, 6, 5, 0, 0, 0, 0, 0, 0, 0, 1, 4, 10, 8, 0, 0, 0, 0, 0, 0, 0, 1, 2, 7, 18, 11, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 14, 31, 18, 0, 0

Offset: 0

Peter Luschny, Dec 05 2021

			Array starts:
[0] 0, 1, 0, -1, 0,  0, 0,  0,  0,  0,  0,  0,   0, ...
[1] 0, 1, 1,  0, 0, -1, 0,  0,  0,  0,  0,  0,   0, ...
[2] 0, 1, 1,  1, 2,  2, 4,  5,  8, 11, 18, 25,  40, ...
[3] 0, 0, 1,  1, 2,  3, 6, 10, 18, 31, 56, 96, 172, ...
[4] 0, 0, 0,  1, 1,  2, 4,  7, 14, 26, 50, 93, 178, ...
[5] 0, 0, 0,  0, 1,  1, 2,  4,  8, 15, 30, 58, 114, ...
[6] 0, 0, 0,  0, 0,  1, 1,  2,  4,  8, 16, 31,  62, ...
[7] 0, 0, 0,  0, 0,  0, 1,  1,  2,  4,  8, 16,  32, ...
[8] 0, 0, 0,  0, 0,  0, 0,  1,  1,  2,  4,  8,  16, ...
[9] 0, 0, 0,  0, 0,  0, 0,  0,  1,  1,  2,  4,   8, ...
.
Compare the rows with the columns of A349802.

Rows are the inverse Euler transforms of A063524, A057427, A000045, A000073, A000078, A001591, A001592.

Cf. A092921, A349802

read transforms;
F := proc(n, k) option remember;
     ifelse(k < 2, k, add(F(n, k-j), j = 1..min(n, k))) end:
Frow := (n, len) -> [seq(0, j = 0..n-3), seq(F(n, k), k = 0..len)]:
Arow := (n, len) -> EULERi(Frow(n, len)):
for n from 0 to 9 do Arow(n, 14 - n) od;

A111435 a(n) = Fibonacci(hexanacci(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 3, 21, 987, 2178309, 6557470319842, 59425114757512643212875125, 3016128079338728432528443992613633888712980904400501

Offset: 0

Jonathan Vos Post, Nov 13 2005

			a(0) = Fibonacci(hexanacci(0)) = A000045(A001592(0)) = A000045(0) = 0.
a(1) = Fibonacci(hexanacci(1)) = A000045(A001592(1)) = A000045(0) = 0.
a(2) = Fibonacci(hexanacci(2)) = A000045(A001592(2)) = A000045(0) = 0.
a(3) = Fibonacci(hexanacci(3)) = A000045(A001592(3)) = A000045(0) = 0.
a(4) = Fibonacci(hexanacci(4)) = A000045(A001592(4)) = A000045(0) = 0.
a(5) = Fibonacci(hexanacci(5)) = A000045(A001592(5)) = A000045(1) = 1.
a(6) = Fibonacci(hexanacci(6)) = A000045(A001592(6)) = A000045(1) = 1.
a(7) = Fibonacci(hexanacci(7)) = A000045(A001592(7)) = A000045(2) = 1.
a(8) = A000045(A001592(8)) = A000045(4) = 3.
a(9) = A000045(A001592(9)) = A000045(8) = 21.
a(10) = A000045(A001592(10)) = A000045(16) = 987.
a(11) = A000045(A001592(11)) = A000045(32) = 2178309.
a(12) = A000045(A001592(12)) = A000045(63) = 6557470319842.

Cf. A000045, A001592, A066178, A007570.

b:= proc(n) option remember; `if`(n<5, 0,
      `if`(n=5, 1, add(b(n-j), j=1..6)))
    end:
a:= n-> (<<0|1>, <1|1>>^b(n))[1,2]:
seq(a(n), n=0..14);  # Alois P. Heinz, Aug 09 2018

a(n) = A000045(A001592(n)).

A111438 Fibonacci(heptanacci(n)), restricted to nonzero heptanacci numbers.

Original entry on oeis.org

1, 1, 1, 3, 21, 987, 2178309, 10610209857723, 155576970220531065681649693, 33449372971981195681356806732944396691005381570580873

Offset: 0

Jonathan Vos Post, Nov 14 2005

			a(0) = Fibonacci(heptanacci(0)) = A000045(A066178(0)) = A000045(1) = 1.
a(1) = Fibonacci(heptanacci(1)) = A000045(A066178(1)) = A000045(1) = 1.
a(2) = Fibonacci(heptanacci(2)) = A000045(A066178(2)) = A000045(2) = 1.
a(3) = Fibonacci(heptanacci(3)) = A000045(A066178(3)) = A000045(4) = 3.
a(4) = Fibonacci(heptanacci(4)) = A000045(A066178(4)) = A000045(8) = 21.
a(5) = Fibonacci(heptanacci(5)) = A000045(A066178(5)) = A000045(16) = 987.
a(6) = A000045(A066178(6)) = A000045(32) = 2178309.
a(7) = A000045(A066178(7)) = A000045(64) = 10610209857723.
a(8) = A000045(A066178(8)) = A000045(127) = 155576970220531065681649693.

Cf. A000045, A001592, A066178, A007570.

a(n) = A000045(A066178(n)).

Corrected by T. D. Noe, Nov 16 2006

A158921 Positions of hexanacci numbers in the EKG sequence.

Original entry on oeis.org

1, 2, 3, 8, 17, 31, 51, 112, 234, 464, 931, 1855, 3687, 4877

Offset: 1

Parthasarathy Nambi, Mar 30 2009

nonn

			The hexanacci number 7617 is located at position 4877 in the EKG sequence.

A001592, A064413, A140897, A144080

cp's OEIS Frontend

A111429 Tribonacci(hexanacci(n)).

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A143662 a(n) is the number of n-tosses having a run of 6 or more heads for a fair coin (i.e., probability is a(n)/2^n).

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A181190 Maximal length of chain-addition sequence mod 10 with window of size n.

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A349903 Array read by ascending antidiagonals. Inverse Euler transform of the right-shifted k-bonacci numbers.

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Maple

A111435 a(n) = Fibonacci(hexanacci(n)).

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A111438 Fibonacci(heptanacci(n)), restricted to nonzero heptanacci numbers.

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A158921 Positions of hexanacci numbers in the EKG sequence.

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