A128209 -id:A128209 - cp's OEIS frontend

A094968 Indices of Fibonacci numbers in Stern's diatomic series A049456 regarded as a single linear sequence.

1, 4, 7, 14, 25, 48, 91, 178, 349, 692, 1375, 2742, 5473, 10936, 21859, 43706, 87397, 174780, 349543, 699070, 1398121, 2796224, 5592427, 11184834, 22369645, 44739268, 89478511, 178956998, 357913969, 715827912, 1431655795, 2863311562

Offset: 0

Paul Barry, May 26 2004

By definition, A049456(a(n))=Fib(n+2).

The rank of Fib(n+2) in row n of A049456 (regarded as an irregular triangle read by rows) is A128209(n) = A001045(n)+1. [Comment edited by N. J. A. Sloane, Nov 23 2016]

Colin Barker, Table of n, a(n) for n = 0..1000
D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A000045, A001045, A049456, A128209.

Vec((1 + x - 4*x^2) / ((1 - x)^2*(1 + x)*(1 - 2*x)) + O(x^30)) \\ Colin Barker, Sep 29 2017

G.f. : (1+x-4*x^2) / ((1-x)*(1-x^2)*(1-2*x)).
a(n) = 2^n + n + Jacobsthal(n).
a(n) = A006127(n) + A001045(n).
From Colin Barker, Sep 29 2017: (Start)
a(n) = ((-1)^(1+n) + 2^(2+n) + 3*n) / 3.
a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4) for n>3.
(End)

A231755 Primes of the form (2^n-1)/3 - n.

Original entry on oeis.org

331, 1398079, 89478457, 393530540239137101071, 1730765619511609209510165443073253, 8173309551284740577911184144801648979299941984979211421, 2142584059011987034055949456454883470029603991710390447068299

Offset: 1

K. D. Bajpai, Nov 13 2013

a(14) has 671 digits. a(15) has 2820 digits (not included in b-file).

Alternately, primes of the form Jacobsthal(n) - n, where Jacobsthal(n) is the n-th Jacobsthal number.

			a(2)= 1398079: n=22: ((2^n-(-1)^n)/3-n)= 1398079, which is prime.
a(4)= 393530540239137101071: n=70: ((2^n-(-1)^n)/3-n)= 393530540239137101071, which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..14

Cf. A001045 (Jacobsthal numbers).
Cf. A107036 (indices of prime Jacobsthal numbers).
Cf. A128209 (Jacobsthal numbers+1).

KD := proc() local a; a:= (2^n -(-1)^n)/3-n; if isprime(a)then RETURN (a); fi; end: seq(KD(),n=1..1000);

for(n=8,500,if(ispseudoprime(t=2^n\/3-n),print1(t", "))) \\ Charles R Greathouse IV, Nov 13 2013

Definition corrected by Charles R Greathouse IV, Nov 13 2013

A364378 Numbers whose representation in Jacobsthal greedy base (A265747) is palindromic.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 20, 22, 27, 36, 41, 44, 60, 68, 84, 86, 97, 112, 123, 132, 143, 158, 169, 172, 204, 220, 252, 260, 292, 308, 340, 342, 363, 396, 417, 432, 453, 486, 507, 516, 537, 570, 591, 606, 627, 660, 681, 684, 748, 780, 844, 860, 924, 956, 1020, 1028

Offset: 1

Amiram Eldar, Jul 21 2023

A128209(n) = A001045(n) + 1 is a term for n >= 3, since its representation is two 1's with n-3 0's between them.
A084639(n) is a term for n >= 1 since its representation is n 1's.
A014825(n) is a term for n >= 1 since its representation is n-1 0's interleaved with n 1's.

			The first 10 terms are:
   n  a(n)  A265747(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     2              2
   4     4             11
   5     6            101
   6     9            111
   7    12           1001
   8    20           1111
   9    22          10001
  10    27          10101

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001045, A014825, A084639, A128209, A265747.

Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341, A364214.

palJacobQ[n_] := PalindromeQ[A265747[n]]; Select[Range[0, 1000], palJacobQ] (* using A265747[n] *)

is(n) = {my(dig = digits(A265747(n))); dig == Vecrev(dig);} \\ using A265747(n)

A340849 a(n) = A001045(n) + A052928(n).

Original entry on oeis.org

0, 1, 3, 5, 9, 15, 27, 49, 93, 179, 351, 693, 1377, 2743, 5475, 10937, 21861, 43707, 87399, 174781, 349545, 699071, 1398123, 2796225, 5592429, 11184835, 22369647, 44739269, 89478513, 178956999, 357913971, 715827913

Offset: 0

Paul Curtz, Jan 24 2021

nonn

a(2*n) is divisible by 3.
a(3*n+2) is divisible by 3.
a(n) is the minimum number of moves to solve a Towers of Hanoi puzzle with 4 pegs and n disks where a disk cannot move away from the destination peg (or symmetrically, a disk cannot return to the initial peg). - Woosuk Kwak, Jan 25 2024

Woosuk Kwak, 1+3 Towers of Hanoi, Puzzling Stack Exchange.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A001045, A052928, A100314, A109613, A128209.

LinearRecurrence[{3, -1, -3, 2}, {0, 1, 3, 5}, 32] (* Robert P. P. McKone, Jan 28 2021 *)

a(n+1) - 2*a(n) = -A109613(n-2), for a(0)=0, a(1)=1. a(n) + a(n+1) = A100314(n).
a(n+1) - a(n) = A128209(n) for n >= 0.
a(n+2) = 1 + 2*A086445(n). - Hugo Pfoertner, Jan 24 2021
From Woosuk Kwak, Jan 25 2024: (Start)
a(n) = n + floor(2^n/3).
a(n) = n + A000975(n-1) for n >= 1. (End)

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A094968 Indices of Fibonacci numbers in Stern's diatomic series A049456 regarded as a single linear sequence.

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A231755 Primes of the form (2^n-1)/3 - n.

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A364378 Numbers whose representation in Jacobsthal greedy base (A265747) is palindromic.

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A340849 a(n) = A001045(n) + A052928(n).

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