A367360 -id:A367360 - cp's OEIS frontend

A368359 Comma transform of Catalan numbers.

11, 12, 25, 51, 44, 21, 24, 91, 4, 21, 65, 62, 27, 2, 9, 53, 1, 4, 1, 6, 2, 9, 3, 1, 44, 21, 26, 42, 1, 83, 41, 95, 82, 8, 43, 21, 24, 41, 6, 2, 1, 3, 1, 5, 2, 8, 3, 1, 5, 21, 67, 62, 21, 4, 41, 26, 22, 41, 4, 21, 66, 62, 29, 53, 1, 5, 2, 8, 3, 1, 5, 2, 7, 3, 1, 4, 1, 7, 2, 1, 4, 1, 6, 2, 1, 4, 1, 6, 2, 1, 3, 1, 6, 2, 9, 3, 1, 5, 2, 8

Offset: 0

N. J. A. Sloane, Jan 02 2024

See A367360 for further information.

Michael S. Branicky, Table of n, a(n) for n = 0..10000

Cf. A000108, A367360.

C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
a:= n-> parse(cat(""||(C(n))[-1], ""||(C(n+1))[1])):
seq(a(n), n=0..99);  # Alois P. Heinz, Jan 03 2024

from math import comb
def A368359(n): return 10*(comb(n<<1,n)//(n+1)%10)+int(str(comb(n+1<<1,n+1)//(n+2))[0]) # Chai Wah Wu, Jan 03 2024

from itertools import count, islice, pairwise
def S(): # generator of Catalan numbers as strings
    C = 1
    for n in count(0):
        yield str(C)
        C = C*(4*n+2)//(n+2)
def agen(): yield from (int(t[-1]+u[0]) for t, u in pairwise(S()))
print(list(islice(agen(), 100))) # Michael S. Branicky, Jan 04 2024

A368360 Comma transform of the even numbers.

Original entry on oeis.org

2, 24, 46, 68, 81, 1, 21, 41, 61, 82, 2, 22, 42, 62, 83, 3, 23, 43, 63, 84, 4, 24, 44, 64, 85, 5, 25, 45, 65, 86, 6, 26, 46, 66, 87, 7, 27, 47, 67, 88, 8, 28, 48, 68, 89, 9, 29, 49, 69, 81, 1, 21, 41, 61, 81, 1, 21, 41, 61, 81, 1, 21, 41, 61, 81, 1, 21, 41, 61, 81, 1, 21, 41, 61, 81, 1, 21, 41, 61, 81, 1, 21, 41, 61, 81, 1, 21, 41, 61

Offset: 0

N. J. A. Sloane, Jan 02 2024

See A367360 for further information.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A005843, A367360.

a:= n-> parse(cat(""||(2*n)[-1], ""||(2*n+2)[1])):
seq(a(n), n=0..88);  # Alois P. Heinz, Jan 03 2024

def A368360(n): return (0,20,40,60,80)[n%5]+int(str(n+1<<1)[0]) # Chai Wah Wu, Jan 03 2024

A368361 Comma transform of the odd numbers.

Original entry on oeis.org

13, 35, 57, 79, 91, 11, 31, 51, 71, 92, 12, 32, 52, 72, 93, 13, 33, 53, 73, 94, 14, 34, 54, 74, 95, 15, 35, 55, 75, 96, 16, 36, 56, 76, 97, 17, 37, 57, 77, 98, 18, 38, 58, 78, 99, 19, 39, 59, 79, 91, 11, 31, 51, 71, 91, 11, 31, 51, 71, 91, 11, 31, 51, 71, 91, 11, 31, 51, 71, 91, 11, 31, 51, 71, 91, 11, 31, 51, 71, 91

Offset: 0

N. J. A. Sloane, Jan 02 2024

See A367360 for further information.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A005408, A367360.

a:= n-> parse(cat(""||(2*n+1)[-1], ""||(2*n+3)[1])):
seq(a(n), n=0..79);  # Alois P. Heinz, Jan 03 2024

def A368361(n): return (10,30,50,70,90)[n%5]+int(str((n<<1)+3)[0]) # Chai Wah Wu, Jan 03 2024

A368782 Comma transform of A366487.

Original entry on oeis.org

12, 35, 94, 15, 16, 28, 31, 34, 37, 41, 45, 55, 55, 55, 55, 61, 67, 74, 71, 89, 98, 97, 18, 19, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 11, 12, 13, 14, 15, 16, 17, 18, 22, 22, 24, 26, 28, 22, 22, 24, 26, 28, 22, 22, 24, 26, 28, 22, 22, 24, 26, 28, 22, 22

Offset: 1

Michael S. Branicky, Jan 05 2024

See A367360 for further information.
Let the comma sequence A121805 be known as S or C0.
A366487, the first differences of A121805, is the same as the comma transform of A121805; call it C1.
This sequence is C2 = C(C(S)), the comma transform C iterated twice.
C4 = C2, C5 = C2, ... once the first term (and the last term if the sequence is finite) are removed from the lower iterates of C.
Theorem: C^{i+2}(S) = C^i(S) for i>=2 in general and for i>=0 when all terms of S have two digits and no least significant digit is zero. See link for proof.
Remark. The lexicographically earliest sequence S with C(S) = S is A010850, all 11's.
The sequence contains 2137451 terms, with a(2137451) = 96. The next term does not exist.

Michael S. Branicky, Table of n, a(n) for n = 1..20000
Michael S. Branicky, Comma Comma Proof

Cf. A010850, A121805, A367360, A366487.

from itertools import islice, pairwise
def S(): # generator of comma sequence
    an = 1
    while True:
        yield an
        an += 10*(an%10)
        children = [an+y for y in range(1, 10) if str(an+y)[0] == str(y)]
        if not children: break
        an = children[0]
def C(g): # generator of comma transform of sequence passed as a generator
    yield from (10*(t%10) + int(str(u)[0]) for t, u in pairwise(g))
def agen(): return C(C(S()))
print(list(islice(agen(), 70))) # Michael S. Branicky, Jan 05 2024

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A368359 Comma transform of Catalan numbers.

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A368360 Comma transform of the even numbers.

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A368361 Comma transform of the odd numbers.

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Maple

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A368782 Comma transform of A366487.

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Python