David F. Marrs - cp's OEIS frontend

A319744 Partial sums of bouncy numbers (A152054).

101, 203, 306, 410, 515, 621, 728, 836, 945, 1065, 1186, 1316, 1447, 1579, 1719, 1860, 2002, 2145, 2295, 2446, 2598, 2751, 2905, 3065, 3226, 3388, 3551, 3715, 3880, 4050, 4221, 4393, 4566, 4740, 4915, 5091, 5271, 5452, 5634, 5817, 6001, 6186, 6372, 6559, 6749, 6940, 7132, 7325, 7519

Offset: 1

David F. Marrs, Oct 21 2018

			a(1) = 101.
a(2) = 101 + 102 = 203.
a(10) = 101 + 102 + 103 + 104 + 105 + 106 + 107 + 108 + 109 + 120 = 1065.

David F. Marrs, Table of n, a(n) for n = 1..10000

Cf. A152054 (bouncy numbers).

for n in range(50):
  a = n
  b = 0
  c = 0
  while a:
    if ''.join(sorted(str(b))) != str(b) and ''.join(sorted(str(b)))[::-1] != str(b): c += b; a -= 1
    b += 1
  print(c)

from itertools import count, islice
def A319744_gen(): # generator of terms
    c = 0
    for n in count(101):
        l = len(s:=tuple(int(d) for d in str(n)))
        for i in range(1,l-1):
            if (s[i-1]-s[i])*(s[i]-s[i+1]) < 0:
                c += n
                yield c
                break
A319744_list = list(islice(A319744_gen(),30)) # Chai Wah Wu, Jul 28 2023

A303029 From a riddle, see Puzzling.SE link.

Original entry on oeis.org

3, 1, 4, 8, 8, 21, 21, 62, 128, 190, 430, 831, 1451, 3030, 6143, 12286, 24361, 48850, 85497, 134347, 268694, 583208, 1071746, 2192342, 3264088, 7514425, 14042601, 24821114, 46378140, 99867664, 171066918, 270934582, 634625444, 1272514976, 2449009584, 0, 2449009584

Offset: 0

David F. Marrs, Aug 16 2018

			a(0,1,2) = 3,1,4
To continue, we use the decimal expansion of Pi = 3.14159...:
a(3) = 3+1+4 (3-bonacci) = 8
a(4) = 8 (1-bonacci) = 8
a(5) = 1+4+8+8 (4-bonacci) = 21
a(6) = 21 (1-bonacci) = 21
a(7) = 21+21+8+8+4 (5-bonacci) = 62
...

Rémy Sigrist, Table of n, a(n) for n = 0..400
Puzzling.SE, What number comes next?

Cf. A000796.

a(n) = 0 for all n > 362. - Alois P. Heinz, Aug 18 2018
From Jianing Song, Dec 25 2022: (Start)
Let d_k = A000796(k+1) be the k-th digit of Pi, then a(n) = a(n-1) + a(n-2) + ... + a(n-d_{n-3}) for n >= 3.
If there exists consecutive 9 digits ...d_{k}d_{k+1}...d_{k+8}... of Pi such that d_{k+i} <= i for i = 0..8, then a(n) = 0 for all n >= k+3. The 360th to 368th digits of Pi are ...001133053..., so a(n) = 0 for all n >= 363. (End)

A305989 Numbers in binary reversed.

Original entry on oeis.org

0, 1, 1, 11, 1, 101, 11, 111, 1, 1001, 101, 1101, 11, 1011, 111, 1111, 1, 10001, 1001, 11001, 101, 10101, 1101, 11101, 11, 10011, 1011, 11011, 111, 10111, 1111, 11111, 1, 100001, 10001, 110001, 1001, 101001, 11001, 111001, 101, 100101, 10101, 110101, 1101, 101101, 11101, 111101, 11, 100011, 10011

Offset: 0

David F. Marrs, Jun 16 2018

			11 is 1011 in binary, and reversing it gives 1101 = a(11).

David F. Marrs, Table of n, a(n) for n = 0..10000

Cf. A305877, A004086, A007088, A030101.

a:= n-> parse(cat(convert(n, base, 2)[])):
seq(a(n), n=0..75);  # Alois P. Heinz, Jun 17 2018

Table[FromDigits@ Reverse@ IntegerDigits[n, 2], {n, 0, 50}]

a(n) = fromdigits(Vecrev(digits(n, 2)), 10);

print(0)
for i in range(1,50):
   print(format(i, 'b')[::-1].strip("0"))

a(n) = A004086(A007088(n)).

A305877 Numbers in base 3 reversed.

Original entry on oeis.org

0, 1, 2, 1, 11, 21, 2, 12, 22, 1, 101, 201, 11, 111, 211, 21, 121, 221, 2, 102, 202, 12, 112, 212, 22, 122, 222, 1, 1001, 2001, 101, 1101, 2101, 201, 1201, 2201, 11, 1011, 2011, 111, 1111, 2111, 211, 1211, 2211, 21, 1021, 2021, 121, 1121, 2121, 221, 1221, 2221, 2

Offset: 0

David F. Marrs, Jun 13 2018

			11 is 102 in base 3, and reversing it gives 201 = a(11).

David F. Marrs, Table of n, a(n) for n = 0..10000

Cf. A004086 (in base 10), A007089, A030102 (when converted in base 10).

a:= n-> parse(cat(convert(n, base, 3)[])):
seq(a(n), n=0..75);  # Alois P. Heinz, Jun 17 2018

Table[FromDigits@ Reverse@ IntegerDigits[n, 3], {n, 0, 54}] (* Giovanni Resta, Jun 13 2018 *)

a(n) = fromdigits(Vecrev(digits(n, 3)), 10); \\ Michel Marcus, Jun 13 2018

a(n) = A004086(A007089(n)). - Felix Fröhlich, Jun 14 2018

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User: David F. Marrs

A319744 Partial sums of bouncy numbers (A152054).

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A303029 From a riddle, see Puzzling.SE link.

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A305989 Numbers in binary reversed.

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A305877 Numbers in base 3 reversed.

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Programs

Maple

Mathematica

PARI

Formula