A007284 -id:A007284 - cp's OEIS frontend

A053701 Vertically symmetric numbers.

0, 1, 8, 11, 25, 52, 88, 101, 111, 181, 205, 215, 285, 502, 512, 582, 808, 818, 888, 1001, 1111, 1251, 1521, 1881, 2005, 2115, 2255, 2525, 2885, 5002, 5112, 5252, 5522, 5882, 8008, 8118, 8258, 8528, 8888, 10001, 10101, 10801, 11011, 11111, 11811

Offset: 1

Henry Bottomley, Feb 14 2000

Numbers that are symmetric about a vertical mirror.

2 and 5 are taken as mirror images (as on calculator displays).

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Nathaniel Johnston)

Cf. A000787, A007284, A018846 (strobogrammatic numbers).

compdig := proc(n) if(n=2)then return 5: elif(n=5)then return 2: elif(n=0 or n=1 or n=8)then return n: else return -1: fi: end: isA053701 := proc(n) local d,l,j: d:=convert(n,base,10): l:=nops(d): for j from 1 to ceil(l/2) do if(not d[j]=compdig(d[l-j+1]))then return false: fi: od: return true: end: for n from 0 to 10000 do if(isA053701(n))then printf("%d, ",n): fi: od: # Nathaniel Johnston, May 17 2011

from itertools import count, islice, product
def lr(s): return s[::-1].translate({ord('2'):ord('5'), ord('5'):ord('2')})
def A053701gen(): # generator of terms
    yield from [0, 1, 8]
    for d in count(2):
        for first in "1258":
            for rest in product("01258", repeat=d//2-1):
                left = first + "".join(rest)
                for mid in [[""], ["0", "1", "8"]][d%2]:
                    yield int(left + mid + lr(left))
print(list(islice(A053701gen(), 45))) # Michael S. Branicky, Jul 09 2022

More terms from Larry Reeves (larryr(AT)acm.org), Oct 01 2001

A242590 Numbers whose representation in Roman numerals is horizontally symmetrical.

Original entry on oeis.org

1, 2, 3, 9, 10, 11, 12, 13, 19, 20, 21, 22, 23, 29, 30, 31, 32, 33, 39, 90, 91, 92, 93, 99, 100, 101, 102, 103, 109, 110, 111, 112, 113, 119, 120, 121, 122, 123, 129, 130, 131, 132, 133, 139, 190, 191, 192, 193, 199, 200, 201, 202, 203, 209, 210, 211, 212, 213, 219, 220, 221, 222, 223, 229, 230, 231, 232, 233, 239, 290

Offset: 1

Philip Mizzi, May 24 2014

The sequence contains only 224 terms and ends with 899.

Roman numerals use the letters I, V, X, L, C, D and M, combinations of which are used to represent numbers. Since the letters V, L and M are not horizontally symmetrical, numbers containing these letters are not part of the sequence. Importantly, Roman numerals for 900 and beyond will always contain the numeral M, so the sequence ends at 899.

			Define two functions:
R(n) converts the number n to Roman number notation.
H[x] takes the argument x and produces a truth value, determining if the argument is horizontally symmetrical.
Hence,
for n = 1, R(n) = I, H[R(n)] = TRUE, so n = 1 is a term,
for n = 5, R(n) = V, H[R(n)] = FALSE, so n = 5 is not a term;
for n = 11, R(n) = XI, H[R(n)] = TRUE, so n = 11 is a term;
for n = 50, R(n) = L, H[R(n)] = FALSE, so n = 50 is not a term;
for n = 100, R(n) = C, H[R(n)] = TRUE, so n = 100 is a term;
for n = 900, R(n) = CM, H[R(n)] = FALSE, so n = 900 is not a term;
for n = 1000, R(n) = M, H[R(n)] = FALSE, so n = 1000 is not a term.

Philip Mizzi, Table of n, a(n) for n = 1..224

Cf. A007284 (horizontally/Arabic), A166874 (vertically/Roman).

Name edited by Jon E. Schoenfield, Sep 12 2017

A321702 Numbers that are still valid after a horizontal reflection on a calculator display.

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 10, 11, 12, 13, 15, 18, 20, 21, 22, 23, 25, 28, 30, 31, 32, 33, 35, 38, 50, 51, 52, 53, 55, 58, 80, 81, 82, 83, 85, 88, 100, 101, 102, 103, 105, 108, 110, 111, 112, 113, 115, 118, 120, 121, 122, 123, 125, 128, 130, 131, 132, 133, 135, 138

Offset: 1

Kritsada Moomuang, Nov 17 2018

Note that these numbers may not be unchanged after a horizontal reflection.
2 and 5 are taken as mirror images (as on calculator displays).
A007284 is a subsequence.
Also, numbers whose all digits are Fibonacci numbers. - Amiram Eldar, Feb 15 2024

			The sequence begins:
0, 1, 2, 3, 5, 8, 10, 11, 12, 13, ...;
0, 1, 5, 3, 2, 8, 10, 11, 15, 13, ...;
23 has its reflection as 53 in a horizontal mirror.
182 has its reflection as 185 in a horizontal mirror.

Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.

Cf. A000787, A007284, A018846.

Select[Range[0, 140], Intersection[IntegerDigits[#], {4, 6, 7, 9}] == {} &] (* Amiram Eldar, Nov 17 2018 *)

a(n, d=[0, 1, 2, 3, 5, 8]) = fromdigits(apply(k -> d[1+k], digits(n-1, #d))) \\ Rémy Sigrist, Nov 17 2018

Sum_{n>=2} 1/a(n) = 4.887249145579262560308470922947674796541485176473171687107616547235128170930... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

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A053701 Vertically symmetric numbers.

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A242590 Numbers whose representation in Roman numerals is horizontally symmetrical.

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A321702 Numbers that are still valid after a horizontal reflection on a calculator display.

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