A002113 -id:A002113 - cp's OEIS frontend

A029804 Numbers that are palindromic in bases 8 and 10.

0, 1, 2, 3, 4, 5, 6, 7, 9, 121, 292, 333, 373, 414, 585, 3663, 8778, 13131, 13331, 26462, 26662, 30103, 30303, 207702, 628826, 660066, 1496941, 1935391, 1970791, 4198914, 55366355, 130535031, 532898235, 719848917, 799535997, 1820330281

Offset: 1

Patrick De Geest

Intersection of A002113 and A029803. - Michel Marcus, Nov 20 2014

Robert G. Wilson v, Table of n, a(n) for n = 1..75
Patrick De Geest, Palindromic numbers beyond base 10
Rick Regan, Code to generate this sequence

Cf. A007632, A007633, A029961, A029962, A029963, A029964, A029965, A029966, A029967, A029968, A029969, A029970, A029731, A097855, A099165.

[n: n in [0..10000000] | Intseq(n, 10) eq Reverse(Intseq(n, 10))and Intseq(n, 8) eq Reverse(Intseq(n, 8))]; // Vincenzo Librandi, Nov 23 2014

b1=8; b2=10; lst={}; Do[d1=IntegerDigits[n, b1];d2=IntegerDigits[n,b2]; If[d1==Reverse[d1]&&d2==Reverse[d2], AppendTo[lst, n]], {n, 0, 100000}]; lst (* Vincenzo Librandi, Nov 13 2014 *)
Select[Range[0,1820331000],PalindromeQ[#]&&IntegerDigits[#,8] == Reverse[ IntegerDigits[#,8]]&] (* Harvey P. Dale, Mar 18 2019 *)

isok(n) = (n==0) || ((d10=digits(n, 10)) && (d10==Vecrev(d10)) && (d8=digits(n, 8)) && (d8==Vecrev(d8))); \\ Michel Marcus, Nov 13 2014

ispal(n,r) = my(d=digits(n,r)); d==Vecrev(d);
for(n=0,10^7,if(ispal(n,10)&&ispal(n,8),print1(n,", "))); \\ Joerg Arndt, Nov 22 2014

def palQ8(n): # check if n is a palindrome in base 8
    s = oct(n)[2:]
    return s == s[::-1]
def palQgen10(l): # unordered generator of palindromes of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,10**l):
            s = str(x)
            yield int(s+s[-2::-1])
            yield int(s+s[::-1])
A029804_list = sorted([n for n in palQgen10(6) if palQ8(n)])
# Chai Wah Wu, Nov 25 2014

More terms from Robert G. Wilson v, Sep 30 2004
Incorrect Mathematica program deleted by N. J. A. Sloane, Sep 01 2009
Terms 33 through 36 corrected by Rick Regan (exploringbinary(AT)gmail.com), Sep 01 2009

A029742 Nonpalindromic numbers.

Original entry on oeis.org

10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 102, 103, 104, 105, 106, 107

Offset: 1

Patrick De Geest

Complement of A002113; A136522(a(n)) = 0.

A064834(a(n)) > 0. - Reinhard Zumkeller, Sep 18 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Patrick De Geest, World!Of Numbers
Index entries for sequences related to palindromes

Cf. A002113. Different from A031955.

a029742 n = a029742_list !! (n-1)
a029742_list = filter ((== 0) . a136522) [1..]
-- Reinhard Zumkeller, Oct 09 2011

[n: n in [0..150] | Intseq(n) ne Reverse(Intseq(n))]; // Bruno Berselli, Apr 01 2015

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]; DeleteCases[ Range[10,110],?palQ] (* _Harvey P. Dale, Jan 28 2012 *)
Table[If[PalindromeQ[n],Nothing,n],{n,120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 13 2019 *)

is(n)=my(d=digits(n)); d!=Vecrev(d) \\ Charles R Greathouse IV, Feb 06 2017

def ok(n): s = str(n); return s != s[::-1]
print(list(filter(ok, range(108)))) # Michael S. Branicky, Oct 12 2021

def A029742(n):
    def f(x): return n+x//10**((l:=len(s:=str(x)))-(k:=l+1>>1))-(int(s[k-1::-1])>x%10**k)+10**(k-1+(l&1^1))-1
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 24 2024

Offset corrected by Reinhard Zumkeller, Oct 09 2011

A029803 Numbers that are palindromic in base 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 27, 36, 45, 54, 63, 65, 73, 81, 89, 97, 105, 113, 121, 130, 138, 146, 154, 162, 170, 178, 186, 195, 203, 211, 219, 227, 235, 243, 251, 260, 268, 276, 284, 292, 300, 308, 316, 325, 333, 341, 349, 357, 365, 373, 381, 390, 398

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

T. D. Noe, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,8], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)

ispal(n,b=8)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

from itertools import chain, count, islice
def A029803_gen(): # generator of terms
    return chain((0,),chain.from_iterable(chain((int((s:=oct(d)[2:])+s[-2::-1],8) for d in range(8**l,8**(l+1))), (int((s:=oct(d)[2:])+s[::-1],8) for d in range(8**l,8**(l+1)))) for l in count(0)))
A029803_list = list(islice(A029803_gen(),20)) # Chai Wah Wu, Jun 23 2022

def A029803(n):
    if n == 1: return 0
    y = (x:=1<<(m:=n.bit_length()-2)-m%3)<<3
    return (c:=n-x)*x+int(oct(c)[-2:1:-1]or'0',8) if nChai Wah Wu, Jun 13 2024

Sum_{n>=2} 1/a(n) = 3.2188878... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A029955 Palindromic in base 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 20, 30, 40, 50, 60, 70, 80, 82, 91, 100, 109, 118, 127, 136, 145, 154, 164, 173, 182, 191, 200, 209, 218, 227, 236, 246, 255, 264, 273, 282, 291, 300, 309, 318, 328, 337, 346, 355, 364, 373, 382, 391, 400, 410, 419, 428, 437

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

T. D. Noe, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,9], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)

ispal(n,b=9)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

from gmpy2 import digits
def palQgen(l,b): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            for y in range(b**(x-1),b**x):
                s = digits(y,b)
                yield int(s+s[-2::-1],b)
            for y in range(b**(x-1),b**x):
                s = digits(y,b)
                yield int(s+s[::-1],b)
A029955_list = list(palQgen(4,9)) # Chai Wah Wu, Dec 01 2014

from gmpy2 import digits
def A029955(n):
    if n == 1: return 0
    y = 9*(x:=9**(len(digits(n>>1,9))-1))
    return int((c:=n-x)*x+int(digits(c,9)[-2::-1]or'0',9) if nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 3.29797695... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A033665 Number of 'Reverse and Add' steps needed to reach a palindrome starting at n, or -1 if n never reaches a palindrome.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 0, 1, 2, 2, 3, 1, 1, 1, 1, 2, 1, 0, 2, 3, 4, 1, 1, 1, 2, 1, 2, 2, 0, 4, 6, 1, 1, 2, 1, 2, 2, 3, 4, 0, 24, 1, 2, 1, 2, 2, 3, 4, 6, 24, 0, 1, 0, 1, 1

Offset: 0

N. J. A. Sloane

Palindromes themselves are not 'Reverse and Add!'ed, so they yield a zero!
Numbers n that may have a(n) = -1 (i.e., potential Lychrel numbers) appear in A023108. - Michael De Vlieger, Jan 11 2018
Record indices and values are given in A065198 and A065199. - M. F. Hasler, Feb 16 2020

			19 -> 19+91 = 110 -> 110+011 = 121 = palindrome, took 2 steps, so a(19)=2.
n = 89 needs 24 steps to end up with the palindrome 8813200023188. See A240510. - _Wolfdieter Lang_, Jan 12 2018

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers Penguin Books, 1987, pp. 142-143.

Kerry Mitchell, Table of n, a(n) for n = 0..195
P. De Geest, Some thematic websources
Jason Doucette, World Records
S. K. Eddins, The Palindromic Order Of A Number [archived page]
Kerry Mitchell, Table of n, a(n) for n = 0..10000 (The -1 entries are only conjectural)
Kerry Mitchell, Table of n, a(n) for n = 0..100000 (The -1 entries are only conjectural)
I. Peter, Search for the biggest numeric palindrome [archived page]
T. Trotter, Jr., Palindrome Power [archived page]
Eric Weisstein's World of Mathematics, 196-Algorithm.
Index entries for sequences related to Reverse and Add!

Cf. A002113, A023108, A023109, A033865, A006960, A016016, A063048, A240510.
Equals A030547(n) - 1.
Cf. A065198, A065199 (record indices & values).

rev[n_]:=FromDigits[Reverse[IntegerDigits[n]]];radd[n_]:=n+rev[n];
pal[n_]:=If[n==rev[n],True,False];
raddN[n_]:=Length[NestWhileList[radd[#]&,n,pal[#]==False&]]-1;
raddN/@Range[0,195] (* Ivan N. Ianakiev, Aug 31 2015 *)
With[{nn = 10^3}, Array[-1 + Length@ NestWhileList[# + IntegerReverse@ # &, #, !PalindromeQ@ # &, 1, nn] /. k_ /; k == nn -> -1 &, 200]] (* Michael De Vlieger, Jan 11 2018 *)

rev(n)={d=digits(n);p="";for(i=1,#d,p=concat(Str(d[i]),p));return(eval(p))}
a(n)=if(n==rev(n),return(0));for(k=1,10^3,i=n+rev(n);if(rev(i)==i,return(k));n=i)
n=0;while(n<100,print1(a(n),", ");n++) \\ Derek Orr, Jul 28 2014

A033665(n,LIM=333)={-!for(i=0,LIM,my(r=A004086(n)); n==r&&return(i); n+=r)} \\ with {A004086(n)=fromdigits(Vecrev(digits(n)))}. The second optional arg is a search limit that could be taken smaller up to very large n, e.g., 99 for n < 10^9, 200 for n < 10^14, 250 for n < 10^18: see A065199 for the records and A065198 for the n's. - M. F. Hasler, Apr 13 2019, edited Feb 16 2020

A033665 = lambda n, LIM=333: next((i for i in range(LIM) if is_A002113(n) or not(n := A004086(n)+n)), -1) # The second, optional argument is a search limit, see above. - M. F. Hasler, May 23 2024

More terms from Patrick De Geest, Jun 15 1998

I truncated the b-file at n=195, since the value of a(196) is not presently known (cf. A006960). The old b-files are now a-files. - N. J. A. Sloane, May 09 2015

A014192 Palindromes in base 4 (written in base 10).

Original entry on oeis.org

0, 1, 2, 3, 5, 10, 15, 17, 21, 25, 29, 34, 38, 42, 46, 51, 55, 59, 63, 65, 85, 105, 125, 130, 150, 170, 190, 195, 215, 235, 255, 257, 273, 289, 305, 325, 341, 357, 373, 393, 409, 425, 441, 461, 477, 493, 509, 514, 530, 546, 562, 582, 598, 614, 630, 650, 666

Offset: 1

N. J. A. Sloane

Rajasekaran, Shallit, & Smith prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

T. D. Noe, Table of n, a(n) for n = 1..10000
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Aayush Rajasekaran, Jeffrey Shallit, and Tim Smith, Sums of palindromes: an approach via automata, arXiv:1706.10206 [cs.FL], 2017.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

[n: n in [0..800] | Intseq(n, 4) eq Reverse(Intseq(n, 4))]; // Vincenzo Librandi, Sep 09 2015

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,4], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
pal4Q[n_]:=Module[{c=IntegerDigits[n,4]},c==Reverse[c]]; Select[Range[ 0,700],pal4Q] (* Harvey P. Dale, Jul 21 2020 *)

ispal(n,b=4)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

from gmpy2 import digits
def A014192(n):
    if n == 1: return 0
    y = (x:=1<<(n.bit_length()-2&-2))<<2
    return (c:=n-x)*x+int(digits(c,4)[-2::-1]or'0',4) if nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 2.7857715... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

More terms from Patrick De Geest

A057148 Palindromes only using 0 and 1 (i.e., base-2 palindromes).

Original entry on oeis.org

0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, 101101, 110011, 111111, 1000001, 1001001, 1010101, 1011101, 1100011, 1101011, 1110111, 1111111, 10000001, 10011001, 10100101, 10111101, 11000011, 11011011, 11100111, 11111111, 100000001

Offset: 1

Henry Bottomley, Aug 14 2000

For each term having fewer than 10 digits, the square will also be a palindrome. - Dmitry Kamenetsky, Oct 21 2008

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A006995 for sequence translated from binary to decimal. A016116 for number of terms of sequence with n+1 binary digits (0 taken to have no digits).

Cf. A002113, A118594, A118595, A118596, A118597, A118598, A118599, A118600.

(* get NextPalindrome from A029965 *)
Select[ NestList[ NextPalindrome, 0, 11110], Max(AT) IntegerDigits(AT)# < 2 &] (* Robert G. Wilson v *)
Select[FromDigits/@Tuples[{0,1},8],IntegerDigits[#]==Reverse[ IntegerDigits[ #]]&] (* Harvey P. Dale, Apr 20 2015 *)

from itertools import count, islice, product
def agen(): # generator of terms
    yield from [0, 1]
    for d in count(2):
        for rest in product("01", repeat=d//2-1):
            left = "1" + "".join(rest)
            for mid in [[""], ["0", "1"]][d%2]:
                yield int(left + mid + left[::-1])
print(list(islice(agen(), 32))) # Michael S. Branicky, Mar 29 2022

def A057148(n):
    if n == 1: return 0
    a = 1<Chai Wah Wu, Jun 10 2024

[int(n.binary()) for n in (0..220) if Word(n.digits(2)).is_palindrome()] # Peter Luschny, Sep 13 2018

A029952 Palindromic in base 5.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 12, 18, 24, 26, 31, 36, 41, 46, 52, 57, 62, 67, 72, 78, 83, 88, 93, 98, 104, 109, 114, 119, 124, 126, 156, 186, 216, 246, 252, 282, 312, 342, 372, 378, 408, 438, 468, 498, 504, 534, 564, 594, 624, 626, 651, 676, 701, 726, 756, 781, 806, 831

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

T. D. Noe, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

Cf. A261917, A261918.

[n: n in [0..900] | Intseq(n, 5) eq Reverse(Intseq(n, 5))]; // Vincenzo Librandi, Sep 09 2015

# test for palindrome in base b, from N. J. A. Sloane, Sep 13 2015
b:=5;
ispal := proc(n) global b; local t1,t2,i;
if n <= b-1 then return(1); fi;
t1:=convert(n,base,b); t2:=nops(t1);
for i from 1 to floor(t2/2) do
if t1[i] <> t1[t2+1-1] then return(-1); fi;
od: return(1); end;
lis:=[]; for n from 0 to 100 do if ispal(n) = 1 then lis:=[op(lis),n]; fi; od: lis;

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,5], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
Select[Range[0,1000],IntegerDigits[#,5]==Reverse[IntegerDigits[#,5]]&] (* Harvey P. Dale, Oct 24 2020 *)

ispal(n,b=5)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

from gmpy2 import digits
def A029952(n):
    if n == 1: return 0
    y = 5*(x:=5**(len(digits(n>>1,5))-1))
    return int((c:=n-x)*x+int(digits(c,5)[-2::-1]or'0',5) if nChai Wah Wu, Jun 13 2024

Sum_{n>=2} 1/a(n) = 2.9200482... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A029954 Palindromic in base 7.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 16, 24, 32, 40, 48, 50, 57, 64, 71, 78, 85, 92, 100, 107, 114, 121, 128, 135, 142, 150, 157, 164, 171, 178, 185, 192, 200, 207, 214, 221, 228, 235, 242, 250, 257, 264, 271, 278, 285, 292, 300, 307, 314, 321, 328, 335, 342, 344, 400, 456

Offset: 1

Patrick De Geest

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 03 2020

T. D. Noe, Table of n, a(n) for n = 1..10000
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
Patrick De Geest, Palindromic numbers beyond base 10.
Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
Index entries for sequences that are an additive basis, order 3.

Palindromes in bases 2 through 10: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113.

f[n_,b_] := Module[{i=IntegerDigits[n,b]}, i==Reverse[i]]; lst={}; Do[If[f[n,7], AppendTo[lst,n]], {n,1000}]; lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
pal7Q[n_]:=Module[{idn7=IntegerDigits[n,7]},idn7==Reverse[idn7]]; Select[ Range[0,500],pal7Q] (* Harvey P. Dale, Jul 30 2015 *)

ispal(n,b=7)=my(d=digits(n,b)); d==Vecrev(d) \\ Charles R Greathouse IV, May 03 2020

from gmpy2 import digits
def palQgen(l,b): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1,l+1):
            for y in range(b**(x-1),b**x):
                s = digits(y,b)
                yield int(s+s[-2::-1],b)
            for y in range(b**(x-1),b**x):
                s = digits(y,b)
                yield int(s+s[::-1],b)
A029954_list = list(palQgen(4,7)) # Chai Wah Wu, Dec 01 2014

from gmpy2 import digits
from sympy import integer_log
def A029954(n):
    if n == 1: return 0
    y = 7*(x:=7**integer_log(n>>1,7)[0])
    return int((c:=n-x)*x+int(digits(c,7)[-2::-1]or'0',7) if nChai Wah Wu, Jun 14 2024

Sum_{n>=2} 1/a(n) = 3.1313768... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020

A003098 Palindromic triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 55, 66, 171, 595, 666, 3003, 5995, 8778, 15051, 66066, 617716, 828828, 1269621, 1680861, 3544453, 5073705, 5676765, 6295926, 35133153, 61477416, 178727871, 1264114621, 1634004361, 5289009825, 6172882716, 13953435931

Offset: 1

N. J. A. Sloane

The only known terms with an even number 2*m of digits that are the concatenation of two palindromes with m digits are 55, 66 and 828828 (see David Wells entry 828828). - Bernard Schott, Apr 29 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Charles W. Trigg, Palindromic Triangular Numbers, J. Rec. Math., 6 (1973), 146-147.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, p. 73 and p. 178, entry 828828 (Rev. ed. 1997)

T. D. Noe, Table of n, a(n) for n = 1..148 (from Patrick De Geest)
Patrick De Geest, Palindromic Triangulars
Shyam Sunder Gupta, Triangular Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.

Cf. A008509.

Intersection of A000217 and A002113.

palQ[n_]:=Module[{idn=IntegerDigits[n]},idn==Reverse[idn]]; Select[ Accumulate[ Range[200000]],palQ]  (* Harvey P. Dale, Mar 23 2011 *)
Select[Accumulate[Range[0,170000]],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 15 2019 *)

list(lim)=my(v=List(),d); for(n=0,(sqrt(8*lim+1)-1)/2, d=digits(n*(n+1)/2); if(d==Vecrev(d), listput(v,n*(n+1)/2))); Vec(v) \\ Charles R Greathouse IV, Jun 23 2017

A003098_list = [m for m in (n*(n+1)//2 for n in range(10**5)) if str(m) == str(m)[::-1]] # Chai Wah Wu, Sep 03 2021

cp's OEIS Frontend

A029804 Numbers that are palindromic in bases 8 and 10.

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A029742 Nonpalindromic numbers.

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A029803 Numbers that are palindromic in base 8.

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A029955 Palindromic in base 9.

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A033665 Number of 'Reverse and Add' steps needed to reach a palindrome starting at n, or -1 if n never reaches a palindrome.

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A014192 Palindromes in base 4 (written in base 10).

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