cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A059868 There exist no palindromic pentagonal numbers of length a(n).

Original entry on oeis.org

3, 9, 11, 12, 24, 30, 32, 33
Offset: 1

Views

Author

Patrick De Geest, Feb 15 2000

Keywords

Links

Crossrefs

Cf. A034822, A059869, A059870, A034307, A000326, A028386, A002069.

Programs

  • Mathematica
    A002069 = {0, 1, 5, 22, 1001, 2882, 15251, 720027, 7081807, 7451547, 26811862, 54177145, 1050660501, 1085885801, 1528888251, 2911771192, 2376574756732, 5792526252975, 5875432345785, 10810300301801, 264571020175462, 5292834004382925, 10808388588380801, 15017579397571051, 76318361016381367, 150621384483126051, 735960334433069537, 1003806742476083001, 1087959810189597801, 2716280733370826172};
A059868[n_] := Length[Select[A002069, IntegerLength[#] == n  || (n == 1 && # == 0) &]];
Select[Range[18], A059868[#] == 0 &] (* Robert Price, Apr 26 2019 *)
  • Python
    def ispal(n): s = str(n); return s == s[::-1]
def penpals(limit):
  for k in range(limit+1):
    if ispal(k*(3*k-1)//2): yield k*(3*k-1)//2
def aupto(limit):
  lengths = set(range(1, limit+1))
  for p in penpals(10**limit):
    lp, minlen = len(str(p)), min(lengths)
    for li in list(lengths):
      if li < lp: print(li, "in A059868"); lengths.discard(li)
    if lp in lengths: lengths.discard(lp); print("... discarding", lp)
    if len(lengths) == 0: return
aupto(15) # Michael S. Branicky, Mar 09 2021

Extensions

Name clarified by David A. Corneth, Apr 26 2019
a(6)-a(8) from Bert Dobbelaere, Apr 15 2025

A059870 Numbers n such that there exist no palindromic octagonal numbers of length n.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 15, 17, 18, 20, 22, 23, 26, 31, 34
Offset: 1

Views

Author

Patrick De Geest, Feb 15 2000

Keywords

Links

Crossrefs

Cf. A034822, A034307, A059868, A059869, A000567, A057106, A057107.

Extensions

a(13)-a(17) from World!Of Numbers link entered by Michel Marcus, Mar 04 2014

A082721 There exist no palindromic hexagonals of length n.

Original entry on oeis.org

3, 8, 9, 12, 22, 24, 27, 30, 36, 38, 40
Offset: 1

Views

Author

Patrick De Geest, Apr 13 2003

Keywords

Links

Crossrefs

Cf. A034822, A054969, A059868, A059869, A059870, A082722, A034307.

Programs

  • Mathematica
    A054969 = {0, 1, 6, 66, 3003, 5995, 15051, 66066, 617716, 828828, 1269621, 1680861, 5073705, 5676765, 1264114621, 5289009825, 6172882716, 13953435931, 1313207023131, 5250178710525, 6874200024786, 61728399382716, 602224464422206, 636188414881636, 1250444114440521, 16588189498188561, 58183932923938185, 66056806460865066, 67898244444289876, 514816979979618415, 3075488771778845703, 6364000440440004636, 15199896744769899151};
A082721[n_] := Length[Select[A054969, IntegerLength[#] == n || (n == 1 && # == 0) &]];
Select[Range[19], A082721[#] == 0 &] (* Robert Price, Apr 27 2019 *)
  • Python
    def ispal(n): s = str(n); return s == s[::-1]
def hexpals(limit):
  yield from (k*(2*k-1) for k in range(limit+1) if ispal(k*(2*k-1)))
def aupto(limit):
  lengths = set(range(1, limit+1))
  for h in hexpals(10**limit):
    if len(lengths) == 0: return
    lh, minlen = len(str(h)), min(lengths)
    if lh > minlen: print(minlen, "in A082721"); lengths.discard(minlen)
    if lh in lengths: lengths.discard(lh); print("... discarding", lh)
aupto(14) # Michael S. Branicky, Mar 08 2021

A082722 Numbers k for which there exist no palindromic 9-gonals (also known as nonagonals or enneagonals) of length k.

Original entry on oeis.org

2, 6, 13, 14, 15, 16, 20, 25, 27, 28, 29, 30, 31, 32
Offset: 1

Views

Author

Patrick De Geest, Apr 13 2003

Keywords

Comments

Previous name was: There exist no palindromic nonagonals (enneagonals) of length n.

Links

Crossrefs

Cf. A034822, A059868, A082721, A059869, A059870, A034307.

Programs

  • Mathematica
    A082723 = {0, 1, 9, 111, 474, 969, 6666, 18981, 67276, 4411144, 6964696, 15444451, 57966975, 448707844, 460595064, 579696975, 931929139, 994040499, 1227667221, 9698998969, 61556965516, 664248842466, 699030030996, 99451743334715499, 428987160061789824, 950178723327871059, 1757445628265447571, 4404972454542794044, 9433971680861793349, 499583536595635385994, 1637992008558002997361, 19874891310701319847891};
A082722[n_] := Length[Select[A082723, IntegerLength[#] == n || (n == 1 && # == 0) &]];
Select[Range[22], A082722[#] == 0 &] (* Robert Price, Apr 29 2019 *)

Extensions

Definition edited by Jon E. Schoenfield, Sep 15 2013

A307790 Number of palindromic heptagonal numbers with exactly n digits.

Original entry on oeis.org

3, 1, 1, 2, 1, 1, 3, 0, 0, 1, 4, 3, 2, 0, 1, 0, 1, 1, 2, 2, 2, 2
Offset: 1

Views

Author

Robert Price, Apr 28 2019

Keywords

Comments

Number of terms in A054910 with exactly n digits.

Examples

			There are only two 4-digit heptagonal numbers that are palindromic, 3553 and 4774. Thus, a(4)=2.

Links

  • G. J. Simmons, Palindromic powers, J. Rec. Math., 3 (No. 2, 1970), 93-98. [Annotated scanned copy] See page 95.

Crossrefs

Cf. A000566, A054910, A054971, A059869, A307765, A307766.

Programs

  • Mathematica
    A054910 = {0, 1, 7, 55, 616, 3553, 4774, 60606, 848848, 4615164, 5400045, 6050506, 7165445617, 62786368726, 65331413356, 73665056637, 91120102119, 345546645543, 365139931563, 947927729749, 3646334336463, 7111015101117, 17685292586717, 19480809790808491, 615857222222758516, 1465393008003935641, 8282802468642082828, 15599378333387399551, 20316023422432061302}; Table[Length[Select[A054910, IntegerLength[#] == n || (n == 1 && # == 0) &]], {n, 19}]
  • Python
    def afind(terms):
  m, n, c = 0, 1, 0
  while n <= terms:
    p = m*(5*m-3)//2
    s = str(p)
    if len(s) == n:
       if s == s[::-1]: c += 1
    else:
      print(c, end=", ")
      n, c = n+1, int(s == s[::-1])
    m += 1
afind(14) # Michael S. Branicky, Mar 01 2021

Extensions

a(20)-a(22) from Michael S. Branicky, Mar 01 2021

A307791 Number of palindromic heptagonal numbers of length n whose index is also palindromic.

Original entry on oeis.org

3, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Robert Price, Apr 28 2019

Keywords

Comments

Is there a nonzero term beyond a(4)?

Examples

			There is only one palindromic heptagonal number of length 4 whose index is also palindromic, 44->4774. Thus, a(4)=1.

Links

Crossrefs

Cf. A000566, A054910, A054971, A059869, A307765, A307766.

Programs

  • Mathematica
    A054910 = {0, 1, 7, 55, 616, 3553, 4774, 60606, 848848, 4615164, 5400045, 6050506, 7165445617, 62786368726, 65331413356, 73665056637, 91120102119, 345546645543, 365139931563, 947927729749, 3646334336463, 7111015101117, 717685292586717, 19480809790808491, 615857222222758516, 1465393008003935641, 8282802468642082828, 15599378333387399551, 20316023422432061302};
A054971 = {0, 1, 2, 5, 16, 38, 44, 156, 583, 1359, 1470, 1556, 53537, 158476, 161656, 171657, 190914, 371778, 382173, 615769, 1207698, 1686537, 16943262, 88274141, 496329416, 765609041, 1820198063, 2497949426, 2850685772}; Table[Length[Select[A054971[[Table[Select[Range[19], IntegerLength[A054910[[#]]] ==  n || (n == 1 && A054910[[#]] == 0) &], {n, 19}][[n]]]], PalindromeQ[#] &]], {n, 19}]
Showing 1-6 of 6 results.

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