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

A059869 Numbers k such that there exist no palindromic heptagonals of length k.

Original entry on oeis.org

8, 9, 14, 16, 32
Offset: 1

Views

Author

Patrick De Geest, Feb 15 2000

Keywords

Links

Crossrefs

Cf. A034822, A034307, A059868, A059870, A000566, A054971, A054910.

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};
A059869[n_] := Length[Select[A054910, IntegerLength[#] == n || (n == 1 && # == 0) &]];
Select[Range[19], A059869[#] == 0 &] (* Robert Price, Apr 28 2019 *)

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

A307801 Number of palindromic octagonal numbers with exactly n digits.

Original entry on oeis.org

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

Views

Author

Robert Price, Apr 29 2019

Keywords

Comments

Number of terms in A057107 with exactly n digits.

Examples

			There is only one 4 digit octagonal number that is palindromic, 8008.  Thus, a(4)=1.

Links

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

Crossrefs

Cf. A000567, A057107, A057106, A059870, A307790, A307791.

Programs

  • Mathematica
    A057107 = {0, 1, 8, 8008, 120232021, 124060421, 161656161, 185464581, 544721127445, 616947749616, 3333169613333, 3333802083333, 6506939396056, 12212500521221, 5466543663456645, 3310988011108890133, 520752145595541257025, 336753352502205253357633, 5882480463134313640842885, 102573006711888117600375201, 8025741496504444056941475208, 18651903272292929227230915681, 33582545421505050512454528533}; Table[Length[Select[A054910, IntegerLength[#] == n || (n == 1 && # == 0) &]], {n, 20}] (* Robert Price, Apr 29 2019 *)

A307802 Number of palindromic octagonal numbers of length n whose index is also palindromic.

Original entry on oeis.org

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

Views

Author

Robert Price, Apr 29 2019

Keywords

Comments

Is there a nonzero term beyond a(1)?

Examples

			There are only three palindromic octagonal numbers of length 1 whose index is also palindromic, 0->0, 1->1, and 2->8. Thus, a(1)=3.

Links

Crossrefs

Cf. A000567, A057107, A057106, A059870, A307790, A307791.

Programs

  • Mathematica
    A057107 = {0, 1, 8, 8008, 120232021, 124060421, 161656161, 185464581, 544721127445, 616947749616, 3333169613333, 3333802083333, 6506939396056, 12212500521221, 5466543663456645, 3310988011108890133, 520752145595541257025, 336753352502205253357633, 5882480463134313640842885, 102573006711888117600375201, 8025741496504444056941475208, 18651903272292929227230915681, 33582545421505050512454528533};
A057106 = {0, 1, 2, 52, 6331, 6431, 7341, 7863, 426115, 453486, 1054067, 1054167, 1472746, 2017631, 42687015, 1050553507, 13175129925, 335038979077, 1400295262095, 5847307263801, 51722791547842, 78849864240621, 105802560494387};
Table[Length[Select[A057106[[Table[Select[Range[20], IntegerLength[A057107[[#]]] ==  n || (n == 1 && A057107[[#]] == 0) &], {n, 20}][[n]]]], PalindromeQ[#] &]], {n, 20}]
Showing 1-6 of 6 results.

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