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.

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A350990 Triangular numbers that are palindromes in base 3.

Original entry on oeis.org

0, 1, 10, 28, 91, 820, 7381, 65341, 66430, 597871, 1633528, 5380840, 48339028, 48427561, 139386556, 435848050, 1178284240, 3529890253, 3922632451, 32614707700, 35296517971, 35303692060, 101891588176, 292358957446, 295883935480, 317733228541, 859413596320, 2649105942220
Offset: 1

Views

Author

Amiram Eldar, Jan 28 2022

Keywords

Comments

This sequence is infinite since A000217((3^k-1)/2) is a term for all k >= 0 (Trigg, 1971).

Examples

			10 is a term since 10 = A000217(4) is a triangular number and also a palindromic number in base 3: 10 = 101_3.
28 is a term since 28 = A000217(7) is a triangular number and also a palindromic number in base 3: 36 = 1001_3.

Links

Crossrefs

Intersection of A000217 and A014190.
The ternary version of A003098.
Cf. A003462, A350987, A350991, A350992, A350993.

Programs

  • Mathematica
    t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 3]] &]

A350991 Triangular numbers that are palindromes in base 5.

Original entry on oeis.org

0, 1, 3, 6, 36, 78, 378, 1953, 20706, 23436, 48828, 147696, 239778, 426426, 449826, 1220703, 2155926, 6011778, 14625936, 30517578, 74218836, 74316336, 149083278, 314290056, 351562386, 762939453, 7897542681, 9141750936, 10201418541, 19073486328, 35952613476, 38218245156
Offset: 1

Views

Author

Amiram Eldar, Jan 28 2022

Keywords

Comments

This sequence is infinite since A000217((5^k-1)/2) is a term for all k >= 0 (Trigg, 1972).

Examples

			6 is a term since 6 = A000217(3) is a triangular number and also a palindromic number in base 5: 6 = 11_5.
36 is a term since 36 = A000217(8) is a triangular number and also a palindromic number in base 5: 36 = 121_5.

Links

Crossrefs

Intersection of A000217 and A029952.
The quinary version of A003098.
Cf. A003098, A125831, A350987, A350990, A350992, A350993.

Programs

  • Mathematica
    t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 5]] &]

A350992 Triangular numbers that are palindromes in base 7.

Original entry on oeis.org

0, 1, 3, 6, 78, 171, 300, 2850, 8256, 9453, 14706, 120786, 208335, 399171, 405450, 416328, 448878, 720600, 5877306, 6046503, 6835753, 9350650, 10122750, 18431556, 19130205, 22596003, 35309406, 499169406, 934394835, 969430528, 999335571, 1059265378, 1730160900
Offset: 1

Views

Author

Amiram Eldar, Jan 28 2022

Keywords

Comments

This sequence is infinite since A000217((7^k-1)/2) is a term for all k >= 0 (Trigg, 1974).

Examples

			78 is a term since 78 = A000217(12) is a triangular number and also a palindromic number in base 7: 78 = 141_7.
171 is a term since 171 = A000217(18) is a triangular number and also a palindromic number in base 7: 171 = 333_7.

Links

Crossrefs

Intersection of A000217 and A029954.
The septenary version of A003098.
Cf. A120741, A350987, A350990, A350991, A350993.

Programs

  • Mathematica
    t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 7]] &]

A350993 Triangular numbers that are palindromes in base 9.

Original entry on oeis.org

0, 1, 3, 6, 10, 91, 136, 300, 528, 820, 4560, 7381, 11476, 20910, 42486, 66430, 552826, 581581, 597871, 1664400, 2001000, 3420420, 3444000, 5070520, 5380840, 48427561, 75995956, 132494781, 134553810, 137158203, 159213090, 290585778, 434520460, 435848050, 669615310
Offset: 1

Views

Author

Amiram Eldar, Jan 28 2022

Keywords

Comments

This sequence is infinite since A000217((9^k-1)/2) is a term for all k >= 0 (Wishard, 1931).
Also, A000217((3 + 5*9^k)/2) is a term for all k>=0 (Trigg, 1984).

Examples

			10 is a term since 10 = A000217(4) is a triangular number and also a palindromic number in base 9: 10 = 11_9.
91 is a term since 91 = A000217(13) is a triangular number and also a palindromic number in base 9: 91 = 111_9.

References

  • Charles W. Trigg, Mathematical Quickies, McGraw Hill Book Co., 1967, Q112, p. 127.

Links

Crossrefs

Intersection of A000217 and A029955.
The nonary version of A003098.
Cf. A191681, A350987, A350990, A350991, A350992.

Programs

  • Mathematica
    t[n_] := n*(n + 1)/2; Select[t /@ Range[0, 3*10^5], PalindromeQ[IntegerDigits[#, 9]] &]

A068641 Smallest n-digit palindromic triangular number, or 0 if no such number exists.

Original entry on oeis.org

1, 55, 171, 3003, 15051, 617716, 1269621, 35133153, 178727871, 1264114621, 13953435931, 0, 1313207023131, 19895044059891, 114401848104411, 1250444114440521, 11121736463712111, 357961407704169753
Offset: 1

Views

Author

Amarnath Murthy, Feb 28 2002

Keywords

Examples

			a(4) = 3003 = 77*78/2 is the smallest 4-digit palindromic triangular number.
a(12) = 0 as no 12-digit palindromic triangular number exists.

Links

Crossrefs

Cf. A068642, A068643, A068644, A000217, A003098, A008509, A008510, A054263.

Extensions

Edited by Patrick De Geest, Mar 24 2002

A068642 Index of the smallest n-digit palindromic triangular number, or 0 if no such number exists.

Original entry on oeis.org

1, 10, 18, 77, 173, 1111, 1593, 8382, 18906, 50281, 167053, 0, 1620621, 6307938, 15126258, 50008881, 149142458, 846122222, 2480116437, 5513600773, 14667896198, 49786655918, 246644446642, 529670494286, 2466444446642
Offset: 1

Views

Author

Amarnath Murthy, Feb 28 2002

Keywords

Examples

			a(4) = 77 as 77*78/2 = 3003 is the smallest 4-digit palindromic triangular number.
a(12) = 0 as no 12-digit palindromic triangular number exists.

Links

Crossrefs

Cf. A068641, A068643, A068644, A000217, A003098, A008509, A008510, A054263.

Extensions

Edited by Patrick De Geest, Mar 24 2002

A068643 Largest n-digit palindromic triangular number, or 0 if no such number exists.

Original entry on oeis.org

6, 66, 666, 8778, 66066, 828828, 6295926, 61477416, 178727871, 6172882716, 87350505378, 0, 68742000024786, 82078700787028, 684866959668486, 8208268228628028, 67898244444289876, 514816979979618415
Offset: 1

Views

Author

Amarnath Murthy, Feb 28 2002

Keywords

Examples

			a(4) = 8778 = 132*133/2 is the largest 4-digit palindromic triangular number.
a(12) = 0 as no 12-digit palindromic triangular number exists.

Links

Crossrefs

Cf. A068641, A068642, A068644, A000217, A003098, A008509, A008510, A054263.

Extensions

Edited by Patrick De Geest, Mar 24 2002

A068644 Index of the largest n-digit palindromic triangular number, or 0 if no such number exists.

Original entry on oeis.org

3, 11, 36, 132, 363, 1287, 3548, 11088, 18906, 111111, 417972, 0, 3707883, 12812392, 37009916, 128127032, 368505751, 1014708805, 3567632391, 11151642876, 36657342048, 104561417190, 417898160427, 1325269593372
Offset: 1

Views

Author

Amarnath Murthy, Feb 28 2002

Keywords

Examples

			a(4) = 132 as 132*133/2 = 8778 is the largest 4-digit palindromic triangular number.
a(12) = 0 as no 12-digit palindromic triangular number exists.

Links

Crossrefs

Cf. A068641, A068642, A068643, A000217, A003098, A008509, A008510, A054263.

Extensions

Edited by Patrick De Geest, Mar 24 2002

A226788 Triangular numbers obtained as the concatenation of n and n+1.

Original entry on oeis.org

45, 78, 4950, 5253, 295296, 369370, 415416, 499500, 502503, 594595, 652653, 760761, 22542255, 49995000, 50025003, 88278828, 1033010331, 1487714878, 4999950000, 5000250003, 490150490151, 499999500000, 500002500003, 509949509950, 33471093347110, 49999995000000, 50000025000003
Offset: 1

Views

Author

Antonio Roldán, Jun 18 2013

Keywords

Examples

			If n=295, n//n+1 = 295296 = 768*769/2, a triangular number.

Crossrefs

Cf. A003098, A068899, A226742, A226772, A226789.

Programs

  • Mathematica
    TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; t = {}; Do[s = FromDigits[Join[IntegerDigits[n], IntegerDigits[n+1]]]; If[TriangularQ[s], AppendTo[t, s]], {n, 100000}]; t (* T. D. Noe, Jun 18 2013 *)
Select[FromDigits[Join[Flatten[IntegerDigits[#]]]]&/@Partition[ Range[ 5000010],2,1], OddQ[Sqrt[8#+1]]&] (* Harvey P. Dale, Jun 11 2015 *)
  • PARI
    concatint(a,b)=eval(concat(Str(a),Str(b)))
istriang(x)=issquare(8*x+1)
{for(n=1,10^7,a=concatint(n,n+1);if(istriang(a),print(a)))}

A226789 Triangular numbers obtained as the concatenation of n+1 and n.

Original entry on oeis.org

10, 21, 26519722651971, 33388573338856, 69954026995401, 80863378086336
Offset: 1

Views

Author

Antonio Roldán, Jun 18 2013

Keywords

Comments

There are only six terms less than 10^20.

Examples

			26519722651971 is the concatenation of 2651972 and 2651971 and a triangular number, because 26519722651971 = 7282818*7282819/2.

Crossrefs

Cf. A003098, A068899, A226742, A226772, A226788.

Programs

  • Mathematica
    TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; t = {}; Do[s = FromDigits[Join[IntegerDigits[n+1], IntegerDigits[n]]]; If[TriangularQ[s], AppendTo[t, s]], {n, 100000}]; t (* T. D. Noe, Jun 18 2013 *)
  • PARI
    concatint(a,b)=eval(concat(Str(a),Str(b)))
istriang(x)=issquare(8*x+1)
{for(n=1,10^7,a=concatint(n+1,n);if(istriang(a),print(a)))}
  • Python
    from math import isqrt
def istri(n): t = 8*n+1; return isqrt(t)**2 == t
def afind(klimit, kstart=0):
    strk = "0"
    for k in range(kstart, klimit+1):
        strkp1 = str(k+1)
        t = int(strkp1 + strk)
        if istri(t):
            print(t, end=", ")
        strk = strkp1
afind(81*10**5) # Michael S. Branicky, Oct 21 2021
  • Python
    # alternate version
def isconcat(n):
    if n < 10: return False
    s = str(n)
    mid = (len(s)+1)//2
    lft, rgt = int(s[:mid]), int(s[mid:])
    return lft - 1 == rgt
def afind(tlimit, tstart=0):
    for t in range(tstart, tlimit+1):
        trit = t*(t+1)//2
        if isconcat(trit):
            print(trit, end=", ")
afind(13*10**6) # Michael S. Branicky, Oct 21 2021
Previous Showing 11-20 of 27 results. Next

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