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 1-5 of 5 results.

A103458 a(n) = 7^n + 1 - 0^n.

Original entry on oeis.org

1, 8, 50, 344, 2402, 16808, 117650, 823544, 5764802, 40353608, 282475250, 1977326744, 13841287202, 96889010408, 678223072850, 4747561509944, 33232930569602, 232630513987208, 1628413597910450, 11398895185373144
Offset: 0

Views

Author

Paul Barry, Feb 07 2005

Keywords

Comments

a(n)^3 is palindromic in base 7 (1_7, 1331_7, 1030301_7, 1003003001_7, ...).

Links

Crossrefs

Cf. A034491, A046233.

Programs

  • Magma
    [1] cat [7^n + 1: n in [1..30]]; // G. C. Greubel, Jun 22 2021
  • Mathematica
    Table[7^n +1 -Boole[n==0], {n,0,30}] (* G. C. Greubel, Jun 22 2021 *)
  • Sage
    [1]+[7^n + 1 for n in (1..30)] # G. C. Greubel, Jun 22 2021

Formula

G.f.: (1-7*x^2)/((1-x)*(1-7*x)).
a(n) = Sum_{k=0..n} binomial(n, k)*0^(k(n-k))*7^k.
a(n) = A034491(n), n > 0. - R. J. Mathar, Aug 28 2008
a(n) = 7*a(n-1) - 6, with a(1)=8. - Vincenzo Librandi, Dec 29 2010
E.g.f.: -1 + exp(x) + exp(7*x). - G. C. Greubel, Jun 22 2021

A103459 a(n) = 8^n + 1 - 0^n.

Original entry on oeis.org

1, 9, 65, 513, 4097, 32769, 262145, 2097153, 16777217, 134217729, 1073741825, 8589934593, 68719476737, 549755813889, 4398046511105, 35184372088833, 281474976710657, 2251799813685249, 18014398509481985, 144115188075855873
Offset: 0

Views

Author

Paul Barry, Feb 07 2005

Keywords

Comments

a(n)^3 is palindromic in base 8 (1_8, 1331_8, 1030301_8, 1003003001_8, ...).

Links

Crossrefs

Cf. A046233, A062395.

Programs

  • Magma
    [1] cat [8^n + 1: n in [1..30]]; // G. C. Greubel, Jun 23 2021
  • Mathematica
    Join[{1},8^Range[20]+1] (* or *) Join[{1},LinearRecurrence[{9,-8},{9,65},20]] (* Harvey P. Dale, Oct 21 2011 *)
  • Sage
    [1]+[8^n+1 for n in (1..30)] # G. C. Greubel, Jun 23 2021

Formula

G.f.: (1-8*x^2)/((1-x)*(1-8*x)).
a(n) = Sum_{k=0..n} binomial(n, k)*0^(k(n-k))*8^k.
a(n) = A062395(n), n > 0. - R. J. Mathar, Aug 28 2008
a(n) = 8*a(n-1) - 7, with a(1)=9. - Vincenzo Librandi, Dec 29 2010
a(n) = 9*a(n-1) - 8*a(n-2); a(0)=1, a(1)=9, a(2)=65. - Harvey P. Dale, Oct 21 2011
E.g.f.: -1 + exp(x) + exp(8*x). - G. C. Greubel, Jun 23 2021

A103462 A triangle with palindromic cubes, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 5, 4, 1, 1, 2, 9, 10, 5, 1, 1, 2, 17, 28, 17, 6, 1, 1, 2, 33, 82, 65, 26, 7, 1, 1, 2, 65, 244, 257, 126, 37, 8, 1, 1, 2, 129, 730, 1025, 626, 217, 50, 9, 1, 1, 2, 257, 2188, 4097, 3126, 1297, 344, 65, 10, 1, 1, 2, 513, 6562, 16385, 15626, 7777
Offset: 0

Views

Author

Paul Barry, Feb 07 2005

Keywords

Examples

			Rows start {1}, {1,1}, {1,2,1}, {1,2,3,1}, {1,2,5,4,1},..

Links

Crossrefs

Columns include A040000, A083318, A103457, A046231, A046233, A103458, A103459, A000533. Cubes of column k are palindromic to base k, k>3 (start with column 0). Row sums are A103480. Diagonal sums are A103481.

Formula

Number triangle T(n, k)=if(k<=n, k^(n-k)+1-0^(n-k), 0); Column k has g.f. x^k(1-kx^2)/((1-x)(1-kx)).

A046234 Cubes which are palindromes in base 5.

Original entry on oeis.org

0, 1, 216, 17576, 2000376, 245314376, 30546884376, 3815429734376, 476855468984376, 59605102540234376, 7450592041021484376, 931322860717802734376, 116415328979492333984376
Offset: 1

Views

Author

Patrick De Geest, May 15 1998

Keywords

Links

Crossrefs

Intersection of A029952 and A000578.
Cf. A046233.

Programs

  • Mathematica
    pb5Q[n_]:=Module[{idn5=IntegerDigits[n,5]},idn5==Reverse[idn5]]; Select[ Range[ 0,49*10^6]^3,pb5Q] (* Harvey P. Dale, Jan 21 2018 *)

Formula

a(n) = A046233(n)^3. - Andrew Howroyd, Aug 10 2024

Extensions

Offset corrected by Andrew Howroyd, Aug 10 2024

A103460 a(n) = 9^n + 1 - 0^n.

Original entry on oeis.org

1, 10, 82, 730, 6562, 59050, 531442, 4782970, 43046722, 387420490, 3486784402, 31381059610, 282429536482, 2541865828330, 22876792454962, 205891132094650, 1853020188851842, 16677181699666570, 150094635296999122
Offset: 0

Views

Author

Paul Barry, Feb 07 2005

Keywords

Comments

a(n)^3 is palindromic in base 9 (1_9, 1331_9, 1030301_9, 1003003001_9, ...).

Links

Crossrefs

Cf. A046233, A062396.

Programs

  • Magma
    [1] cat [9^n +1: n in [1..40]]; // G. C. Greubel, Jun 26 2021
  • Mathematica
    Table[9^n + 1 - Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 26 2021 *)
  • Sage
    [1]+[9^n +1 for n in (1..40)] # G. C. Greubel, Jun 26 2021

Formula

G.f.: (1-9*x^2)/((1-x)*(1-9*x)).
a(n) = Sum_{k=0..n} binomial(n, k)*0^(k*(n-k))*9^k.
a(n) = A062396(n), n > 0. - R. J. Mathar, Aug 28 2008
a(n) = 9*a(n-1) - 8, with a(1)=10. - Vincenzo Librandi, Dec 29 2010
E.g.f.: -1 + exp(x) + exp(9*x). - G. C. Greubel, Jun 26 2021
Showing 1-5 of 5 results.

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