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 1-8 of 8 results.

A087218 Satisfies A(x) = 1 + x*A(x)*f(x)^2, where f(x) = Sum_{k>=0} x^((3^n-1)/2) and f(x)^2 = 2 - f(x^2) + 2*Sum_{n>0} x^A023745(n).

Original entry on oeis.org

1, 1, 3, 6, 13, 30, 66, 147, 327, 726, 1614, 3588, 7974, 17725, 39399, 87573, 194655, 432669, 961716, 2137659, 4751490, 10561392, 23475378, 52179987, 115983270, 257802273, 573031011, 1273706934, 2831137095, 6292921101, 13987615113
Offset: 0

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Author

Paul D. Hanna, Aug 26 2003

Keywords

Examples

			Given f(x) = 1 + x + x^4 + x^13 + x^40 + x^121 + ... so that f(x)^2 = 1 + 2x + x^2 + 2x^4 + 2x^5 + x^8 + 2*x^13 + ... then A(x) = 1 + x*A(x)*(1 + 2x + x^2 + 2x^4 + 2x^5 + x^8 + ...) = 1 + x + 3x^2 + 6x^3 + 13x^4 + 30x^5 + ...

Crossrefs

Cf. A078932, A023745, A087219.

Programs

  • PARI
    a(n)=local(A,m); if(n<1,1,m=1; A=1+O(x); while(m<=2*n,m*=3; A=1/(1/subst(A,x,x^3)-x)); polcoeff(A,2*n));

Formula

a(n) = A078932(2n). a(m) = 1 (mod 3) when m = (3^n - 1)/2, otherwise a(m) = 0 (mod 3).

A087219 Satisfies A(x) = f(x) + x*A(x)*f(x)^2, where f(x) = Sum_{k>=0} x^((3^n-1)/2) and f(x)^2 = 2 - f(x^2) + 2*Sum_{n>0} x^A023745(n). Also, A(x) = f(x)*B(x), where B(x) = Sum_{k>=0} A087218(k)*x^k.

Original entry on oeis.org

1, 2, 4, 9, 20, 44, 99, 219, 487, 1083, 2406, 5349, 11889, 26426, 58739, 130563, 290208, 645062, 1433814, 3187014, 7083951, 15745878, 34999212, 77794638, 172918335, 384354909, 854326387, 1898957331, 4220914872, 9382055124
Offset: 0

Views

Author

Paul D. Hanna, Aug 27 2003

Keywords

Examples

			Given f(x) = 1 + x + x^4 + x^13 + x^40 + x^121 + ... so that f(x)^2 = 1 + 2x + x^2 + 2x^4 + 2x^5 + x^8 + 2*x^13 + ... then A(x) = (1 + x + x^4 + ...) + x*A(x)*(1 + 2x + x^2 + 2x^4 + 2x^5 + ...) = 1 + 2x + 4x^2 + 9x^3 + 20x^4 + 44x^5 + ...

Crossrefs

Cf. A078932, A087218.

Programs

  • PARI
    a(n)=local(A,m); if(n<1,1,m=1; A=1+O(x); while(m<=2*n+1,m*=3; A=1/(1/subst(A,x,x^3)-x)); polcoeff(A,2*n+1));

Formula

a(n) = A078932(2n+1). a(m) = 1 (mod 3) when m = (3^n-1)/2 (mod 3), else a(m) = 2 (mod 3) when m = A023745(n), otherwise a(m) = 0 (mod 3).

A031988 Duplicate of A023745.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 13, 14, 17, 26, 40, 41, 44, 53, 80, 121, 122, 125, 134, 161, 242
Offset: 1

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Author

Keywords

A073216 The terms of A055235 (sums of two powers of 3) divided by 2.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 14, 15, 18, 27, 41, 42, 45, 54, 81, 122, 123, 126, 135, 162, 243, 365, 366, 369, 378, 405, 486, 729, 1094, 1095, 1098, 1107, 1134, 1215, 1458, 2187, 3281, 3282, 3285, 3294, 3321, 3402, 3645, 4374, 6561, 9842, 9843, 9846, 9855, 9882, 9963, 10206, 10935, 13122, 19683
Offset: 0

Views

Author

Jeremy Gardiner, Jul 21 2002

Keywords

Comments

n such that 3 is the largest power of 3 dividing binomial(3n,n). - Benoit Cloitre, Jan 01 2004
Equals A023745 + 1.
This sequence is A007051 together with its (successive) multiples by (powers of) 3. - R. K. Guy, Oct 08 2011

Examples

			T(2,0) = 5 = (3^2 + 3^0) / 2.
Triangle begins:
     1;
     2,    3;
     5,    6,    9;
    14,   15,   18,   27;
    41,   42,   45,   54,   81;
   122,  123,  126,  135,  162,  243;
   365,  366,  369,  378,  405,  486,  729;
  1094, 1095, 1098, 1107, 1134, 1215, 1458, 2187;
  ...

Links

Crossrefs

Cf. A000244 (main diagonal), A055235, A007051 (first column), A023745.
T(2n,n) gives A025551.

Programs

  • Python
    from math import isqrt
def A073216(n): return 3**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+3**(n-1-(a*(a+1)>>1))>>1 # Chai Wah Wu, Apr 08 2025

Formula

T(n,m) = (3^n + 3^m) / 2, n = 0, 1, 2, 3 ..., m = 0, 1, 2, 3, ... n.

Extensions

Edited by Jeremy Gardiner, Oct 08 2011
Offset changed by Alois P. Heinz, Apr 08 2025

A023753 Plaindromes: numbers whose digits in base 12 are in nondecreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 39, 40, 41, 42, 43, 44, 45, 46, 47, 52, 53, 54, 55, 56, 57, 58, 59, 65, 66, 67, 68, 69, 70, 71, 78, 79, 80, 81, 82, 83, 91, 92, 93, 94, 95, 104
Offset: 1

Views

Author

Olivier Gérard

Keywords

Links

Crossrefs

Cf. A000225, A023745, ..., A023757 (nondecreasing digits in base 2, 3, ..., 16).
Cf. A023758, A023759, ..., A023771 (nonincreasing digits in base 2, 3, ..., 16).

Programs

  • Mathematica
    Select[Range[0,104],LessEqual@@IntegerDigits[#,12]&] (* Ray Chandler, Jan 06 2014 *)
Select[Range[0,120],Min[Differences[IntegerDigits[#,12]]]>-1&] (* Harvey P. Dale, Jul 10 2023 *)
  • PARI
    is(n)=vecsort(n=digits(n,12))==n
for(n=0,2,forvec(d=vector(n,i,[1,11]),print1(fromdigits(d,12)","),1)) \\ M. F. Hasler, May 05 2020
  • Python
    from itertools import count, islice, combinations_with_replacement
def A023753_gen(): # generator of terms
    yield 0
    yield from (int(''.join(c),12) for l in count(1) for c in combinations_with_replacement('123456789ab',l))
A023753_list = list(islice(A023753_gen(),30)) # Chai Wah Wu, Apr 08 2025

Extensions

Change offset to 1 by Ray Chandler, Jan 06 2014

A273040 Least k >= 2 such that the base-k digits of n are nondecreasing.

Original entry on oeis.org

2, 3, 2, 3, 3, 4, 2, 3, 5, 4, 4, 5, 3, 3, 2, 6, 3, 5, 5, 7, 4, 4, 4, 5, 7, 3, 4, 6, 6, 8, 2, 5, 5, 5, 6, 8, 5, 5, 5, 3, 3, 4, 4, 3, 6, 6, 4, 7, 5, 6, 6, 6, 3, 8, 8, 10, 6, 6, 6, 7, 7, 5, 2, 5, 6, 7, 7, 5, 5, 9, 6, 11, 7, 5, 7, 7, 8, 8, 8, 3, 7, 7, 7, 8, 4
Offset: 1

Views

Author

Robert Israel, May 13 2016

Keywords

Comments

a(n) = 2 iff n is in A000225.
a(n) = 3 iff n is in A023745 but not A000225.
a(n) <= floor(n/2)-1 if n > 9.

Examples

			a(6) = 4 because 6 is 110 in base 2 and 20 in base 3, which do not have nondecreasing digits, but 12 in base 4 has nondecreasing digits.

Links

Crossrefs

Cf. A000225, A009994, A023745 to A023757.

Programs

  • Maple
    F:= proc(n) local k;
   for k from 2 do if ListTools:-Sorted(convert(n,base,k),`>`) then return k fi od:
end proc:
map(f, [$1..1000]);
  • Mathematica
    Table[k = 2; While[Sort@ # != # &@ IntegerDigits[n, k], k++]; k, {n, 1, 120}] (* Michael De Vlieger, May 14 2016 *)
lk[n_]:=Module[{k=2},While[Min[Differences[IntegerDigits[n,k]]]<0,k++]; k]; Array[lk,90] (* Harvey P. Dale, May 24 2016 *)

A032341 Duplicate of A023761.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 11, 12, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 30, 31, 50, 55, 56, 60, 61, 62, 75, 80, 81, 85, 86, 87, 90, 91, 92, 93, 100, 105, 106, 110, 111, 112
Offset: 1

Views

Author

Keywords

Links

Crossrefs

Cf. A000225, A023745, ..., A023757 (nondecreasing digits in base 2, 3, ..., 16).
Cf. A023758, A023759, ..., A023771 (nonincreasing digits in base 2, 3, ..., 16).

A032342 Duplicate of A023762.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 12, 13, 14, 18, 19, 20, 21, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 42, 43, 72, 78, 79, 84, 85, 86, 108, 114, 115, 120, 121, 122, 126, 127, 128, 129
Offset: 1

Views

Author

Keywords

Comments

Was: Nonincreasing base 6 digits.

Crossrefs

Cf. A000225, A023745, ..., A023757 (nondecreasing digits in base 2, 3, ..., 16).
Cf. A023758, A023759, ..., A023771 (nonincreasing digits in base 2, 3, ..., 16).
Showing 1-8 of 8 results.

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