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.

A015473 q-Fibonacci numbers for q=2, scale a(n-1).

Original entry on oeis.org

0, 1, 2, 9, 74, 1193, 38250, 2449193, 313534954, 80267397417, 41097221012458, 42083634584154409, 86187324725569242090, 353023324159566199755049, 2891967157702491033962603498, 47381990264820937260009495466281
Offset: 0

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Author

Olivier Gérard

Keywords

Comments

a(1) = 1, a(n+1) = denominator of continued fraction [1;2,4,8,...,2^n]. - Amarnath Murthy, May 02 2001
The difference equation y(n, x, s) = q^(n-1)*x*y(n-1, x, s) + s*y(n-2, x, s) yields a type of two variable q-Fibonacci polynomials in the form F(n, x, s, q) = Sum_{j=0..floor((n-1)/2)} q-binomial(n-j-1,j, q^2)*q^binomial(n-2*j,2)* x^(n-2*j)*s^j. When x=s=1 these polynomials reduce to q-Fibonacci numbers. This family of q-Fibonacci numbers is different from that of the q-Fibonacci numbers defined in A015459. - G. C. Greubel, Dec 17 2019

Links

Crossrefs

Cf. A061377.
q-Fibonacci numbers: A000045 (q=1), this sequence (q=2), A015474 (q=3), A015475 (q=4), A015476 (q=5), A015477 (q=6), A015479 (q=7), A015480 (q=8), A015481 (q=9), A015482 (q=10), A015484 (q=11), A015485 (q=12).

Programs

  • GAP
    q:=2;; a:=[0,1];; for n in [3..20] do a[n]:=q^(n-2)*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 17 2019
  • Magma
    [0] cat [n le 2 select n else 2^(n-1)*Self(n-1) + Self(n-2): n in [1..16]]; // Vincenzo Librandi, Nov 09 2012
  • Maple
    q:=2; seq(add((product((1-q^(2*(n-j-1-k)))/(1-q^(2*k+2)), k=0..j-1))* q^binomial(n-2*j,2), j = 0..floor((n-1)/2)), n = 0..20); # G. C. Greubel, Dec 17 2019
  • Mathematica
    RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]*2^(n-1)+a[n-2]},  a, {n, 30}] (* Vincenzo Librandi, Nov 09 2012 *)
Join[{0},Denominator[Table[FromContinuedFraction[2^Range[0,n]],{n,0,20}]]] (* Harvey P. Dale, Feb 09 2013 *)
F[n_, q_]:= Sum[QBinomial[n-j-1,j,q^2]*q^Binomial[n-2*j,2], {j,0,Floor[(n-1)/2] }]; Table[F[n, 2], {n, 0, 20}] (* G. C. Greubel, Dec 17 2019 *)
  • PARI
    q=2; m=20; v=concat([0,1], vector(m-2)); for(n=3, m, v[n]=q^(n-2)*v[n-1]+v[n-2]); v \\ G. C. Greubel, Dec 17 2019
  • Sage
    def F(n,q): return sum( q_binomial(n-j-1, j, q^2)*q^binomial(n-2*j,2) for j in (0..floor((n-1)/2)))
[F(n,2) for n in (0..20)] # G. C. Greubel, Dec 17 2019

Formula

a(n) = 2^(n-1)*a(n-1) + a(n-2).

A280219 a(1) = 1, a(n+1) is the numerator of n-th partial fraction of the continued fraction [1; 3, 9, 27, ..., 3^n].

Original entry on oeis.org

1, 4, 37, 1003, 81280, 19752043, 14399320627, 31491333963292, 206614656532479439, 4066796316020126761129, 240140255871287121650385760, 42540125910897696055021012987849, 22607567054453522745047709284925846169, 36043764129000043869363596706325850854686436, 172396206472341818392860586297603696245873653954653
Offset: 1

Views

Author

Seiichi Manyama, Dec 29 2016

Keywords

Examples

			G.f. = x + 4*x^2 + 37*x^3 + 1003*x^4 + 81280*x^5 + 19752043*x^6 + ...
a(3) = 37, the numerator of 1 + 1/(3 + 1/9) = 37/28.

Links

Crossrefs

Denominators are in A015474.
Cf. A061377, A280220.

Programs

  • Mathematica
    f[n_] := Numerator[ FromContinuedFraction[ Reverse[3^Range[0, n -1]] ]]; Array[f, 14] (* Robert G. Wilson v, Dec 30 2016 *)

Formula

a(n) = 3^(n-1)*a(n-1) + a(n-2).

A127611 a(n) = numerator of the continued fraction which has the positive divisors of n as its terms.

Original entry on oeis.org

1, 3, 4, 13, 6, 63, 8, 107, 37, 163, 12, 3259, 14, 311, 319, 1725, 18, 10449, 20, 13928, 613, 751, 24, 638475, 151, 1043, 1003, 37306, 30, 1513023, 32, 55307, 1489, 1771, 1511, 19381852, 38, 2207, 2071, 4538318, 42, 5649833, 44, 142046, 131413, 3223, 48
Offset: 1

Views

Author

Leroy Quet, Jan 19 2007

Keywords

Comments

The divisors can be written either from largest to smallest or from smallest to largest and the numerator of the continued fraction would remain unchanged.

Examples

			The divisors of 6 are 1,2,3,6. So a(6) is the numerator of 1 +1/(2 +1/(3 +1/6)) = 63/44. a(6) is also the numerator of 6 +1/(3 +1/(2+1/1)) = 63/10.

Links

Crossrefs

Cf. A127612 (denominators), A127613 (denominators).
Cf. A061377, A280219.

Programs

  • Maple
    f:= n -> numer(numtheory:-cfrac(sort(convert(numtheory:-divisors(n),list)))):
map(f, [$1..100]); # Robert Israel, Jan 17 2023
  • Mathematica
    f[n_] := Numerator[FromContinuedFraction[Divisors[n]]];Table[f[n], {n, 47}] (* Ray Chandler, Jan 22 2007 *)
  • PARI
    a(n) = contfracpnqn(divisors(n))[1,1]; \\ Kevin Ryde, Jan 19 2023

Formula

If p is prime, a(p^k) = p^k * a(p^(k-1)) + a(p^(k-2)), with a(p^0) = a(1) = 1 and a(p^1) = p+1. - Robert Israel, Jan 17 2023

Extensions

Extended by Ray Chandler, Jan 22 2007

A280220 a(1) = 1, a(n+1) is the numerator of n-th partial fraction of the continued fraction [1; 4, 16, 64, ..., 4^n].

Original entry on oeis.org

1, 5, 81, 5189, 1328465, 1360353349, 5572008645969, 91291791015909445, 5982898821590650033489, 1568381028778351153394849861, 1644566705638271237843748737881425, 6897812711726991987001765057444407253061, 115726093792191122162903443021235072225308939601
Offset: 1

Views

Author

Seiichi Manyama, Dec 29 2016

Keywords

Examples

			G.f. = x + 5*x^2 + 81*x^3 + 5189*x^4 + 1328465*x^5 + 1360353349*x^6 + ...
a(3) = 81, the numerator of 1 + 1/(4 + 1/16) = 81/65.

Links

Crossrefs

Denominators are in A015475.
Cf. A061377, A280219.

Programs

  • Mathematica
    f[n_] := Numerator[ FromContinuedFraction[ Reverse[4^Range[0, n -1]] ]]; Array[f, 12] (* Robert G. Wilson v, Dec 29 2016 *)

Formula

a(n) = 4^(n-1)*a(n-1) + a(n-2).

A280042 a(1) = 1, a(n+1) is the numerator of n-th partial fraction of the continued fraction [1; 10, 100, 1000, ..., 10^n].

Original entry on oeis.org

1, 11, 1101, 1101011, 11010111101, 1101011111201011, 1101011111212021111101, 11010111112121312122121201011, 1101011111212132313223231313121111101, 1101011111212132324233342425242423223121201011
Offset: 1

Views

Author

Seiichi Manyama, Dec 31 2016

Keywords

Examples

			G.f. = x + 11*x^2 + 1101*x^3 + 1101011*x^4 + 11010111101*x^5 + 1101011111201011*x^6 + ...
a(3) = 1101, the numerator of 1 + 1/(10 + 1/100) = 1101/1001.

Links

Crossrefs

Denominators are in A015482.
Cf. A061377, A280219, A280220.

Formula

a(n) = 10^(n-1)*a(n-1) + a(n-2).
Showing 1-5 of 5 results.

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