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 1-3 of 3 results.

A292032 a(n) is the value k such that A292030(A292031(n), k) = n.

Original entry on oeis.org

0, 2, 3, 4, 1, 5, 1, 2, 6, 2, 1, 3, 1, 7, 3, 2, 1, 3, 4, 2, 3, 8, 1, 4, 1, 2, 3, 2, 4, 5, 1, 2, 3, 4, 9, 3, 1, 5, 4, 2, 1, 3, 1, 4, 3, 5, 1, 6, 4, 2, 3, 2, 1, 5, 1, 10, 3, 2, 4, 3, 6, 5, 3, 4, 1, 3, 1, 2, 4, 5, 1, 3, 1, 6, 3, 2, 7, 5, 4, 2, 3, 2, 1, 4, 1, 5, 6, 2, 4, 11
Offset: 0

Views

Author

Ely Golden, Sep 08 2017

Keywords

Comments

a(n) > 0 for all n > 0.
for n != 1 the value is unique. For n = 1 A292030(A292031(n), 1) = A292030(A292031(1), 2) = 1. a(n) = 2 by convention.

Examples

			a(2)=3 since A292030(A292031(2),3) = A292030(0,3) = 2.

Links

Crossrefs

Cf. A292031, A292030.

Programs

  • Python
    def smallestSeq(n):
  if(n<0): return []
  if(n==0): return [0,0]
  j,r0,r1=0,0,1
  while(r1<=n): r0,r1=r1,r0+r1 ; j+=1
  while(r1>1):
    if(n%r1==r0): return [n//r1,j]
    r1,r0=r0,r1-r0
    j-=1
  return [n-1, j]
for i in range(10001):
  print(str(i)+" "+str(smallestSeq(i)[1]))

A292030 Table read by ascending antidiagonals: T(n,k) = A000045(k+1)*n + A000045(k).

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 3, 3, 3, 2, 4, 4, 5, 5, 3, 5, 5, 7, 8, 8, 5, 6, 6, 9, 11, 13, 13, 8, 7, 7, 11, 14, 18, 21, 21, 13, 8, 8, 13, 17, 23, 29, 34, 34, 21, 9, 9, 15, 20, 28, 37, 47, 55, 55, 34, 10, 10, 17, 23, 33, 45, 60, 76, 89, 89, 55, 11, 11, 19, 26, 38, 53, 73, 97, 123, 144, 144, 89
Offset: 0

Views

Author

Ely Golden, Sep 07 2017

Keywords

Comments

T(n,k) is entry k in the Fibonacci-like sequence which starts n, n+1, i.e., T(n,0) = n, T(n,1) = n+1, T(n,k) = T(n,k-1) + T(n,k-2).
Therefore, T(0,k), T(1,k), and T(2,k) are equivalent to A000045(k), A000045(k+2), and A000045(k+3) respectively.
Any two rows T(x,.) and T(y,.) contain infinitely many differing terms for any values of x and y, provided that max(x,y) >= 3.
Column k is the list of all positive integers congruent to A000045(k) mod A000045(k+1), with the exception of row 0 which is the same as row 1 with 0 included.

Examples

			T(4,2) = 9 because 9 is element 2 in the Fibonacci-like sequence starting with 4 and 5.
The array begins:
  0, 1,  1,  2,  3,  5,  8, ...
  1, 2,  3,  5,  8, 13, 21, ...
  2, 3,  5,  8, 13, 21, 34, ...
  3, 4,  7, 11, 18, 29, 47, ...
  4, 5,  9, 14, 23, 37, 60, ...
  5, 6, 11, 17, 28, 45, 73, ...
  6, 7, 13, 20, 33, 53, 86, ...

Links

Crossrefs

Cf. A000045, A292031, A292032.
A023548 gives the antidiagonal sums.

Programs

  • Mathematica
    T[n_,k_]:=SeriesCoefficient[(x+y-x*y)/((1-x)^2(1-y-y^2)),{x,0,n},{y,0,k}]; Table[T[n-k,k],{n,0,11},{k,0,n}]//Flatten (* Stefano Spezia, May 21 2025 *)
  • PARI
    T(n, k) = fibonacci(k+1)*n + fibonacci(k);
tabl(nn) = matrix(nn+1,nn+1, i,j, T(i-1,j-1)); \\ Michel Marcus, Sep 27 2017
  • Python
    def nextAntidiagonal(d):
  if(d[0]==0): return [d[1]+1,0]
  return [d[0]-1,d[1]+1]
def fibonacciM(m,n):
  f0,f1=0,1
  if(n<0):
    for _ in range(-n): f1,f0=f0,f1-f0
  else:
    for _ in range(n): f0,f1=f1,f0+f1
  return f1*m+f0
d=[0,0]
for i in range(10001):
  print(str(i)+" "+str(fibonacciM(d[0],d[1])))
  d=nextAntidiagonal(d)

Formula

G.f.: (x + y - x*y)/((1 - x)^2*(1 - y - y^2)). - Stefano Spezia, May 21 2025

A292794 Numbers not congruent to A000045(k) mod A000045(k+1) for all k > 1.

Original entry on oeis.org

0, 4, 6, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 64, 66, 70, 72, 82, 84, 90, 94, 96, 100, 102, 106, 114, 120, 124, 126, 130, 132, 136, 142, 150, 154, 156, 162, 166, 172, 174, 180, 184, 186, 192, 196, 204, 210, 214, 220, 222, 226, 232, 234, 240, 246, 250, 252, 256
Offset: 0

Views

Author

Ely Golden, Sep 23 2017

Keywords

Comments

For n > 0, also numbers n such that A292032(n) = 1.
It is conjectured that A035105(n) is always a member of this sequence for n >= 4 but this remains unproved.
This is the complement of (1 + 2Z) U (2 + 3Z) U (3 + 5Z) U (5 + 8Z) U ..., see also the Example section. - M. F. Hasler, Feb 25 2018

Examples

			a(2) = 6 since 6 mod 2 = 0, 6 mod 3 = 0, 6 mod 5 = 1, and 6 mod 8 = 6. (No other terms of A000045 need to be checked since the "illegal congruences" are all greater than 6, yet 6 is always congruent to 6 for those terms.)
From _M. F. Hasler_, Feb 26 2018: (Start)
This set can be constructed using a sieve which removes:
- first all numbers == 1 (mod 2), there remain the even numbers 0, 2, 4...;
- then all numbers == 2 (mod 3), i.e., == 2 (mod 6), there remain the numbers == 0 or 4 (mod 6): 0, 4, 6, 10, 12, 16, 18, 22, 24, 28, ...;
- then all numbers == 3 (mod 5), i.e., == 8 (mod 10), these are the numbers == 18 or 28 (mod 30), there remain numbers == 0, 4, 6, 10, 12, 16, 22 or 24 (mod 30);
- then all those == 5 (mod 8), but all these are odd;
- then all those == 8 (mod 13), i.e., == 8 (mod 26): there are 8 of these in [1..30*13], and there remain 8*(13-1) residue classes mod 30*13.
- then all those == 13 (mod 21): there are 48 of these left in [1..30*13*7], and there remain 8*12*7-48 = 48*(14-1) residue classes mod 30*13*7.
- then again there are none to remove == 21 (mod 34);
- then those == 34 (mod 55): these are 12*13 of the remaining 48*13*11 residue classes mod 30*13*7*11, so there remain 12*13*(4*11-1) of these; and so on.
This yields as upper bound of the asymptotic density: 1/2 * 2/3 * 4/5 * 12/13 * 13*14 * 43/44 ~ 0.223, the actual value is 0.2187...
(End)

Links

Crossrefs

Cf. A292030, A292031, A292032.
Cf. A300004 for the sequence of first differences.

Programs

  • Mathematica
    {0}~Join~Select[Range[3, 250], Function[n, NoneTrue[Block[{k = {1, 1}}, While[Last@ k <= n, AppendTo[k, Total@ Take[k, -2]]]; Partition[Most@ k, 2, 1]], Mod[n, #2] == #1 & @@ # &]]] (* Michael De Vlieger, Mar 19 2018 *)
  • PARI
    is_A292794(n,F=1)=!for(k=3,oo,F==n%(F=fibonacci(k))&&return;F>n&&break) \\ M. F. Hasler, Feb 25 2018

Formula

a(10^7) = 45721410, a(10^8) = 457214230, a(10^9) = 4572142416. - Jacques Tramu, Feb 26 2018
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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