A292030 -id:A292030 - cp's OEIS frontend

A292031 a(n) is the smallest value m such that n appears in row m of A292030.

0, 0, 0, 0, 3, 0, 5, 3, 0, 4, 9, 3, 11, 0, 4, 7, 15, 5, 3, 9, 6, 0, 21, 4, 23, 12, 8, 13, 5, 3, 29, 15, 10, 6, 0, 11, 35, 4, 7, 19, 39, 13, 41, 8, 14, 5, 45, 3, 9, 24, 16, 25, 51, 6, 53, 0, 18, 28, 11, 19, 4, 7, 20, 12, 63, 21, 65, 33, 13, 8, 69, 23, 71, 5, 24, 37, 3, 9, 15, 39, 26

Offset: 0

Ely Golden, Sep 08 2017

nonn

a(n) = 0 if and only if n is a member of A000045.

a(n) is never 1 nor 2 since row 1 and 2 are equal to row 0 with a shift. - Michel Marcus, Sep 27 2017, amended by M. F. Hasler, Feb 26 2018

			a(3)=0 since A292030(0,5) = 3.
a(7)=3 since A292030(3,2) = 7.

Ely Golden, Table of n, a(n) for n = 0..10000

Cf. A292030.

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)[0]))

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

Ely Golden, Sep 08 2017

nonn

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.

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

Ely Golden, Table of n, a(n) for n = 0..10000

Cf. A292031, A292030.

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]))

A023548 Convolution of natural numbers >= 2 and Fibonacci numbers.

Original entry on oeis.org

2, 5, 11, 21, 38, 66, 112, 187, 309, 507, 828, 1348, 2190, 3553, 5759, 9329, 15106, 24454, 39580, 64055, 103657, 167735, 271416, 439176, 710618, 1149821, 1860467, 3010317, 4870814, 7881162, 12752008, 20633203, 33385245, 54018483, 87403764, 141422284

Offset: 1

Clark Kimberling

Minimal cost of maximum height Huffman tree of size n for strictly "worst case height" sequences. (A strictly "worst case height" sequence generates only maximum height Huffman trees; a non-strictly "worst case height" sequence can generate also non-maximum height Huffman trees.) - Alex Vinokur (alexvn(AT)barak-online.net), Oct 26 2004
Record-positions for A107910: A107910(a(n+2)) = A005578(n), A107910(m) < A005578(n) for m < a(n+2). - Reinhard Zumkeller, May 28 2005
From Jianing Song, Apr 28 2025: (Start)
For n >= 4, a(n-3) is the number of subsets of {1,2,...,n} with at least 2 elements that contain no consecutive elements modulo n. Note that:
 - the number of subsets of {1,2,...,n} with k elements such that the difference of successive elements is at least 2 is binomial(n+1-k,k);
 - the number of such subsets of {1,2,...,n} with k elements that contain both 1 and n is equal to the number of such subsets of {3,...,n-2} with k-2 elements, which is binomial(n-3-(k-2),k-2),
hence a(n) = Sum_{k=2..floor((n+1)/2)} binomial(n+1-k,k) - Sum_{k=0..floor((n-3)/2)} binomial(n-3-k,k) = (F(n+2) - binomial(n+1,0) - binomial(n,1)) - F(n-2) = F(n+1) + F(n-1) - (n+1).
If subsets of {1,2,...,n} are only required to contain no consecutive elements, then the result is A001924(n-2). (End)

Colin Barker, Table of n, a(n) for n = 1..1000
N.-N. Cao and F.-Z. Zhao, Some Properties of Hyperfibonacci and Hyperlucas Numbers, J. Int. Seq. 13 (2010) # 10.8.8
Ligia L. Cristea, Ivica Martinjak, and Igor Urbiha, Hyperfibonacci Sequences and Polytopic Numbers, Journal of Integer Sequences, Volume 19, 2016, Issue 7, #16.7.6.
A. B. Vinokur, Huffman trees and Fibonacci numbers, Kibernetika Issue 6 (1986) 9-12 (in Russian); English translation in Cybernetics 21, Issue 6 (1986), 692-696.
Alex Vinokur, Fibonacci connection between Huffman codes and Wythoff array, arXiv:cs/0410013 [cs.DM], 2004-2005.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A000032, A000045, A001924, A006327.

Antidiagonal sums of A292030.

List([1..40], n-> Lucas(1,-1,n+3)[2] -n-4); # G. C. Greubel, Jul 08 2019

[4*(Fibonacci(n+1)-1)+3*Fibonacci(n)-n: n in [1..40]]; // Vincenzo Librandi, Sep 16 2017

Table[4(Fibonacci[n+1] -1) +3Fibonacci[n] -n, {n, 40}] (* Vincenzo Librandi, Sep 16 2017 *)

a(n) = 4*fibonacci(n+1) + 3*fibonacci(n) - n - 4; \\ Michel Marcus, Sep 08 2016

Vec(x*(2-x) / ((1-x-x^2)*(1-x)^2) + O(x^40)) \\ Colin Barker, Mar 11 2017

[lucas_number2(n+3,1,-1) -n-4 for n in (1..40)] # G. C. Greubel, Jul 08 2019

From Wolfdieter Lang: (Start)
Convolution of natural numbers n >= 1 with Lucas numbers (A000032).
a(n) = 4*(F(n+1) - 1) + 3*F(n) - n, F(n)=A000045 (Fibonacci).
G.f.: x*(2-x)/((1-x-x^2)*(1-x)^2). (End)
For n >= 1, a(n) = L(n+3) - (n+4), where L(n) are Lucas numbers. - Mario Catalani (mario.catalani(AT)unito.it), Jul 22 2004
a(n) = F(n+4) + F(n+2) - (n+4) for n >= 1. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 26 2004 [Offset corrected by Jianing Song, Apr 28 2025]
a(n) = (-4 + (2^(-n)*((1-sqrt(5))^n*(-5+2*sqrt(5)) + (1+sqrt(5))^n*(5+2*sqrt(5)))) / sqrt(5) - n). - Colin Barker, Mar 11 2017
a(n) = Sum_{i=1..n} C(n-i+2,i+1) + C(n-i+1,i). - Wesley Ivan Hurt, Sep 13 2017
E.g.f.: 2*exp(x/2)*(2*cosh((sqrt(5)*x)/2) + sqrt(5)*sinh((sqrt(5)*x)/2)) - exp(x)*(4 + x). - 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

Ely Golden, Sep 23 2017

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

			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)

Ely Golden, Table of n, a(n) for n = 0..10000
Ely Golden, Python program for generating terms of this sequence

Cf. A292030, A292031, A292032.

Cf. A300004 for the sequence of first differences.

{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 *)

is_A292794(n,F=1)=!for(k=3,oo,F==n%(F=fibonacci(k))&&return;F>n&&break) \\ M. F. Hasler, Feb 25 2018

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

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A292031 a(n) is the smallest value m such that n appears in row m of A292030.

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

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A023548 Convolution of natural numbers >= 2 and Fibonacci numbers.

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

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