Floor van Lamoen - cp's OEIS frontend

A248898 Numbers k such that A246964(k) = 1.

0, 2, 4, 6, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 32, 34, 36, 38, 40, 42, 46, 48, 50, 52, 55, 59, 61, 63, 65, 69, 71, 73, 75, 77, 79, 82, 84, 86, 88, 90, 92, 96, 98, 100, 102, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 128, 132, 134, 136, 138, 142, 144, 146, 148, 150

Offset: 1

Floor van Lamoen, Mar 06 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A246964.

A248797 Number of terms > 1 in Fibonacci-variation of Collatz sequence starting with (1, 2n+1).

Original entry on oeis.org

0, 1, 2, 1, 5, 5, 4, 1, 8, 3, 2, 3, 3, 5, 5, 1, 10, 4, 7, 3, 9, 7, 5, 4, 7, 2, 3, 11, 10, 12, 7, 1, 12, 6, 4, 6, 17, 10, 9, 6, 13, 5, 2, 4, 10, 10, 7, 7, 5, 6, 11, 8, 7, 15, 7, 10, 15, 9, 12, 9, 17, 8, 14, 1, 16, 8, 6, 5, 11, 12, 8, 8, 11, 13, 8, 9, 8, 12, 4

Offset: 0

Floor van Lamoen, Mar 03 2015

nonn

In a Fibonacci-variation of Collatz sequence the next term is the odd part of the sum of the preceding two terms. The sequence terminates when 1 is reached. All sequences with initial values {1, 2n+1} terminate.

Proof: Let FC be Fibonacci-variation of Collatz sequence, then FC_{n+1} <= max(FC_{n},FC_{n-1}), with = only if FC_{n}=FC_{n-1}. Therefore FC cannot get into a loop of length greater than 1 (because for all i>n FC_{i}< max(FC_{n},FC_{n+1}). When FC_{n} and FC_{n-1} are coprime, FC_{n} and FC_{n+1} are coprime as well. We conclude that with initial values 1 and 2n+1 (n>0) all pairs of consecutive terms must be coprime, hence cannot become constant (loop of length 1) different from 1.

			a(8)=8 as the Fibonacci-Collatz sequence starting with 1, 17 becomes 1, 17, 9, 13, 11, 3, 7, 5, 3, 1 and has 8 terms > 1.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000045, A006577, A174429.

b:= proc(i, j) local m, r; m:= i+j;
      while irem(m, 2, 'r')=0 do m:=r od; m
    end:
a:= proc(n) local i, j, k; i, j:= 1, 2*n+1;
      for k from 0 while j<>1 do i, j:= j, b(i, j) od; k
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 15 2015

b[i_, j_] := Module[{m, r}, m = i+j; While[Mod[m, 2] == 0, r = Quotient[m, 2]; m = r]; m];
a[n_] := Module[{i, j, k}, {i, j} = {1, 2*n+1}; For[k = 0, j != 1, {i, j} = {j, b[i, j]}; k++]; k];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 21 2025, after Alois P. Heinz *)

A246964 Limiting sequence of transformations when we start with the all 1's sequence a=A000012 and at step n>=0 replace a(n+a(n)) with Sum_{k=n..n+a(n)} a(k).

Original entry on oeis.org

1, 2, 1, 5, 1, 2, 1, 5, 10, 1, 2, 1, 23, 1, 2, 1, 5, 1, 39, 1, 2, 47, 50, 1, 2, 1, 5, 1, 2, 1, 5, 10, 1, 2, 1, 105, 1, 2, 1, 5, 1, 121, 1, 2, 129, 132, 1, 2, 1, 5, 1, 2, 1, 5, 10, 1, 2, 206, 432, 1, 2, 1, 5, 1, 449, 1, 2, 457, 889, 1, 2, 1, 820, 1, 2, 1, 5, 1

Offset: 0

Floor van Lamoen, Mar 02 2015

			Start . . . . . . . . . . . . . . . . .       : 1,1,1,1,1,...
Step 0: a(0+a(0)) = a(1)<- a(0)+a(1) = 2      : 1,2,1,1,1,...
Step 1: a(1+a(1)) = a(3)<- a(1)+a(2)+a(3) = 4 : 1,2,1,4,1,...
Step 2: a(2+a(2)) = a(3)<- a(2)+a(3) = 5      : 1,2,1,5,1,...

Alois P. Heinz, Table of n, a(n) for n = 0..20000

mx:= 20000:  # maximal index needed
b:= proc() 1 end:
a:= proc(n) option remember; global mx; local t;
      if n<0 then 0 else a(n-1); t:= b(n);
        if n+t<= mx then b(n+t):= add(b(k), k=n..n+t) fi; t
      fi
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 04 2015

mx = 20000; (* Maximal index needed *)
b[_] = 1;
a[n_] := a[n] = Module[{t}, If[n<0, 0, t = b[n]; If[n+t <= mx, b[n+t] = Sum[b[k], {k, n, n+t}]]; t]];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz *)

A111024 Triangle read by rows: T(n,s), 0<=s

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Floor van Lamoen, Oct 05 2005

			1; 1,1; 1,1,1; 1,1,1,1; 1,1,1,1,1; 1,1,0,1,1,1

B. Gruenbaum, 'Points on lines' revisited, Nieuw Tijdschr. Wisk., 68 (1980-81), 209-213.
F. Heierman, Punten op lijnen, Nieuw Tijdschr. Wisk., 67 (1979-80) 113-120.

A113036 Number of solutions to +- 1 +- 2 +- .. +- n = 2.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 0, 8, 13, 0, 0, 69, 123, 0, 0, 719, 1313, 0, 0, 8215, 15260, 0, 0, 99774, 187615, 0, 0, 1264854, 2399207, 0, 0, 16544234, 31587644, 0, 0, 221625505, 425313967, 0, 0, 3025271756, 5829531261, 0, 0, 41929052284, 81066732018, 0

Offset: 0

Floor van Lamoen, Oct 11 2005

nonn

Ray Chandler, Table of n, a(n) for n = 0..3339 (terms < 10^1000)

Cf. A058377, A063865, A063866.

A113036:= proc(n) local i,j,p,t; t:= NULL; for j to n do p:=1; for i to j do p:=p*(x^(-i)+x^i); od; t:=t,coeff(p,x,2); od; t; end;

nmax = 50; d = {1}; a1 = {};
Do[
  i = Ceiling[Length[d]/2] + 2;
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 14 2014 *)

a(n) is the coefficient of x^2 in product(x^(-k)+x^k, k=1..n).

A113037 Number of solutions to +- 1 +- 2 +- .. +- n = 3.

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 5, 0, 0, 23, 39, 0, 0, 219, 396, 0, 0, 2406, 4435, 0, 0, 28431, 53167, 0, 0, 353500, 667874, 0, 0, 4557831, 8675836, 0, 0, 60382450, 115601178, 0, 0, 816998489, 1571272955, 0, 0, 11242173783, 21701318843, 0, 0, 156841667096

Offset: 0

Floor van Lamoen, Oct 11 2005

nonn

Ray Chandler, Table of n, a(n) for n = 0..3340 (terms < 10^1000)

Cf. A058377, A063865, A063866.

A113037:= proc(n) local i,j,p,t; t:= NULL; for j to n do p:=1; for i to j do p:=p*(x^(-i)+x^i); od; t:=t,coeff(p,x,3); od; t; end;

nmax = 50; d = {1}; a1 = {};
Do[
  i = Ceiling[Length[d]/2] + 3;
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 14 2014 *)

a(n) is the coefficient of x^3 in product(x^(-k)+x^k, k=1..n).

A113044 Number of ways you can split the set of the first n primes into two proper subsets of which the sum of one is thrice the sum of the other.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 0, 5, 0, 11, 0, 0, 0, 75, 0, 203, 0, 558, 0, 1559, 0, 0, 0, 12786, 0, 37147, 0, 108491, 0, 321551, 0, 964713, 0, 2904950, 0, 8775407, 0, 0, 0, 0, 0, 0, 0, 760875083, 0, 0, 0, 7272292133, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2063638853745109

Offset: 1

Floor van Lamoen, Oct 12 2005

nonn

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A022894.

A113044:=proc(n) local i,j,p,t; t:=0; for j from 2 to n do p:=1; for i to j do p:=p*(x^(-3*ithprime(i))+x^(ithprime(i))); od; t:=t,coeff(p,x,0); od; t; end;
# second Maple program
sp:= proc(n) option remember; `if` (n=1, 2, sp(n-1) +ithprime(n)) end: b:= proc() option remember; local i, j, t; `if` (args[1]=0, `if` (nargs=2, 1, b(args[t] $t=2..nargs)), add (`if` (args[j] -ithprime (args[nargs]) <0, 0, b(sort ([seq (args[i] -`if` (i=j, ithprime (args[nargs]), 0), i=1..nargs-1)])[], args[nargs]-1)), j=1..nargs-1)) end: a:= proc(n) local m; m:= sp(n); `if` (irem(m, 4)=0, b(m/4, 3*m/4, n), 0) end: seq (a(n), n=1..70); # Alois P. Heinz, Nov 02 2011

d = {1}; nMax = 100; Lst = {};
Do[
  p = Prime[n];
  d = PadLeft[d, Length[d] + 4 p] + PadRight[d, Length[d] + 4 p];
  AppendTo[Lst, d[[-Ceiling[Length[d]/4]]]];
  , {n, 1, nMax}];
Lst(* Ray Chandler, Mar 09 2014 *)

More terms from Alois P. Heinz, Nov 02 2011

A113445 Triangle T(n,k) read by rows: consider the sequence a(m) = a(m-1) + sum_{0

Original entry on oeis.org

1, 3, -1, 5, -6, 1, 7, -15, 11, -1, 9, -28, 38, -20, 1, 11, -45, 90, -90, 37, -1, 13, -66, 175, -260, 207, -70, 1, 15, -91, 301, -595, 707, -469, 135, -1, 17, -120, 476, -1176, 1862, -1848, 1052, -264, 1, 19, -153, 708, -2100, 4158, -5502, 4692, -2340, 521, -1

Offset: 0

Floor van Lamoen, Nov 04 2005

The sequence a(m) is also the expansion of (1-x^n)/(1-x-2x^n+x^{n+1}).

Instead of b(i) = a(n*i) one can take b(i) = a(n*i+p) for p=1..n-1.

			For n=5 (A113444) the recurrence relation is b(i) = 11b(i-1)-45b(i-2) +90b(i-3)-90b(i-4)+37b(i-5)-b(i-6), so the fifth row reads 11, -45, 90, -90, 37, -1.

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0..2 give: A005408, -A000384, A007585(n-1) for n>=1. - Alois P. Heinz, Jul 16 2009

T:= (n,k)-> (-1)^k /(k+1)! *(1+k +(n-k) *2^(k+1)) *mul (n+j-k, j=1..k):
seq(seq(T(n,k), k=0..n), n=0..11);  # Alois P. Heinz, Jul 16 2009

From Alois P. Heinz, Jul 16 2009: (Start)
T(n,k) = (-1)^k/(k+1)! * (1+k+(n-k)*2^(k+1)) * Product_{j=1..k}(n+j-k).
G.f. of column k: (-1)^k * x^k * (1+(2^(k+1)-1)*x)/(1-x)^(k+2). (End)

More terms from Alois P. Heinz, Jul 16 2009

A113722 A variant of Golomb's sequence using odd numbers: a(n) is the number of times 2*n+1 occurs, starting with a(1) = 1.

Original entry on oeis.org

1, 3, 5, 5, 5, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 13, 15, 15, 15, 15, 15, 15, 15, 17, 17, 17, 17, 17, 17, 17, 19, 19, 19, 19, 19, 19, 19, 21, 21, 21, 21, 21, 21, 21, 23, 23, 23, 23, 23, 23, 23, 23, 23, 25, 25, 25, 25, 25, 25, 25, 25, 25, 27

Offset: 0

Floor van Lamoen and Paul D. Hanna, Nov 08 2005

nonn

a(n) is taken to be the smallest number >= a(n-1) which is compatible with the description.

			Start with 1 in row 1 and form a triangle where row n is generated from row n-1 by the rule given in the description. Then row 2 will have (1) 3, row 3 will have (3) 5's, row 4 will have (5) 7's, (5) 9's and (5) 11's, etc.
The triangle begins:
1;
3;
5,5,5;
7,7,7,7,7,9,9,9,9,9,11,11,11,11,11; ...
The number of terms in each row (also row sums with offset) is given by A113723: [1,1,3,15,135,3845,769605,3821696361,...].

Cf. A001462 (Golomb's sequence), A113723, A113724, A113676.

a=[1,3,5,5,5];for(n=3,20, for(i=1,a[n],a=concat(a,2*n+1)));a

A113723 The number of terms in row n of A113722 when interpreted as a triangle.

Original entry on oeis.org

1, 1, 3, 15, 135, 3845, 769605, 3821696361

Offset: 1

Floor van Lamoen and Paul D. Hanna, Nov 08 2005

nonn

A113722 is a variant of Golomb's sequence using odd numbers: a(n) is the number of times 2*n+1 occurs, starting with a(1) = 1.

Cf. A001462 (Golomb's sequence), A113722, A113724, A113725, A113676.

cp's OEIS Frontend

User: Floor van Lamoen

A248898 Numbers k such that A246964(k) = 1.

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A248797 Number of terms > 1 in Fibonacci-variation of Collatz sequence starting with (1, 2n+1).

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A246964 Limiting sequence of transformations when we start with the all 1's sequence a=A000012 and at step n>=0 replace a(n+a(n)) with Sum_{k=n..n+a(n)} a(k).

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A111024 Triangle read by rows: T(n,s), 0<=s

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A113036 Number of solutions to +- 1 +- 2 +- .. +- n = 2.

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Maple

Mathematica

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A113037 Number of solutions to +- 1 +- 2 +- .. +- n = 3.

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A113044 Number of ways you can split the set of the first n primes into two proper subsets of which the sum of one is thrice the sum of the other.

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Maple

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Extensions

A113445 Triangle T(n,k) read by rows: consider the sequence a(m) = a(m-1) + sum_{0

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Maple

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A113722 A variant of Golomb's sequence using odd numbers: a(n) is the number of times 2*n+1 occurs, starting with a(1) = 1.

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