A022912 -id:A022912 - cp's OEIS frontend

A022911 Arrange the nontrivial binomial coefficients C(m,k) (2 <= k <= m/2) in increasing order (not removing duplicates); record the sequence of m's.

4, 5, 6, 6, 7, 8, 7, 9, 10, 11, 8, 12, 8, 13, 9, 14, 15, 16, 10, 9, 17, 18, 11, 19, 20, 21, 10, 12, 22, 10, 23, 24, 13, 25, 26, 11, 27, 14, 28, 29, 30, 15, 11, 31, 12, 32, 33, 16, 34, 35, 36, 37, 17, 38, 13, 39, 40, 12, 18, 41, 42, 43, 12, 44, 19, 45, 14, 46, 47, 48

Offset: 1

Clark Kimberling

nonn

In case of duplicates, the m values are listed in decreasing order. Thus a(18)=16 and a(19)=10 corresponding to binomial(16,2)=binomial(10,3)=120. - Robert Israel, Sep 18 2018

Robert Israel, Table of n, a(n) for n = 1..10000
Sean A. Irvine, Java program (github)

Cf. A003015, A006987, A022912, A319382.

N:= 10000: # for binomial(n,k) values <= N
Res:= NULL:
for n from 2 while n*(n-1)/2 <= N do
  for k from 2 to n/2 do
    v:= binomial(n,k);
    if v > N then break fi;
    Res:= Res,[v,n,k]
od od:
Res:= sort([Res],proc(p,q) if p[1]<>q[1] then  p[1]q[2] then p[2]>q[2]
  fi end proc):
map(t -> t[2], Res); # Robert Israel, Sep 18 2018

A319382(n) = binomial(a(n),A022912(n)). - Robert Israel, Sep 18 2018

A319382 Binomial coefficients binomial(m,k) for 2 <= k <= m/2 in sorted order.

Original entry on oeis.org

6, 10, 15, 20, 21, 28, 35, 36, 45, 55, 56, 66, 70, 78, 84, 91, 105, 120, 120, 126, 136, 153, 165, 171, 190, 210, 210, 220, 231, 252, 253, 276, 286, 300, 325, 330, 351, 364, 378, 406, 435, 455, 462, 465, 495, 496, 528, 560, 561, 595, 630, 666, 680, 703, 715, 741, 780, 792, 816, 820, 861, 903, 924

Offset: 1

Robert Israel, Sep 18 2018

nonn

In contrast to A006987, here the duplicates are not removed. Thus 120 = binomial(10,3) = binomial(16,2) appears twice.

			The first three terms are binomial(4,2) = 6, binomial(5,2) = 10, binomial(6,2) = 15.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003015, A006987, A022911 (values of m), A022912 (values of k).

N:= 10^3: # to get terms <= N
Res:= NULL:
for n from 2 while n*(n-1)/2 <= N do
  for k from 2 to n/2 do
    v:= binomial(n,k);
    if v > N then break fi;
    Res:= Res,v
od od:
sort([Res]);

M = 10^3;
Reap[For[n = 2, n(n-1)/2 <= M, n++, For[k = 2, k <= n/2, k++, v = Binomial[n, k]; If[v > N, Break[]]; Sow[v]]]][[2, 1]] // Sort (* Jean-François Alcover, Apr 27 2019, from Maple *)

a(n) = binomial(A022911(n),A022912(n)).

A318955 Binomial(A319500(n),a(n)) = A006917(n) with 2 <= a(n) <= A319500(n)/2.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 2, 2, 3, 4, 2, 2, 3, 2, 2, 4, 3, 2, 5, 2, 2, 3, 2, 2, 4, 2, 3, 2, 2, 2, 3, 5, 2, 4, 2, 2, 3, 2, 2, 2, 2, 3, 2, 4, 2, 2, 5, 3, 2, 2, 2, 6, 2, 3, 2, 4, 2, 2, 2, 3, 2, 2, 2, 5, 2, 3, 4, 2, 2, 2, 3, 2, 2, 2, 6, 2, 3, 4, 2, 2, 2, 5, 2, 3, 2, 2, 2, 2, 3, 2

Offset: 1

Robert Israel, Sep 20 2018

nonn

First differs from A022912 at n=18.

			a(3) = 6 because A006987(3) = 15 = binomial(6,2).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006987, A022912, A319500.

N:= 10000: # for binomial(n, k) values <= N
S:= {}:
for n from 2 while n*(n-1)/2 <= N do
  for k from 2 to n/2 do
    v:= binomial(n, k);
    if v > N then break fi;
    if not member(v,S) then
      S:= S union {v};
      K[v]:= k;
    fi
od od:
A006987:= sort(convert(S,list)):
seq(K[A006987[i]],i=1..nops(A006987));

A374691 The smallest m+k such that n can be written as n=binomial(m,k).

Original entry on oeis.org

0, 3, 4, 5, 6, 6, 8, 9, 10, 7, 12, 13, 14, 15, 8, 17, 18, 19, 20, 9, 9, 23, 24, 25, 26, 27, 28, 10, 30, 31, 32, 33, 34, 35, 10, 11, 38, 39, 40, 41, 42, 43, 44, 45, 12, 47, 48, 49, 50, 51, 52, 53, 54, 55, 13, 11, 58, 59, 60, 61, 62, 63, 64, 65, 66, 14, 68, 69, 70, 12, 72, 73, 74, 75, 76, 77, 78, 15, 80, 81

Offset: 1

R. J. Mathar, Jul 16 2024

This is most often a(n) = n+1 because the n that do not appear in the "main" body of the Pascal Triangle appear at last at k=1.

			Searching along upwards diagonals, the 6 appears first at 6=binomial(4,2) with m+k=4+2=6, so a(6)=6. The 10 appears first at 10=binomial(5,2) with m+k=7, so a(10)=7.

Cf. A022911, A022912.

A374691 := proc(n)
    local mk,k,m ;
    for mk from 0 to n+1 do
        for k from 0 to mk/2 do
            m := mk-k ;
            if binomial(m,k) = n then
                return mk ;
            end if;
        end do:
    end do:
    return -1 ;
end proc:
seq( A374691(n),n=1..80) ;

A022913 Arrange the nontrivial binomial coefficients C(m,k) (2 <= k <= m-2) in increasing order; record the positions of the central binomial coefficients.

Original entry on oeis.org

1, 4, 13, 30, 63, 124, 233, 431, 798, 1480, 2772, 5232, 9964, 19120, 36931, 71711, 139843, 273633, 536868, 1055596, 2079074, 4100581, 8096797, 16002508, 31652202, 62648233, 124068507

Offset: 1

Clark Kimberling

Cf. A022911, A022912.

Terms corrected by Sean A. Irvine, May 27 2019

cp's OEIS Frontend

A022911 Arrange the nontrivial binomial coefficients C(m,k) (2 <= k <= m/2) in increasing order (not removing duplicates); record the sequence of m's.

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A319382 Binomial coefficients binomial(m,k) for 2 <= k <= m/2 in sorted order.

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A318955 Binomial(A319500(n),a(n)) = A006917(n) with 2 <= a(n) <= A319500(n)/2.

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A374691 The smallest m+k such that n can be written as n=binomial(m,k).

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Maple

A022913 Arrange the nontrivial binomial coefficients C(m,k) (2 <= k <= m-2) in increasing order; record the positions of the central binomial coefficients.

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