A014631 Numbers in order in which they appear in Pascal's triangle.

1, 2, 3, 4, 6, 5, 10, 15, 20, 7, 21, 35, 8, 28, 56, 70, 9, 36, 84, 126, 45, 120, 210, 252, 11, 55, 165, 330, 462, 12, 66, 220, 495, 792, 924, 13, 78, 286, 715, 1287, 1716, 14, 91, 364, 1001, 2002, 3003, 3432, 105, 455, 1365, 5005, 6435, 16, 560, 1820, 4368, 8008, 11440

Offset: 1

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

Comments

A permutation of the natural numbers. - Robert G. Wilson v, Jun 12 2014

In Pascal's triangle a(n) occurs the first time in row A265912(n). - Reinhard Zumkeller, Dec 18 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.
• F. Richman, Combinations and Permutations
• Index entries for triangles and arrays related to Pascal's triangle
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A034356, A074909, A119629.

Cf. A034868, A119629 (inverse), A265912.

Programs

Haskell

import Data.List (nub)
a014631 n = a014631_list !! (n-1)
a014631_list = 1 : (nub $ concatMap tail a034868_tabf)
-- Reinhard Zumkeller, Dec 19 2015

Mathematica

lst = {1}; t = Flatten[Table[Binomial[n, m], {n, 16}, {m, Floor[n/2]}]]; Do[ If[ !MemberQ[lst, t[[n]]], AppendTo[lst, t[[n]] ]], {n, Length@t}]; lst (* Robert G. Wilson v *)
DeleteDuplicates[Flatten[Table[Binomial[n,m],{n,20},{m,0,Floor[n/2]}]]] (* Harvey P. Dale, Apr 08 2013 *)

Python

from itertools import count, islice
def A014631_gen(): # generator of terms
    s, c =(1,), set()
    for i in count(0):
        for d in s:
            if d not in c:
                yield d
                c.add(d)
        s=(1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1)) + ((s[-1]<<1,) if i&1 else ())
A014631_list = list(islice(A014631_gen(),30)) # Chai Wah Wu, Oct 17 2023

Extensions

More terms from Erich Friedman

Offset changed by Reinhard Zumkeller, Dec 18 2015