A265912 -id:A265912 - cp's OEIS frontend

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

Mohammad K. Azarian

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

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

Cf. A034356, A074909, A119629.

Cf. A034868, A119629 (inverse), A265912.

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

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 *)

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

More terms from Erich Friedman

Offset changed by Reinhard Zumkeller, Dec 18 2015

A126257 Number of distinct new terms in row n of Pascal's triangle.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 5, 7, 8, 9, 9, 9, 8, 11, 11, 12, 12, 13, 13, 13, 14, 15, 15, 16, 16, 17, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 21, 23, 23, 24, 24, 25, 25, 26, 26, 27, 26, 26, 28, 29, 29, 30, 30, 31, 31, 32, 32, 32, 33, 34, 34, 34, 35, 36, 36, 37, 37, 38

Offset: 0

Nick Hobson, Dec 24 2006

Partial sums are in A126256.

n occurs a(n) times in A265912. - Reinhard Zumkeller, Dec 18 2015

			Row 6 of Pascal's triangle is: 1, 6, 15, 20, 15, 6, 1. Of these terms, only 15 and 20 do not appear in rows 0-5. Hence a(6)=2.

Nick Hobson, Table of n, a(n) for n = 0..1000
Nick Hobson, Python program for this sequence

Cf. A007318, A062854, A126254, A126255, A126256.

Cf. A034868, A265912.

import Data.List.Ordered (minus, union)
a126257 n = a126257_list !! n
a126257_list = f [] a034868_tabf where
   f zs (xs:xss) = (length ys) : f (ys `union` zs) xss
                   where ys = xs `minus` zs
-- Reinhard Zumkeller, Dec 18 2015

lim=77; z=listcreate(1+lim^2\4); print1(1, ", "); r=1; for(a=1, lim, for(b=1, a\2, s=Str(binomial(a, b)); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); print1(1+#z-r, ", "); r=1+#z)

def A126257(n):
    if n:
        s, c = (1,), {1}
        for i in range(n-1):
            c.update(set(s:=(1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1))+(1,)))
        return len(set((1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1))+(1,))-c)
    return 1 # Chai Wah Wu, Oct 17 2023

A119629 Inverse permutation to sequence A014631.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 10, 13, 17, 7, 25, 30, 36, 42, 8, 54, 61, 69, 78, 9, 11, 104, 115, 126, 138, 150, 163, 14, 189, 203, 218, 233, 249, 265, 12, 18, 315, 333, 352, 371, 391, 411, 432, 453, 21, 496, 519, 542, 566, 590, 615, 640, 666, 692, 26, 15, 771, 799, 828, 857, 887

Offset: 1

Leroy Quet, Jun 08 2006

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A014631.

Cf. A265912.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a119629 = (+ 1) . fromJust . (`elemIndex` a014631_list)
-- Reinhard Zumkeller, Dec 18 2015

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}]; Flatten@Table[ Position[lst, n], {n, 61}] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jun 08 2006

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

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A126257 Number of distinct new terms in row n of Pascal's triangle.

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A119629 Inverse permutation to sequence A014631.

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