A208279 -id:A208279 - cp's OEIS frontend

A008975 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 10.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 0, 0, 5, 1, 1, 6, 5, 0, 5, 6, 1, 1, 7, 1, 5, 5, 1, 7, 1, 1, 8, 8, 6, 0, 6, 8, 8, 1, 1, 9, 6, 4, 6, 6, 4, 6, 9, 1, 1, 0, 5, 0, 0, 2, 0, 0, 5, 0, 1, 1, 1, 5, 5, 0, 2, 2, 0, 5, 5, 1, 1, 1, 2, 6, 0, 5, 2, 4, 2, 5, 0, 6, 2, 1, 1, 3, 8, 6, 5, 7, 6, 6

Offset: 0

N. J. A. Sloane

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened
Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m)
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A047999, A083093, A034931, A034930, A034932.
Cf. A208278 (row sums), A208279 (central terms), A208134 (number of zeros per row), A208280 (distinct terms per row).
Sequences based on the triangles formed by reading Pascal's triangle mod m: A047999 (m = 2), A083093 (m = 3), A034931 (m = 4), A095140 (m = 5), A095141 (m = 6), A095142 (m = 7), A034930 (m = 8), A095143 (m = 9), (this sequence) (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

a008975 n k = a008975_tabl !! n !! k
a008975_row n = a008975_tabl !! n
a008975_tabl = iterate
   (\row -> map (`mod` 10) $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Reinhard Zumkeller, Feb 24 2012

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 10] (* Robert G. Wilson v, May 26 2004 *)

from math import isqrt, comb
from sympy.ntheory.modular import crt
def A008975(n):
    w, c = n-((r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))*(r+1)>>1), 1
    d = int(not ~r & w)
    while True:
        r, a = divmod(r,5)
        w, b = divmod(w,5)
        c = c*comb(a,b)%5
        if r<5 and w<5:
            c = c*comb(r,w)%5
            break
    return crt([5,2],[c,d])[0] # Chai Wah Wu, May 01 2025

T(i, j) = binomial(i, j) mod 10.

A037453 Positive numbers whose base-5 representation contains no 3 or 4.

Original entry on oeis.org

1, 2, 5, 6, 7, 10, 11, 12, 25, 26, 27, 30, 31, 32, 35, 36, 37, 50, 51, 52, 55, 56, 57, 60, 61, 62, 125, 126, 127, 130, 131, 132, 135, 136, 137, 150, 151, 152, 155, 156, 157, 160, 161, 162, 175, 176, 177, 180, 181, 182, 185, 186, 187

Offset: 1

Clark Kimberling

5 divides neither C(2s-1,s) = A001700(s) (nor C(2s,s) = A000984(s), central column of Pascal's Triangle) if and only if s is one of the terms in this sequence.
k such that binomial(2k,k) != 0 (mod 10). - Benoit Cloitre, Aug 18 2002
Let us recall the plan of Apery's irrationality proof. Consider the recurrence (n+1)^3 * u_(n+1) = (34n^3 + 51n^2 + 27n + 5)u_n - n^3 * u_(n-1). The solution with starting values u_0 = 1; u_1 = 5 has the peculiar property that it has integral terms, despite the fact that at every recursion step we divide by (n+1)^3. The n-th term is given by f(n) = Sum_{i=0..n} binomial(n+i,i)^2 * binomial(n,i)^2 = A005259(n) (see Beukers link) and m such that f(m) mod 5 <> 0 equals 2*a(m). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 08 2004
Numbers k such that A208279(k) <> 0. A073095 is a subsequence. - Chai Wah Wu, Dec 08 2023

			From _David A. Corneth_, Dec 23 2023: (Start)
27_10 = 102_5 is a term since its base-5 representation contains no 3 and no 4.
28_10 = 103_5 is not a term since its base-5 representation contains a 3.
(End)

Clark Kimberling, Table of n, a(n) for n = 1..1000
Frits Beukers Consequences of Apery's work on zeta(3), Rencontres Arithmétiques de Caen, zeta(3) irrationnel: les retombées, 1995.
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.
W. Shur, The last digit of C(2*n,n) and sigma C(n,i)*C(2*n-2*i,n-i), The Electronic Journal of Combinatorics, R16, Volume 4, Issue 2 (1997).

Cf. A050607, A005836, A007091.

Cf. A000984, A073095, A208279.

function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 3)
        r += b * q
        b *= 5
    end
r end; [a(n) for n in 1:53] |> println # Peter Luschny, Jan 03 2021

a:= proc(t) option remember; 5*procname(floor(t/3))+ (t mod 3) end proc:
a(0):= 0:
seq(a(n),n=1..100); # Robert Israel, Sep 02 2014

Table[FromDigits[IntegerDigits[k,3],5], {k,60}] (* T. D. Noe, Apr 18 2007 *)
Rest[FromDigits[#,5]&/@Tuples[{0,1,2},4]] (* Harvey P. Dale, Aug 31 2016 *)
Select[Range[187], !Divisible[Binomial[2#, #], 10]&] (* Stefano Spezia, Dec 09 2023 *)

f(n)=sum(i=0,n,binomial(n+i,i)^2*binomial(n,i)^2); for (i=1,1000,if(Mod(f(i),5)<>0,print1(i/2,",")))

isok(k) = binomial(2*k, k) % 10; \\ Michel Marcus, Dec 08 2023

is(n) = my(s = Set(digits(n, 5))); s[#s] < 3 \\ David A. Corneth, Dec 23 2023

a(n) = fromdigits(digits(n, 3), 5) \\ David A. Corneth, Dec 23 2023

from itertools import count, islice
from sympy.ntheory.factor_ import digits
def A037453_gen(startvalue=1): # generator of terms >= startvalue
    if startvalue <= 0: yield 0
    yield from filter(lambda n: all(x<3 for x in digits(n, 5)[1:]), count(max(startvalue, 1)))
A037453_list = list(islice(A037453_gen(), 30)) # Chai Wah Wu, Dec 08 2023

from gmpy2 import digits
def A037453(n): return int(digits(n,3),5) # Chai Wah Wu, Aug 10 2025

a(3n)=5a(n), a(3n+1)=5a(n)+1, a(3n+2)=5a(n)+2, where by definition a(0)=0. - Emeric Deutsch, Mar 23 2004

G.f. satisfies g(x) = 5*(1+x+x^2)*g(x^3) + (x + 2*x^2)/(1-x^3). - Robert Israel, Sep 02 2014

Better definition from T. D. Noe, Apr 18 2007

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A008975 Triangle, read by rows, formed by reading Pascal's triangle (A007318) mod 10.

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A037453 Positive numbers whose base-5 representation contains no 3 or 4.

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