A051638 -id:A051638 - cp's OEIS frontend

A098534 Mod 3 analog of Stern's diatomic series.

0, 1, 1, 2, 3, 2, 2, 4, 3, 4, 7, 5, 6, 5, 5, 4, 6, 4, 4, 8, 6, 8, 8, 7, 6, 10, 7, 8, 15, 11, 14, 10, 12, 10, 13, 11, 12, 11, 11, 10, 12, 10, 10, 11, 9, 8, 14, 10, 12, 10, 10, 8, 12, 8, 8, 16, 12, 16, 13, 14, 12, 17, 14, 16, 18, 16, 16, 17, 15, 14, 17, 13, 12, 22, 16, 20, 18, 17, 14, 22

Offset: 0

Paul Barry, Sep 13 2004

Essentially diagonal sums of Pascal's triangle modulo 3.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A002487, A051638.

[0] cat [(&+[Binomial(n-k-1,k) mod 3: k in [0..Floor((n-1)/2)]]): n in [1..100]]; // G. C. Greubel, Jan 17 2018

Table[Sum[Mod[Binomial[n - k - 1, k], 3], {k, 0, Floor[(n - 1)/2]}], {n, 0, 100}] (* G. C. Greubel, Jan 17 2018 *)

for(n=0,100, print1(sum(k=0,floor((n-1)/2), lift(Mod(binomial(n-k-1,k),3))), ", ")) \\ G. C. Greubel, Jan 17 2018

a(n) = Sum_{k=0..floor((n-1)/2)} mod(binomial(n-k-1, k), 3).

A113045 Number triangle binomial(n,floor((n-k)/2)) mod 3.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1

Offset: 0

Paul Barry, Oct 11 2005

Row sums appear to be A051638. First row appears to be A039971. Diagonal sums are A113046.

			Rows begin
1;
1,1;
2,1,1;
0,0,1,1;
0,1,1,1,1;
1,1,2,2,1,1;
2,0,0,0,0,1,1;

A385828 a(n) = Sum_{k=0..n} (binomial(n, k) mod 11).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 31, 40, 36, 50, 56, 2, 4, 8, 16, 32, 64, 62, 80, 72, 100, 112, 4, 8, 16, 32, 53, 106, 91, 116, 100, 156, 169, 8, 16, 32, 64, 62, 124, 116, 188, 134, 224, 228, 16, 32, 53, 62, 91, 182, 144, 222, 202, 272, 291, 32, 64, 106, 124, 182, 188, 244

Offset: 0

Chai Wah Wu, Jul 09 2025

nonn

Sum of n-th row of Pascal's triangle mod 11, A095144.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Richard Garfield and Herbert S. Wilf, The distribution of the binomial coefficients modulo p, Journal of Number Theory, 41 (1) (1992), 1-5.

Cf. A001316, A051638, A384715, A385285, A385741, A385825, A385826.

from gmpy2 import digits
from sympy.abc import x
from sympy import Poly, rem
def A385828(n):
    s = digits(n,11)
    t = tuple(s.count(digits(i,11)) for i in range(1,11))
    G = 2**t[0]*(x+2)**t[1]*(5*x**5+6)**t[9]*(2*x**8+2)**t[2]*(2*x**5+2*x**4+2)**t[4]*(x**9+2*x**2+2)**t[3]*(2*x**7+2*x**5+2*x+2)**t[6]*(2*x**9+2*x**3+x**2+4)**t[7]*(2*x**9+x**6+2*x**2+2)**t[5]*(2*x**8+2*x**7+2*x**6+2*x**4+2)**t[8]
    c = Poly(rem(G,x**10-1),x).all_coeffs()[::-1]
    return int(sum(pow(2,i,11)*c[i] for i in range(len(c)) if c[i]))

A385859 a(n) = Sum_{k=0..n} (C(n,k) mod 3)^2.

Original entry on oeis.org

1, 2, 6, 2, 4, 12, 6, 12, 21, 2, 4, 12, 4, 8, 24, 12, 24, 42, 6, 12, 21, 12, 24, 42, 21, 42, 66, 2, 4, 12, 4, 8, 24, 12, 24, 42, 4, 8, 24, 8, 16, 48, 24, 48, 84, 12, 24, 42, 24, 48, 84, 42, 84, 132, 6, 12, 21, 12, 24, 42, 21, 42, 66, 12, 24, 42, 24, 48, 84, 42

Offset: 0

Chai Wah Wu, Jul 10 2025

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A008459, A051638.

a[n_]:=Sum[Mod[Binomial[n,k],3]^2,{k,0,n}]; Array[a,70,0] (* Stefano Spezia, Jul 10 2025 *)

from gmpy2 import digits
def A385859(n): return 5*3**(s:=digits(n,3)).count('2')-3<>1

If n has k '1' digits and m '2' digits in base 3, then a(n) = 2^(k-1)*(5*3^m - 3).

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A098534 Mod 3 analog of Stern's diatomic series.

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A113045 Number triangle binomial(n,floor((n-k)/2)) mod 3.

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A385828 a(n) = Sum_{k=0..n} (binomial(n, k) mod 11).

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A385859 a(n) = Sum_{k=0..n} (C(n,k) mod 3)^2.

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