Adriano Caroli - cp's OEIS frontend

User: Adriano Caroli

Adriano Caroli has authored 3 sequences.

A175880 a(1)=1, a(2)=2. If n >= 3: if n/2 is in the sequence, a(n)=0, otherwise a(n)=n.

1, 2, 3, 0, 5, 0, 7, 8, 9, 0, 11, 12, 13, 0, 15, 0, 17, 0, 19, 20, 21, 0, 23, 0, 25, 0, 27, 28, 29, 0, 31, 32, 33, 0, 35, 36, 37, 0, 39, 0, 41, 0, 43, 44, 45, 0, 47, 48, 49, 0, 51, 52, 53, 0, 55, 0, 57, 0, 59, 60, 61, 0, 63, 0, 65, 0, 67, 68, 69, 0, 71, 0, 73, 0, 75, 76, 77, 0, 79, 80

Offset: 1

Adriano Caroli, Dec 05 2010

If n > 0 and n is in the sequence, then a(2*n) = 0. Example: 5 is in the sequence, so a(2*5) = a(10) = 0.

Is this a(n) = n*A039982(n-1), n > 1? [R. J. Mathar, Dec 07 2010]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A053661.

A000040, A001749, A002001, A002042, A002063, A002089, A003947, A004171 and A081294 are subsequences.

import Data.List (delete)
a175880 n = a175880_list !! (n-1)
a175880_list = 1 : f [2..] [2..] where
   f (x:xs) (y:ys) | x == y    = x : (f xs $ delete (2*x) ys)
                   | otherwise = 0 : (f xs (y:ys))
for_bFile = take 10000 a175880_list
-- Reinhard Zumkeller, Feb 09 2011

A110654 := proc(n) 2*n+1-(-1)^n ; %/4 ;end proc:
A175880 := proc(n) if n <=2 then n; else if type(n,'even') then n-2*procname(A110654(n)) ; else n; end if; end if; end proc:
seq(A175880(n),n=1..40) ; # R. J. Mathar, Dec 07 2010

a(n) = n - (1 + (-1)^n) * a((2*n + 1 - (-1)^n)/4), n >= 3.

a(n) = n - A010673(n+1)*a(A110654(n)).

A173711 Nonnegative integers, six even followed by two odd.

Original entry on oeis.org

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

Offset: 0

Adriano Caroli, Nov 25 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,-1,1,1,-1).

I:=[0,0,0,0,0,0,1]; [n le 7 select I[n] else Self(n-1) + Self(n-2) - Self(n-3)-Self(n-4)+Self(n-5)+Self(n-6)-Self(n-7): n in [1..80]]; // Vincenzo Librandi, Nov 24 2016

LinearRecurrence[{1,1,-1,-1,1,1,-1},{0,0,0,0,0,0,1},50] (* G. C. Greubel, Nov 23 2016 *)
CoefficientList[Series[x^6 / ((x + 1) (x^4 + 1) (x - 1)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 24 2016 *)
Table[If[EvenQ[n],PadRight[{},6,n],{n,n}],{n,0,20}]//Flatten (* Harvey P. Dale, Nov 07 2020 *)

a(n) = A180969(3,n) + A180969(3,n+2).
G.f.: x^6 / ((x+1)*(x^4+1)*(x-1)^2).
a(n) = a(n-1) + a(n-2) - a(n-3) - a(n-4) + a(n-5) + a(n-6) - a(n-7). - G. C. Greubel, Nov 23 2016

A180969 Array read by antidiagonals: a(k,n) = natural numbers each repeated 2^k times.

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 1, 0, 0, 4, 1, 0, 0, 0, 5, 2, 0, 0, 0, 0, 6, 2, 1, 0, 0, 0, 0, 7, 3, 1, 0, 0, 0, 0, 0, 8, 3, 1, 0, 0, 0, 0, 0, 0, 9, 4, 1, 0, 0, 0, 0, 0, 0, 0, 10, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 11, 5, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 12, 5, 2, 1, 0, 0, 0, 0

Offset: 0

Adriano Caroli, Nov 17 2010

Generalization of P. Barry's (2003) formula in A004526.

			Sequence gives the antidiagonals of the infinite square array with rows indexed by k and columns indexed by n:
0  1  2  3  4  5  6  7  8  9 10 11 12 13...
0  0  1  1  2  2  3  3  4  4   5   5   6...
0  0  0  0  1  1  1  1  2  2   2   2   3...
0  0  0  0  0  0  0  0  1  1   1   1   1...
0  0  0  0  0  0  0  0  0  0   0   0   0...
...........................................

Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Cf. A001477, A004526, A002265, A132292.

function v=A180969(k,n,q)
% n=vector of natural numbers 0,1,...,n
% v=vector in which each n is repeated k times
% q=q-th term of v from where to start
if k==0;v=n+q;return;end
v=A180969(k-1,n,q);
% calculate repetition only if v terms are not all zeros
if any(v); v=v/2+((-1).^v-1)/4;end
% Adriano Caroli, Nov 28 2010

Table[Floor[#/2^k] &[n - k], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 30 2019 *)

matrix(10, 20, k, n, k--; n--; floor(n/2^k)) \\ Michel Marcus, Sep 09 2019

a(k,n) = (n/2^k) + Sum_{j=1..k} ((-1)^a(j-1,n) - 1)/2^(k-j+2).

a(k,n) = floor(n/2^k). - Adriano Caroli, Sep 30 2019

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User: Adriano Caroli

A175880 a(1)=1, a(2)=2. If n >= 3: if n/2 is in the sequence, a(n)=0, otherwise a(n)=n.

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A173711 Nonnegative integers, six even followed by two odd.

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A180969 Array read by antidiagonals: a(k,n) = natural numbers each repeated 2^k times.

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