cp's OEIS Frontend

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446. - Jeremy Gardiner, Mar 16 2003

The ordinal transform of a sequence b_0, b_1, b_2, ... is the sequence a_0, a_1, a_2, ... where a_n is the number of times b_n has occurred in {b_0 ... b_n}.

From Artur Jasinski, Mar 07 2010: (Start)

This sequence is the even subset of A003983 for odd p=2,4,6,8,....

For the odd subset of A003983 see A004739. (End)

From Gary W. Adamson, Mar 30 2010: (Start)

Given the triangle rows: (1; 1,2,1; 1,2,3,2,1; ...) as polcoeff with offset 0:

q = (1 + 2x + x^2), r = (1 + 2x + 3x^2 + 2x^3 +x^4), etc.; then

(1 + 2x + 3x^2 + ...) = q(x) * q(x^2) * q(x^4) * q(x^8) * ...

..................... = r(x) * r(x^3) * r(x^9) * r(x^27) * ...

..................... = s(x) * s(x^4) * s(x^16)* s(x^64) * ...

... (End)

From L. Edson Jeffery, Jan 13 2012: (Start)

Let U_1(t)=1, U_2(t)=2*t, and U_r(t)=2*t*U_(r-1)(t)-U(r-2)(t), r>2, be Chebyshev polynomials of the second kind. For q>1 an integer, let N=2*q and x_k=cos((2*k-1)*Pi/N), and define the ordered column vectors V_k=[U_k(x_1), U_k(x_2), ..., U_k(x_q)]^T, k=1,...,q, where A^T denotes the transpose of matrix A. Let E_N=[V_1, V_2, ..., V_q] be the q X q matrix formed from the ordered components of the V_k. E_N contains the joint spectra of the Danzer basis (see [Jeffery]) associated with N. Let M_N=(1/q)*[E_N]^T*E_N. For the trivial case q=1, let M_2=[1]. CONJECTURE: E_N and M_N are always integral and symmetric, with M_N having diagonal entries {1,2,...} beginning at entries 1,j (j odd) in the first row and i,1 (i odd) in the first column and with zeros elsewhere. If N is allowed to increase without bound, and assuming the conjecture is true, then triangle A004737 emerges in its entirety from the successive antidiagonals containing those entries [M_N]_(i,j) such that i+j=2*v, for each v in {1,2,...,floor((q+1)/2)}. For example, for N=18 and q=9 (omitting the zeros for clarity),

M_18=[

(1 1 1 1 1);

( 2 2 2 2 );

(1 3 3 3 3);

( 2 4 4 4 );

(1 3 5 5 5);

( 2 4 6 6 );

(1 3 5 7 7);

( 2 4 6 8 );

(1 3 5 7 9)],

from which the first five rows of the sequence can be read off in succession. (End)

T(n,k) = min(n,k). The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013

Expanded form of T(2,k) k=0,1,...,2m for ascending m-nomial triangles. - Bob Selcoe, Feb 07 2014

Terms in the first nine rows of the triangle can be duplicated by performing (111...)^2 with <= nine ones. By way of example, (11111)^2 = 123454321. - Gary W. Adamson, Mar 27 2015

From Johannes W. Meijer, May 29 2010: (Start)

a(n) is the number of ways White can force checkmate in exactly (n+1) moves, n >= 0, ignoring the fifty-move and the triple repetition rules, in the following chess position: White Ka1, Ra8, Bc1, Nb8, pawns a6, a7, b2, c6, d2, f6, g5 and h6; Black Ke8, Nh8, pawns b3, c7, d3, f7, g6 and h7. (After Noam D. Elkies, see link; diagram 5).

Counts all paths of length n, n >= 0, starting at the third node on the path graph P_5, see the Maple program. (End)

From Alec Jones, Feb 25 2016: (Start)

The a(n) are the n-th terms in a "Fibonacci snake" drawn on a rectilinear grid. The n-th term is computed as the sum of the previous terms in cells adjacent to the n-th cell (diagonals included). (This sequence excludes the first term of the snake.)

For example:

1 ... 1 1 ... 1 4 1 4 6 ... 1 4 6 1 4 6 ... and so on.

1 ... 1 2 1 2 ... 1 2 1 2 12 ... 1 2 12 18 (End)

From Gus Wiseman, Oct 06 2023: (Start)

Also the number of subsets of {1..n} containing no two distinct elements summing to n. The a(0) = 1 through a(4) = 12 subsets are:

{} {} {} {} {}

{1} {1} {1} {1}

{2} {2} {2}

{1,2} {3} {3}

{1,3} {4}

{2,3} {1,2}

{1,4}

{2,3}

{2,4}

{3,4}

{1,2,4}

{2,3,4}

For n+1 instead of n we have A038754, complement A167762.

Including twins gives A117855, complement A366131.

The complement is counted by A365544.

For all subsets (not just pairs) we have A365377, complement A365376. (End)

Also the number of integer partitions of n that are of length 2 or contain n/2. The first condition alone is A004526, complement A058984. The second condition alone is A035363, complement A086543, ranks A344415. - Gus Wiseman, Oct 07 2023

Rows are palindromic.

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

From Robert Israel, Nov 06 2023: (Start)

Positive integers divisible by the product of four primes, prime(i)*prime(j)*prime(k)*prime(l), i < j <= k < l, with i + l = j + k.

All positive multiples of terms are terms. (End)

First differs from A086971 at a(90) = 3, A086971(90) = 4.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

Inverse binomial transform yields two zeros followed by A077917 (a signed variant of A127864).

a(n) mod 10 is zero followed by a sequence with period length 8: 0, 1, 2, 7, 4, 7, 4, 5 (repeat).

a(n) is the number of length n+1 binary words with some prefix w such that w contains three more 1's than 0's and no prefix of w contains three more 0's than 1's. - Geoffrey Critzer, Dec 13 2013

From Gus Wiseman, Oct 06 2023: (Start)

Also the number of subsets of {1..n} with two distinct elements summing to n + 1. For example, the a(2) = 1 through a(5) = 14 subsets are:

{1,2} {1,3} {1,4} {1,5}

{1,2,3} {2,3} {2,4}

{1,2,3} {1,2,4}

{1,2,4} {1,2,5}

{1,3,4} {1,3,5}

{2,3,4} {1,4,5}

{1,2,3,4} {2,3,4}

{2,4,5}

{1,2,3,4}

{1,2,3,5}

{1,2,4,5}

{1,3,4,5}

{2,3,4,5}

{1,2,3,4,5}

The complement is counted by A038754.

Allowing twins gives A167936, complement A108411.

For n instead of n + 1 we have A365544, complement A068911.

The version for all subsets (not just pairs) is A366130.

(End)