cp's OEIS Frontend

Areas of perfect 4:3 rectangles (for n > 0).

Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A069190 in the same spiral. - Omar E. Pol, Sep 16 2011

(x,y,z) = (-a(n), 1 + n*a(n), 1 - n*a(n)) are solutions of the Diophantine equation x^3 + 2*y^3 + 2*z^3 = 4. - XU Pingya, Apr 30 2022

Exponents e such that x^e + x^2 - 1 is reducible.

If Y is a 4-subset of an (2n+1)-set X then, for n>=3, a(n-2) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007

Also the number of cube units that frame a cube of edge length n+1. - Peter M. Chema, Mar 27 2016

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 44 ).

Or, with a different offset, 12*n - 1. In any case, numbers congruent to -1 (mod 12). - Alonso del Arte, May 29 2011

Numbers congruent to 2 (mod 3) and 3 (mod 4). - Bruno Berselli, Jul 06 2017

a(n+12) = a(n);

a(n) = A000007(A010881(n));

a(A168186(n)) = 1; a(A008594(n)) = 0;

A033444(n) = Sum_{k=0..n} a(k)*(n-k).

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(71).

A089911(2*a(n)) = 7. - Reinhard Zumkeller, Jul 05 2013

Equivalently, intersection of A016813 and A016789. - Bruno Berselli, Jan 24 2018

Partial sums give A051867. - Leo Tavares, Mar 19 2023

T(n,k) = A003991(n-1,k) for 1 <= k < n;

T(n,k) = T(n,n-1-k) for k < n;

T(n,1) = n-1; T(n,n) = 0; T(n,2) = A005843(n-2) for n > 1;

T(n,3) = A008585(n-3) for n>2; T(n,4) = A008586(n-4) for n > 3;

T(n,5) = A008587(n-5) for n>4; T(n,6) = A008588(n-6) for n > 5;

T(n,7) = A008589(n-7) for n>6; T(n,8) = A008590(n-8) for n > 7;

T(n,9) = A008591(n-9) for n>8; T(n,10) = A008592(n-10) for n > 9;

T(n,11) = A008593(n-11) for n>10; T(n,12) = A008594(n-12) for n > 11;

T(n,13) = A008595(n-13) for n>12; T(n,14) = A008596(n-14) for n > 13;

T(n,15) = A008597(n-15) for n>14; T(n,16) = A008598(n-16) for n > 15;

T(n,17) = A008599(n-17) for n>16; T(n,18) = A008600(n-18) for n > 17;

T(n,19) = A008601(n-19) for n>18; T(n,20) = A008602(n-20) for n > 19;

Row sums give A000292; triangle sums give A000332;

All numbers m > 0 occur A000005(m) times;

A002378(n) = T(A005408(n),n+1) = n*(n+1).

k-th columns are arithmetic progressions with step k, starting with 0. If a zero is prefixed to the sequence, then we get a new table where the columns are again arithmetic progressions with step k, but starting with k, k=0,1,2,...: 1st column = (0,0,0,...), 2nd column = (1,2,3,...), 3rd column = (2,4,6,8,...), etc. - M. F. Hasler, Feb 02 2013

Construct the infinite-dimensional matrix representation of angular momentum operators (J_1,J_2,J_3) in the Jordan-Schwinger form (cf. Harter, Klee, Schwinger). The triangle terms T(n,k) = T(2j,j+m) satisfy: (1/2)T(2j,j+m)^(1/2) = = = i = -i . Matrices for J_1 and J_2 are sparse. These equalities determine the only nonzero entries. - Bradley Klee, Jan 29 2016

T(n+1,k+1) is the number of degrees of freedom of a k-dimensional affine subspace within an n-dimensional vector space. This is most readily interpreted geometrically: e.g. in 3 dimensions a line (1-dimensional subspace) has T(4,2) = 4 degrees of freedom and a plane has T(4,3) = 3. T(n+1,1) = n indicates that points in n dimensions have n degrees of freedom. T(n,n) = 0 for any n as all n-dimensional spaces in an n-dimensional space are equivalent. - Daniel Leary, Apr 29 2020