cp's OEIS Frontend

If X is an n-set and Y_i (i=1,2,3,4) mutually disjoint 2-subsets of X then a(n-6) is equal to the number of 5-subsets of X intersecting each Y_i (i=1,2,3,4). - Milan Janjic, Aug 26 2007

Sequence found by reading the line from 0, in the direction 0, 14, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line and direction in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 10 2011

Sequence found by reading the line from 1, in the direction 1, 18, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. This is one of the diagonals in the spiral. - Omar E. Pol, Sep 10 2011

Also sequence found by reading the line from 1, in the direction 1, 18, ..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. This is a line perpendicular to the main axis A195015 in the same spiral. - Omar E. Pol, Oct 14 2011

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

a(n+3) = 14*n + 49 is the sum of seven consecutive odd numbers starting with 2*n+1. - Wesley Ivan Hurt, Apr 11 2015

Numbers k such that 3^k + 1 is divisible by 547. - Bruno Berselli, Aug 22 2018

Sum of the numbers from 2*(n-1) to 2*(n+2). - Bruno Berselli, Oct 25 2018

Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.

In this case the sequence S is A000203.

The n-th diagonal of this triangle gives n times A000203.

The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.

T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.

In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.

The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.

For an illustration of the pyramid, see the Links section.

The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.

Column k lists the positive multiples of sigma(k).

The k-th term in the n-th diagonal is equal to n*sigma(k).

Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.

Left column of triangle A131876.

Binomial transform of (1, 14, 0, 0, 0, ...).

Partial sums give A051868. - Leo Tavares, Mar 19 2023

Partial sums give the generalized 18-gonal numbers (A274979).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 18-gonal numbers.