cp's OEIS Frontend

The multiples of 20 are exactly those integers which do not have a multiple whose decimal digits are of alternating parity. (International Mathematical Olympiad 2004, problem 6, see A110303) - Joseph Myers, Jul 13 2004

Also, k such that Fibonacci(k) mod 27 = 0. - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 18 2004

A033183(a(n)) = n+1. - Reinhard Zumkeller, Nov 07 2009

A122841(a(n)) > 1 for n > 0. - Reinhard Zumkeller, Nov 10 2013

Sum of the numbers from 4*(n-1) to 4*(n+1). - Bruno Berselli, Oct 25 2018

Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - Gary W. Adamson, Aug 24 2010

Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011

From Lamine Ngom, Aug 19 2021: (Start)

Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section).

a(n) is a bisection of A195042.

a(n) is a trisection of A028387.

a(n) + 1 is promic (A002378).

a(n) + 2 is a trisection of A002061.

a(n) + 9 is the arithmetic mean of its neighbors.

4*a(n) + 5 is a square: A016945(n)^2. (End)

Sequence found by reading the line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Semi-axis opposite to A195316 in the same spiral.

Area of a square with diagonal 6n. - Wesley Ivan Hurt, Jun 19 2014

Number of identical tessellation tiles that are composed of 48 equilateral edge joined triangles that can be formed into a order n hexagon. The example tiles shown in the link below are tessellated with eight sphinx tiles. See A291582. - Craig Knecht, Sep 02 2017

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

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.

Solutions to (11^x + 13^x) mod 19 = 17.