cp's OEIS Frontend

Binomial transform of (1, 6, 9, 4, 0, 0, 0, ...). - Gary W. Adamson, Oct 16 2007

a(n) is the sum of the maximum(m,n) over {(m,n): m,n in positive integers, m<=n}. - Geoffrey Critzer, Oct 11 2009

We obtain these numbers for d=2 in the identity n*(n*(d*n-d+2)/2)-sum(k*(d*k-d+2)/2, k=0..n-1) = n*(n+1)*(2*d*n-2*d+3)/6 (see Klaus Strassburger in Formula lines). - Bruno Berselli, Apr 21 2010, Nov 16 2010

q^a(n) is the Hankel transform of the q-Catalan numbers. - Paul Barry, Dec 15 2010

Row 1 of the convolution array A213835. - Clark Kimberling, Jul 04 2012

From Ant King, Oct 24 2012: (Start)

For n>0, the digital roots of this sequence A010888(A002412(n)) form the purely periodic 27-cycle {1,7,4,5,5,8,9,3,3,4,1,7,8,8,2,3,6,6,7,4,1,2,2,5,6,9,9}.

For n>0, the units' digits of this sequence A010879(A002412(n)) form the purely periodic 20-cycle {1,7,2,0,5,1,2,2,5,5,6,2,7,5,0,6,7,7,0,0}.

(End)

Partial sums of A000384. - Omar E. Pol, Jan 12 2013

Row sums of A094728. - J. M. Bergot, Jun 14 2013

Number of orbits of Aut(Z^7) as function of the infinity norm (n+1) of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 40320. - Philippe A.J.G. Chevalier, Dec 28 2015

Coefficients in the hypergeometric series identity 1 - 7*(x - 1)/(3*x + 1) + 22*(x - 1)*(x - 2)/((3*x + 1)*(3*x + 2)) - 50*(x - 1)*(x - 2)*(x - 3)/((3*x + 1)*(3*x + 2)*(3*x + 3)) + ... = 0, valid for Re(x) > 1. Cf. A000326 and A002418. Column 3 of A103450. - Peter Bala, Mar 14 2019

Write 0, 1, 2, ... in a clockwise spiral; sequence gives numbers on one of 4 diagonals.

Also, least m > n such that T(m)*T(n) is a square and more precisely that of A055112(n). {T(n) = A000217(n)}. - Lekraj Beedassy, May 14 2004

Also sequence found by reading the line from 0, in the direction 0, 8, ... and the same line from 0, in the direction 0, 24, ..., in the square spiral whose vertices are the generalized decagonal numbers A074377. Axis perpendicular to A195146 in the same spiral. - Omar E. Pol, Sep 18 2011

Number of diagonals with length sqrt(5) in an (n+1) X (n+1) square grid. Every 1 X 2 rectangle has two such diagonals. - Wesley Ivan Hurt, Mar 25 2015

Imagine a board made of squares (like a chessboard), one of whose squares is completely surrounded by square-shaped layers made of adjacent squares. a(n) is the total number of squares in the first to n-th layer. a(1) = 8 because there are 8 neighbors to the unit square; adding them gives a 3 X 3 square. a(2) = 24 = 8 + 16 because we need 16 more squares in the next layer to get a 5 X 5 square: a(n) = (2*n+1)^2 - 1 counting the (2n+1) X (2n+1) square minus the central square. - R. J. Cano, Sep 26 2015

The three platonic solids (the simplex, hypercube, and cross-polytope) with unit side length in n dimensions all have rational volume if and only if n appears in this sequence, after 0. - Brian T Kuhns, Feb 26 2016

The number of active (ON, black) cells in the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood. - Robert Price, May 19 2016

The square root of a(n), n>0, has continued fraction [2n; {1,4n}] with whole number part 2n and periodic part {1,4n}. - Ron Knott, May 11 2017

Numbers k such that k+1 is a square and k is a multiple of 4. - Bruno Berselli, Sep 28 2017

a(n) is the number of vertices of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - Emeric Deutsch, May 13 2018

a(n) is the number of vertices in conjoined n X n octagons which are arranged into a square array, a.k.a. truncated square tiling. - Donghwi Park, Dec 20 2020

a(n-2) is the number of ways to place 3 adjacent marks in a diagonal, horizontal, or vertical row on an n X n tic-tac-toe grid. - Matej Veselovac, May 28 2021

a(n-3) is the number of aperiodic necklaces (Lyndon words) with 3 black beads and n-3 white beads.

Number of triangular partitions (see Almkvist).

Consists of arithmetic progression quadruples of common difference n+1 starting at A045943(n). Refers to the least number of coins needed to be rearranged in order to invert the pattern of a (n+1)-rowed triangular array. For instance, a 5-rowed triangular array requires a minimum of a(4)=5 rearrangements (shown bracketed here) for it to be turned upside down.

.....{*}..................{*}*.*{*}{*}

.....*.*....................*.*.*.{*}

....*.*.*....---------\......*.*.*

..{*}*.*.*...---------/.......*.*

{*}{*}*.*{*}..................{*}

- Lekraj Beedassy, Oct 13 2003

Partial sums of 1,1,1,2,2,2,3,3,3,4,4,4,... - Jon Perry, Mar 01 2004

Sum of three successive terms is a triangular number in natural order starting with 3: a(n)+a(n+1)+a(n+2) = T(n+2) = (n+2)*(n+3)/2. - Amarnath Murthy, Apr 25 2004

Apply Riordan array (1/(1-x^3),x) to n. - Paul Barry, Apr 16 2005

Absolute values of numbers that appear in A145919. - Matthew Vandermast, Oct 28 2008

In the Moree definition, (-1)^n*a(n) is the 3rd Witt transform of A033999 and (-1)^n*A004524(n) with 2 leading zeros dropped is the 2nd Witt transform of A033999. - R. J. Mathar, Nov 08 2008

Column sums of:

1 2 3 4 5 6 7 8 9.....

1 2 3 4 5 6.....

1 2 3.....

........................

----------------------

1 2 3 5 7 9 12 15 18 - Jon Perry, Nov 16 2010

a(n) is the sum of the positive integers <= n that have the same residue modulo 3 as n. They are the additive counterpart of the triple factorial numbers. - Peter Luschny, Jul 06 2011

a(n+1) is the number of 3-tuples (w,x,y) with all terms in {0,...,n} and w=3*x+y. - Clark Kimberling, Jun 04 2012

a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x-y = (1 mod 3), and x+y < n. - Clark Kimberling, Jul 02 2012

a(n+1) is the number of partitions of n into two sorts of part(s) 1 and one sort of (part) 3. - Joerg Arndt, Jun 10 2013

Arrange A004523 in rows successively shifted to the right two spaces and sum the columns:

1 2 2 3 4 4 5 6 6...

1 2 2 3 4 4 5...

1 2 2 3 4...

1 2 2...

1...

------------------------------

1 2 3 5 7 9 12 15 18... - L. Edson Jeffery, Jul 30 2014

a(n) = A258708(n+1,1) for n > 0. - Reinhard Zumkeller, Jun 23 2015

Also the number of triples of positive integers summing to n + 4, the first less than each of the other two. Also the number of triples of positive integers summing to n + 2, the first less than or equal to each of the other two. - Gus Wiseman, Oct 11 2020

Also the lower matching number of the (n+1)-triangular honeycomb king graph = n-triangular grid graph (West convention). - Eric W. Weisstein, Dec 14 2024

Number of edges of a complete 4-partite graph of order 4n, K_n,n,n,n. - Roberto E. Martinez II, Oct 18 2001

Number of edges of the complete bipartite graph of order 7n, K_n, 6n. - Roberto E. Martinez II, Jan 07 2002

Number of edges in the line graph of the product of two cycle graphs, each of order n, L(C_n x C_n). - Roberto E. Martinez II, Jan 07 2002

Total surface area of a cube of edge length n. See A000578 for cube volume. See A070169 and A071399 for surface area and volume of a regular tetrahedron and links for the other Platonic solids. - Rick L. Shepherd, Apr 24 2002

a(n) can represented as n concentric hexagons (see example). - Omar E. Pol, Aug 21 2011

Sequence found by reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A003154 in the same spiral. - Omar E. Pol, Sep 08 2011

Together with 1, numbers m such that floor(2*m/3) and floor(3*m/2) are both squares. Example: floor(2*150/3) = 100 and floor(3*150/2) = 225 are both squares, so 150 is in the sequence. - Bruno Berselli, Sep 15 2014

a(n+1) gives the number of vertices in a hexagon-like honeycomb built from A003215(n) congruent regular hexagons (see link). Example: a hexagon-like honeycomb consisting of 7 congruent regular hexagons has 1 core hexagon inside a perimeter of six hexagons. The perimeter has 18 vertices. The core hexagon has 6 vertices. a(2) = 18 + 6 = 24 is the total number of vertices. - Ivan N. Ianakiev, Mar 11 2015

a(n) is the area of the Pythagorean triangle whose sides are (3n, 4n, 5n). - Sergey Pavlov, Mar 31 2017

More generally, if k >= 5 we have that the sequence whose formula is a(n) = (2*k - 4)*n^2 is also the sequence found by reading the line from 0, in the direction 0, (2*k - 4), ..., in the square spiral whose vertices are the generalized k-gonal numbers. In this case k = 5. - Omar E. Pol, May 13 2018

The sequence also gives the number of size=1 triangles within a match-made hexagon of size n. - John King, Mar 31 2019

For hexagons, the number of matches required is A045945; thus number of size=1 triangles is A033581; number of larger triangles is A307253 and total number of triangles is A045949. See A045943 for analogs for Triangles; see A045946 for analogs for Stars. - John King, Apr 04 2019

Write the integers in groups: 0; 1,2; 3,4,5; 6,7,8,9; ... and add the groups: a(n) = Sum_{j=0..n} (A000217(n)+j), row sums of the triangular view of A001477. - Asher Auel, Jan 06 2000

With offset = 2, a(n) is the number of edges of the line graph of the complete graph of order n, L(K_n). - Roberto E. Martinez II, Jan 07 2002

Also the total number of pips on a set of dominoes of type n. (A "3" domino set would have 0-0, 0-1, 0-2, 0-3, 1-1, 1-2, 1-3, 2-2, 2-3, 3-3.) - Gerard Schildberger, Jun 26 2003. See A129533 for generalization to n-armed "dominoes". - N. J. A. Sloane, Jan 06 2016

Common sum in an (n+1) X (n+1) magic square with entries (0..n^2-1).

Alternate terms of A057587. - Jeremy Gardiner, Apr 10 2005

If Y is a 3-subset of an n-set X then, for n >= 5, a(n-5) is the number of 4-subsets of X which have exactly one element in common with Y. Also, if Y is a 3-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-5)-subsets of X which have exactly one element in common with Y. - Milan Janjic, Dec 28 2007

These numbers, starting with 3, are the denominators of the power series f(x) = (1-x)^2 * log(1/(1-x)), if the numerators are kept at 1. This sequence of denominators starts at the term x^3/3. - Miklos Bona, Feb 18 2009

a(n) is the number of triples (w,x,y) having all terms in {0..n} and satisfying at least one of the inequalities x+y < w, y+w < x, w+x < y. - Clark Kimberling, Jun 14 2012

From Martin Licht, Dec 04 2016: (Start)

Let b(n) = (n+1)(n+2)(n+3)/2 (the same sequence, but with a different offset). Then (see Arnold et al., 2006):

b(n) is the dimension of the Nédélec space of the second kind of polynomials of order n over a tetrahedron.

b(n-1) is the dimension of the curl-conforming Nédélec space of the first kind of polynomials of order n with tangential boundary conditions over a tetrahedron.

b(n) is the dimension of the divergence-conforming Nédélec space of the first kind of polynomials of order n with normal boundary conditions over a tetrahedron. (End)

After a(0), the digital root has period 9: repeat [3, 3, 3, 6, 6, 6, 9, 9, 9]. - Peter M. Chema, Jan 19 2017

From Omar E. Pol, Aug 20 2011: (Start)

Cellular automaton on the hexagonal net. The sequence gives the number of "ON" cells in the structure after n-th stage. A007310 gives the first differences. For a definition without words see the illustration of initial terms in the example section. Note that the cells become intermittent. A083577 gives the primes of this sequences.

A033581 and A003154 interleaved.

Row sums of an infinite square array T(n,k) in which column k lists 2*k-1 zeros followed by the numbers A008458 (see example). (End)

Sequence found by reading the line from 0, in the direction 0, 1, ... and the same line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Main axis perpendicular to A045943 in the same spiral. - Omar E. Pol, Sep 08 2011

Deleting the least significant digit yields the (n-1)-st triangular number: a(n) = 10*A000217(n-1) + 1. - Amarnath Murthy, Dec 11 2003

All divisors of a(n) are congruent to 1 or -1, modulo 10; that is, they end in the decimal digit 1 or 9. Proof: If p is an odd prime different from 5 then 5n^2 - 5n + 1 == 0 (mod p) implies 25(2n - 1)^2 == 5 (mod p), whence p == 1 or -1 (mod 10). - Nick Hobson, Nov 13 2006

Centered decagonal numbers. - Omar E. Pol, Oct 03 2011

The partial sums of this sequence give A004466. - Leo Tavares, Oct 04 2021

The continued fraction expansion of sqrt(5*a(n)) is [5n-3; {2, 2n-2, 2, 10n-6}]. For n=1, this collapses to [2; {4}]. - Magus K. Chu, Sep 12 2022

Numbers m such that 20*m + 5 is a square. Also values of the Fibonacci polynomial y^2 - x*y - x^2 for x = n and y = 3*n - 1. This is a subsequence of A089270. - Klaus Purath, Oct 30 2022

All terms can be written as a difference of two consecutive squares a(n) = A005891(n-1)^2 - A028895(n-1)^2, and they can be represented by the forms (x^2 + 2mxy + (m^2-1)y^2) and (3x^2 + (6m-2)xy + (3m^2-2m)y^2), both of discriminant 4. - Klaus Purath, Oct 17 2023

From Floor van Lamoen, Jul 21 2001: (Start)

Write 1,2,3,4,... in a hexagonal spiral around 0; then a(n) is the sequence found by reading the line from 0 in the direction 0, 6, ...

The spiral begins:

85--84--83--82--81--80

/ \

86 56--55--54--53--52 79

/ / \ \

87 57 33--32--31--30 51 78

/ / / \ \ \

88 58 34 16--15--14 29 50 77

/ / / / \ \ \ \

89 59 35 17 5---4 13 28 49 76

/ / / / / \ \ \ \ \

<==90==60==36==18===6===0 3 12 27 48 75

/ / / / / / / / / /

61 37 19 7 1---2 11 26 47 74

\ \ \ \ / / / /

62 38 20 8---9--10 25 46 73

\ \ \ / / /

63 39 21--22--23--24 45 72

\ \ / /

64 40--41--42--43--44 71

\ /

65--66--67--68--69--70

(End)

If Y is a 4-subset of an n-set X then, for n >= 5, a(n-5) is the number of (n-4)-subsets of X having exactly two elements in common with Y. - Milan Janjic, Dec 28 2007

a(n) is the maximal number of points of intersection of n+1 distinct triangles drawn in the plane. For example, two triangles can intersect in at most a(1) = 6 points (as illustrated in the Star of David configuration). - Terry Stickels (Terrystickels(AT)aol.com), Jul 12 2008

Also sequence found by reading the line from 0, in the direction 0, 6, ... and the same line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Axis perpendicular to A195143 in the same spiral. - Omar E. Pol, Sep 18 2011

Partial sums of A008588. - R. J. Mathar, Aug 28 2014

Also the number of 5-cycles in the (n+5)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017

a(n-4) is the maximum irregularity over all maximal 3-degenerate graphs with n vertices. The extremal graphs are 3-stars (K_3 joined to n-3 independent vertices). (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023

Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative integer lattice point of the orbit, when the cardinality of the orbit is equal to 6720. - Philippe A.J.G. Chevalier, Dec 28 2015

a(n) is the sum of the numerator and denominator of the reduced fraction resulting from the sum A000217(n-2)/A000217(n-1) + A000217(n-1)/A000217(n), n>1. - J. M. Bergot, Jun 10 2017

For n > 1, a(n) is also the number of (not necessarily maximal) cliques in the (n-1)-Andrásfai graph. - Eric W. Weisstein, Nov 29 2017

a(n+1) is the sum of the lengths of all the segments used to draw a square of side n representing the most classic pattern for walls made of 2 X 1 bricks, known as a 1-over-2 pattern, where each joint between neighboring bricks falls over the center of the brick below. - Stefano Spezia, Jun 05 2021