cp's OEIS Frontend

Suppose that a number n has base-b digits b(m), b(m-1), ..., b(0). The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1). The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b). Guide to related sequences and partitions of the natural numbers:

***

Base b {DV(n,b)} {UV(n,b)} {TV(n,b)}

2 A033264 A037800 A037834

3 A037853 A037844 A037835

4 A037854 A037845 A037836

5 A037855 A037846 A037837

6 A037856 A037847 A037838

7 A037857 A037848 A037839

8 A037858 A037849 A037840

9 A037859 A037850 A037841

10 A037860 A037851 A297330

11 A297231 A297232 A297233

12 A297234 A297235 A297236

13 A297237 A297238 A297239

14 A297240 A297241 A297242

15 A297243 A297244 A297245

16 A297246 A297247 A297247

For each b, let u = {n : UV(n,b) < DV(n,b)}, e = {n : UV(n,b) = DV(n,b)}, and d = {n : UV(n,b) > DV(n,b)}. The sets u,e,d partition the natural numbers. A guide to the matching sequences for u, e, d follows:

***

Base b Sequence u Sequence e Sequence d

2 A005843 A005408 (none)

3 A297249 A297250 A297251

4 A297252 A297253 A297254

5 A297255 A297256 A297257

6 A297258 A297259 A297260

7 A297261 A297262 A297263

8 A297264 A297265 A297266

9 A297267 A297268 A297269

10 A297270 A297271 A297272

11 A297273 A297274 A297275

12 A297276 A297277 A297278

13 A297279 A297280 A297281

14 A297282 A297283 A297284

15 A297285 A297286 A297287

16 A297288 A297289 A297290

Not a duplicate of A151950: e.g., a(100)=1 but A151950(100)=11. - Robert Israel, Feb 06 2018

The number of rounds in a round-robin tournament with n competitors. - A. Timothy Royappa, Aug 13 2011

Diagonal sums of number triangle A113126. - Paul Barry, Oct 14 2005

When partitioning a convex n-gon by all the diagonals, the maximum number of sides in resulting polygons is 2*floor(n/2)+1 = a(n-1) (from Moscow Olympiad problem 1950). - Tanya Khovanova, Apr 06 2008

The inverse values of the coefficients in the series expansion of f(x) = (1/2)*(1+x)*log((1+x)/(1-x)) lead to this sequence; cf. A098557. - Johannes W. Meijer, Nov 12 2009

From Reinhard Zumkeller, Dec 05 2009: (Start)

First differences: A010673; partial sums: A000982;

A059329(n) = Sum_{k = 0..n} a(k)*a(n-k);

A167875(n) = Sum_{k = 0..n} a(k)*A005408(n-k);

A171218(n) = Sum_{k = 0..n} a(k)*A005843(n-k);

A008794(n+2) = Sum_{k = 0..n} a(k)*A059841(n-k). (End)

Dimension of the space of weight 2n+4 cusp forms for Gamma_0(5). - Michael Somos, May 29 2013

For n > 4: a(n) = A230584(n) - A230584(n-2). - Reinhard Zumkeller, Feb 10 2015

The arithmetic function v+-(n,2) as defined in A290988. - Robert Price, Aug 22 2017

For n > 0, also the chromatic number of the (n+1)-triangular (Johnson) graph. - Eric W. Weisstein, Nov 17 2017

a(n-1), for n >= 1, is also the upper bound a_{up}(b), where b = 2*n + 1, in the first (top) row of the complete coach system Sigma(b) of Hilton and Pedersen [H-P]. All odd numbers <= a_{up}(b) of the smallest positive restricted residue system of b appear once in the first rows of the c(2*n+1) = A135303(n) coaches. If b is an odd prime a_{up}(b) is the maximum. See a comment in the proof of the quasi-order theorem of H-P, on page 263 ["Furthermore, every possible a_i < b/2 ..."]. For an example see below. - Wolfdieter Lang, Feb 19 2020

Satisfies the nested recurrence a(n) = a(a(n-2)) + 2*a(n-a(n-1)) with a(0) = a(1) = 1. Cf. A004001. - Peter Bala, Aug 30 2022

The binomial transform is 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560,.. (see A057711). - R. J. Mathar, Feb 25 2023

This is permutation of natural numbers larger than 1.

From Antti Karttunen, Dec 19 2014: (Start)

If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.

For navigating in this array:

A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.

A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.

First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).

(End)

The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015

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

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

This array can be obtained by taking every second column from array A242378, starting from its column 2.

Permutation of natural numbers larger than 1.

The terms on row n are all divisible by n-th prime, A000040(n).

Each column is strictly growing, and the terms in the same column have the same prime signature.

A055396(n) gives the row number of row where n occurs,

and A246277(n) gives its column number, both starting from 1.

From Antti Karttunen, Jan 03 2015: (Start)

A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.

If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252752 gives the inverse permutation. See also A246276.

(End)

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).

Taking p(S) = 1 - S gives the INVERT transform of s, so that p-INVERT is a generalization of the INVERT transform (e.g., A033453).

Guide to p-INVERT sequences using p(S) = 1 - S - S^2:

t(A000012) = t(1,1,1,1,1,1,1,...) = A001906

t(A000290) = t(1,4,9,16,25,36,...) = A289779

t(A000027) = t(1,2,3,4,5,6,7,8,...) = A289780

t(A000045) = t(1,2,3,5,8,13,21,...) = A289781

t(A000032) = t(2,1,3,4,7,11,14,...) = A289782

t(A000244) = t(1,3,9,27,81,243,...) = A289783

t(A000302) = t(1,4,16,64,256,...) = A289784

t(A000351) = t(1,5,25,125,625,...) = A289785

t(A005408) = t(1,3,5,7,9,11,13,...) = A289786

t(A005843) = t(2,4,6,8,10,12,14,...) = A289787

t(A016777) = t(1,4,7,10,13,16,...) = A289789

t(A016789) = t(2,5,8,11,14,17,...) = A289790

t(A008585) = t(3,6,9,12,15,18,...) = A289795

t(A000217) = t(1,3,6,10,15,21,...) = A289797

t(A000225) = t(1,3,7,15,31,63,...) = A289798

t(A000578) = t(1,8,27,64,625,...) = A289799

t(A000984) = t(1,2,6,20,70,252,...) = A289800

t(A000292) = t(1,4,10,20,35,56,...) = A289801

t(A002620) = t(1,2,4,6,9,12,16,...) = A289802

t(A001906) = t(1,3,8,21,55,144,...) = A289803

t(A001519) = t(1,1,2,5,13,34,...) = A289804

t(A103889) = t(2,1,4,3,6,5,8,7,,...) = A289805

t(A008619) = t(1,1,2,2,3,3,4,4,...) = A289806

t(A080513) = t(1,2,2,3,3,4,4,5,...) = A289807

t(A133622) = t(1,2,1,3,1,4,1,5,...) = A289809

t(A000108) = t(1,1,2,5,14,42,...) = A081696

t(A081696) = t(1,1,3,9,29,97,...) = A289810

t(A027656) = t(1,0,2,0,3,0,4,0,5...) = A289843

t(A175676) = t(1,0,0,2,0,0,3,0,...) = A289844

t(A079977) = t(1,0,1,0,2,0,3,...) = A289845

t(A059841) = t(1,0,1,0,1,0,1,...) = A289846

t(A000040) = t(2,3,5,7,11,13,...) = A289847

t(A008578) = t(1,2,3,5,7,11,13,...) = A289828

t(A000142) = t(1!, 2!, 3!, 4!, ...) = A289924

t(A000201) = t(1,3,4,6,8,9,11,...) = A289925

t(A001950) = t(2,5,7,10,13,15,...) = A289926

t(A014217) = t(1,2,4,6,11,17,29,...) = A289927

t(A000045*) = t(0,1,1,2,3,5,...) = A289975 (* indicates prepended 0's)

t(A000045*) = t(0,0,1,1,2,3,5,...) = A289976

t(A000045*) = t(0,0,0,1,1,2,3,5,...) = A289977

t(A290990*) = t(0,1,2,3,4,5,...) = A290990

t(A290990*) = t(0,0,1,2,3,4,5,...) = A290991

t(A290990*) = t(0,0,01,2,3,4,5,...) = A290992

a(n) is also the volume of a special stepped pyramid with n levels related to the symmetric representation of sigma. Note that starting at the top of the pyramid, the total area of the horizontal regions at the n-th level is equal to A239050(n), and the total area of the vertical regions at the n-th level is equal to 8*n.

From Omar E. Pol, Sep 19 2015: (Start)

Also, consider that the area of the central square in the top of the pyramid is equal to 1, so the total area of the horizontal regions at the n-th level starting from the top is equal to sigma(n) = A000203(n), and the total area of the vertical regions at the n-th level is equal to 2*n.

Also note that this stepped pyramid can be constructed with four copies of the stepped pyramid described in A245092 back-to-back (one copy in every quadrant). (End)

From Omar E. Pol, Jan 20 2021: (Start)

Convolution of A000203 and the nonzero terms of A008586.

Convolution of A074400 and the nonzero terms of A005843.

Convolution of A340793 and the nonzero terms of A046092.

Convolution of A239050 and A000027.

(End)

a(n) is also the binary rank of the complete graph K(n). - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 07 2009

Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n >= 6, a(n) is the number of (0,1) n X n matrices A <= P^(-1)+I+P having exactly two 1's in every row and column with perA=2. - Vladimir Shevelev, Apr 12 2010

a(n+2) is the number of symmetry allowed, linearly independent terms at n-th order in the series expansion of the (E+A)xe vibronic perturbation matrix, H(Q) (cf. Eisfeld & Viel). - Bradley Klee, Jul 21 2015

The arithmetic function v_2(n,1) as defined in A289187. - Robert Price, Aug 22 2017

For n > 1, also the chromatic number of the n X n white bishop graph. - Eric W. Weisstein, Nov 17 2017

For n > 2, also the maximum vertex degree of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 23 2018

For n >= 2, a(n+2) gives the minimum weight of a Boolean function of algebraic degree at most n-2 whose support contains n linearly independent elements. - Christof Beierle, Nov 25 2019

Same as Pisot sequences E(1, 16), L(1, 16), P(1, 16), T(1, 16). Essentially same as Pisot sequences E(16, 256), L(16, 256), P(16, 256), T(16, 256). See A008776 for definitions of Pisot sequences.

Convolution-square (auto-convolution) of A098430. - R. J. Mathar, May 22 2009

Subsequence of A161441: A160700(a(n)) = 1. - Reinhard Zumkeller, Jun 10 2009

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 16-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

If X is an n-set and Y a fixed (n-3)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007

Bisection of A165157. - Jaroslav Krizek, Sep 05 2009

a(n) is the number of (w,x,y) having all terms in {0,...,n} and w=x+y-1. - Clark Kimberling, Jun 02 2012

Numbers m >= 0 such that 8m+25 is a square. - Bruce J. Nicholson, Jul 26 2017

a(n-1) = 3*(n-1) + (n-1)*(n-2)/2 is the number of connected, loopless, non-oriented, multi-edge vertex-labeled graphs with n edges and 3 vertices. Labeled multigraph analog of A253186. There are 3*(n-1) graphs with the 3 vertices on a chain (3 ways to label the middle graph, n-1 ways to pack edges on one of connections) and binomial(n-1,2) triangular graphs (one way to label the graphs, pack 1 or 2 or ...n-2 on the 1-2 edge, ...). - R. J. Mathar, Aug 10 2017

a(n) is also the number of vertices of the quiver for PGL_{n+1} (see Shen). - Stefano Spezia, Mar 24 2020

Starting from a(2) = 7, this is the 4th column of the array: natural numbers written by antidiagonals downwards. See the illustration by Kival Ngaokrajang and the cross-references. - Andrey Zabolotskiy, Dec 21 2021