cp's OEIS Frontend

Numbers n such that phi(4n) = phi(3n). - Benoit Cloitre, Aug 06 2003

Or, numbers relatively prime to 2 and 3, or coprime to 6, or having only prime factors >= 5; also known as 5-rough numbers. (Edited by M. F. Hasler, Nov 01 2014: merged with comments from Zak Seidov, Apr 26 2007 and Michael B. Porter, Oct 09 2009)

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 38 ).

Numbers k such that k mod 2 = 1 and (k+1) mod 3 <> 1. - Klaus Brockhaus, Jun 15 2004

Also numbers n such that the sum of the squares of the first n integers is divisible by n, or A000330(n) = n*(n+1)*(2*n+1)/6 is divisible by n. - Alexander Adamchuk, Jan 04 2007

Numbers n such that the sum of squares of n consecutive integers is divisible by n, because A000330(m+n) - A000330(m) = n*(n+1)*(2*n+1)/6 + n*(m^2+n*m+m) is divisible by n independent of m. - Kaupo Palo, Dec 10 2016

A126759(a(n)) = n + 1. - Reinhard Zumkeller, Jun 16 2008

Terms of this sequence (starting from the second term) are equal to the result of the expression sqrt(4!*(k+1) + 1) - but only when this expression yields integral values (that is when the parameter k takes values, which are terms of A144065). - Alexander R. Povolotsky, Sep 09 2008

For n > 1: a(n) is prime if and only if A075743(n-2) = 1; a(2*n-1) = A016969(n-1), a(2*n) = A016921(n-1). - Reinhard Zumkeller, Oct 02 2008

A156543 is a subsequence. - Reinhard Zumkeller, Feb 10 2009

Numbers n such that ChebyshevT(x, x/2) is not an integer (is integer/2). - Artur Jasinski, Feb 13 2010

If 12*k + 1 is a perfect square (k = 0, 2, 4, 10, 14, 24, 30, 44, ... = A152749) then the square root of 12*k + 1 = a(n). - Gary Detlefs, Feb 22 2010

A089128(a(n)) = 1. Complement of A047229(n+1) for n >= 1. See A164576 for corresponding values A175485(a(n)). - Jaroslav Krizek, May 28 2010

Cf. property described by Gary Detlefs in A113801 and in Comment: more generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (with h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 6). Also a(n)^2 - 1 == 0 (mod 12). - Bruno Berselli, Nov 05 2010 - Nov 17 2010

Numbers n such that ( Sum_{k = 1..n} k^14 ) mod n = 0. (Conjectured) - Gary Detlefs, Dec 27 2011

From Peter Bala, May 02 2018: (Start)

The above conjecture is true. Apply Ireland and Rosen, Proposition 15.2.2. with m = 14 to obtain the congruence 6*( Sum_{k = 1..n} k^14 )/n = 7 (mod n), true for all n >= 1. Suppose n is coprime to 6, then 6 is a unit in Z/nZ, and it follows from the congruence that ( Sum_{k = 1..n} k^14 )/n is an integer. On the other hand, if either 2 divides n or 3 divides n then the congruence shows that ( Sum_{k = 1..n} k^14 )/n cannot be integral. (End)

A126759(a(n)) = n and A126759(m) < n for m < a(n). - Reinhard Zumkeller, May 23 2013

(a(n-1)^2 - 1)/24 = A001318(n), the generalized pentagonal numbers. - Richard R. Forberg, May 30 2013

Numbers k for which A001580(k) is divisible by 3. - Bruno Berselli, Jun 18 2014

Numbers n such that sigma(n) + sigma(2n) = sigma(3n). - Jahangeer Kholdi and Farideh Firoozbakht, Aug 15 2014

a(n) are values of k such that Sum_{m = 1..k-1} m*(k-m)/k is an integer. Sums for those k are given by A062717. Also see Detlefs formula below based on A062717. - Richard R. Forberg, Feb 16 2015

a(n) are exactly those positive integers m such that the sequence b(n) = n*(n + m)*(n + 2*m)/6 is integral, and also such that the sequence c(n) = n*(n + m)*(n + 2*m)*(n + 3*m)/24 is integral. Cf. A007775. - Peter Bala, Nov 13 2015

Along with 2, these are the numbers k such that the k-th Fibonacci number is coprime to every Lucas number. - Clark Kimberling, Jun 21 2016

This sequence is the Engel expansion of 1F2(1; 5/6, 7/6; 1/36) + 1F2(1; 7/6, 11/6; 1/36)/5. - Benedict W. J. Irwin, Dec 16 2016

The sequence a(n), n >= 4 is generated by the successor of the pair of polygonal numbers {P_s(4) + 1, P_(2*s - 1)(3) + 1}, s >= 3. - Ralf Steiner, May 25 2018

The asymptotic density of this sequence is 1/3. - Amiram Eldar, Oct 18 2020

Also, the only vertices in the odd Collatz tree A088975 that are branch values to other odd nodes t == 1 (mod 2) of A005408. - Heinz Ebert, Apr 14 2021

From Flávio V. Fernandes, Aug 01 2021: (Start)

For any two terms j and k, the product j*k is also a term (the same property as p^n and smooth numbers).

From a(2) to a(phi(A033845(n))), or a((A033845(n))/3), the terms are the totatives of the A033845(n) itself. (End)

Also orders n for which cyclic and semicyclic diagonal Latin squares exist (see A123565 and A342990). - Eduard I. Vatutin, Jul 11 2023

If k is in the sequence, then k*2^m + 3 is also in the sequence, for all m > 0. - Jules Beauchamp, Aug 29 2024

a(n) is the size of largest conjugacy class in D_2n, the dihedral group with 2n elements. - Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002

a(n+1) is the composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type D_4 (quaternion group). - Paul Boddington, Oct 23 2003

For n > 1, a(n) is the greatest common divisor of all permutations of {0, 1, ..., n} treated as base n + 1 integers. - David Scambler, Nov 08 2006 (see the Mathematics Stack Exchange link below).

From Dimitrios Choussos (choussos(AT)yahoo.de), May 11 2009: (Start)

Sequence A075888 and the above sequence are fitting together.

First 2 entries of this sequence have to be taken out.

In some cases two three or more sequenced entries of this sequence have to be added together to get the next entry of A075888.

Example: Sequences begin with 1, 3, 2, 5, 3, 7, 4, 9 (4 + 9 = 13, the next entry in A075888).

But it works out well up to primes around 50000 (haven't tested higher ones).

As A075888 gives a very regular graph. There seems to be a regularity in the primes. (End)

Starting with 1 = triangle A115359 * [1, 2, 3, ...]. - Gary W. Adamson, Nov 27 2009

From Gary W. Adamson, Dec 11 2009: (Start)

Let M be an infinite lower triangular matrix with (1, 1, 1, 0, 0, 0, ...) in every column, shifted down twice. This sequence starting with 1 = M * (1, 2, 3, ...)

M =

1;

1, 0;

1, 1, 0;

0, 1, 0, 0;

0, 1, 1, 0, 0;

0, 0, 1, 0, 0, 0;

0, 0, 1, 1, 0, 0, 0;

...

A026741 = M * (1, 2, 3, ...); but A002487 = lim_{n->infinity} M^n, a left-shifted vector considered as a sequence. (End)

A particular case of sequence for which a(n+3) = (a(n+2) * a(n+1)+q)/a(n) for every n > n0. Here n0 = 1 and q = -1. - Richard Choulet, Mar 01 2010

For n >= 2, a(n+1) is the smallest m such that s_n(2*m*(n-1))/(n-1) is even, where s_b(c) is the sum of digits of c in base b. - Vladimir Shevelev, May 02 2011

A001477 and A005408 interleaved. - Omar E. Pol, Aug 22 2011

Numerator of n/((n-1)*(n-2)). - Michael B. Porter, Mar 18 2012

Number of odd terms of n-th row in the triangles A162610 and A209297. - Reinhard Zumkeller, Jan 19 2013

For n >= 3, a(n) is the periodic of integer of spiral length ratio of spiral that have (n-1) circle centers. See illustration in links. - Kival Ngaokrajang, Dec 28 2013

This is the sequence of Lehmer numbers u_n(sqrt(R), Q) with the parameters R = 4 and Q = 1. It is a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all natural numbers n and m. Cf. A005013 and A108412. - Peter Bala, Apr 18 2014

The sequence of convergents of the 2-periodic continued fraction [0; 1, -4, 1, -4, ...] = 1/(1 - 1/(4 - 1/(1 - 1/(4 - ...)))) = 2 begins [0/1, 1/1, 4/3, 3/2, 8/5, 5/3, 12/7, ...]. The present sequence is the sequence of denominators; the sequence of numerators of the continued fraction convergents [0, 1, 4, 3, 8, 5, 12, ...] is A022998, also a strong divisibility sequence. - Peter Bala, May 19 2014

For n >= 3, (a(n-2)/a(n))*Pi = vertex angle of a regular n-gon. See illustration in links. - Kival Ngaokrajang, Jul 17 2014

For n > 1, the numerator of the harmonic mean of the first n triangular numbers. - Colin Barker, Nov 13 2014

The difference sequence is a permutation of the integers. - Clark Kimberling, Apr 19 2015

From Timothy Hopper, Feb 26 2017: (Start)

Given the function a(n, p) = n/p if n mod p = 0, else n, then a possible formula is: a(n, p) = n*(1 + (p-1)*((n^(p-1)) mod p))/p, p prime, (n^(p-1)) mod p = 1, n not divisible by p. (Fermat's Little Theorem). Examples: p = 2; a(n), p = 3; A051176(n), p = 5; A060791(n), p = 7; A106608(n).

Conjecture: lcm(n, p) = p*a(n, p), gcd(n, p) = n/a(n, p). (End)

Let r(n) = (a(n+1) + 1)/a(n+1) if n mod 2 = 1, a(n+1)/(a(n+1) + 2) otherwise; then lim_{k->oo} 2^(k+2) * Product_{n=0..k} r(n)^(k-n) = Pi. - Dimitris Valianatos, Mar 22 2021

Number of integers k from 1 to n such that gcd(n,k) is odd. - Amiram Eldar, May 18 2025

For n >= 1, a(n) is also the number of conjugacy classes in the automorphism group of the n-dimensional hypercube. This automorphism group is the wreath product of the cyclic group C_2 and the symmetric group S_n, its order is in sequence A000165. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 04 2001

Also, number of noncongruent matrices in GL_n(Z): each Jordan block can only have +1 or -1 on the diagonal. - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 15 2004

a(n) = Sum (k(1)+1)*(k(2)+1)*...*(k(n)+1), where the sum is taken over all (k(1),k(2),...,k(n)) such that k(1)+2*k(2)+...+n*k(n) = n, k(i)>=0, i=1..n, cf. A104510, A077285. - Vladeta Jovovic, Apr 21 2005

Convolution of partition numbers (A000041) with itself. - Graeme McRae, Jun 07 2006

Number of one-to-one partial endofunctions on n unlabeled points. Connected components are either cycles or "lines", hence two for each size. - Franklin T. Adams-Watters, Dec 28 2006

Equals A000716: (1, 3, 9, 22, 561, 108, ...) convolved with A010815. A000716 = the number of partitions of n into parts of 3 kinds = the Euler transform of [3,3,3,...]. - Gary W. Adamson, Oct 26 2008

Paraphrasing the g.f.: 1 + 2x + 5x^2 + ... = s(x) * s(x^2) * s(x^3) * s(x^4) * ...; where s(x) = 1 + 2x + 3x^2 + 4x^3 + ... is (up to a factor x) the g.f. of A000027. - Gary W. Adamson, Apr 01 2010

Also equals number of partitions of 2n in which the odd parts appear as many times in even as in odd positions. - Wouter Meeussen, Apr 17 2013

Also number of ordered pairs (R,S) with R a partition of r, S a partition of s, and r+s=n; see example. This corresponds to the formula a(n) = sum(r+s==n, p(r)*p(s) ) = Sum_{k=0..n} p(k)*p(n-k). - Joerg Arndt, Apr 29 2013

Also the number of all multi-graphs with exactly n-edges and with vertex degrees 1 or 2. - Ebrahim Ghorbani, Dec 02 2013

If one decomposes k-permutations into cycles and so-called paths, the number of different type of decompositions equals to a(k); see the paper by Chen, Ghorbani, and Wong. - Ebrahim Ghorbani, Dec 02 2013

Let T(n,k) be the number of partitions of n having parts 1 through k of two kinds, with T(n,0) = A000041(n), the number of partitions of n. Then a(n) = T(n,0) + T(n-1,1) + T(n-2,2) + T(n-3,3) + ... - Gregory L. Simay, May 18 2019

Also the number of orbits of projections in the partition monoid P_n under conjugation by permutations. - James East, Jul 21 2020

Number of edges in the join of the complete graph and the cycle graph, both of order n, K_n * C_n. - Roberto E. Martinez II, Jan 07 2002

Also number of cards to build an n-tier house of cards. - Martin Wohlgemuth, Aug 11 2002

The modular form Delta(q) = q*Product_{n>=1} (1-q^n)^24 = q*(1 + Sum_{n>=1} (-1)^n*(q^(n*(3*n-1)/2)+q^(n*(3*n+1)/2)))^24 = q*(1 + Sum_{n>=1} A033999(n)*(q^A000326(n)+q^a(n)))^24. - Jonathan Vos Post, Mar 15 2006

Row sums of triangle A134403.

Bisection of A001318. - Omar E. Pol, Aug 22 2011

Sequence found by reading the line from 0 in the direction 0, 7, ... and the line from 2 in the direction 2, 15, ... in the square spiral whose vertices are the generalized pentagonal numbers, A001318. - Omar E. Pol, Sep 08 2011

A general formula for the n-th second k-gonal number is given by T(n, k) = n*((k-2)*n+k-4)/2, n>=0, k>=5. - Omar E. Pol, Aug 04 2012

Partial sums give A006002. - Denis Borris, Jan 07 2013

A002260 is the following array A read by antidiagonals:

0, 1, 2, 3, 4, 5, ...

0, 1, 2, 3, 4, 5, ...

0, 1, 2, 3, 4, 5, ...

0, 1, 2, 3, 4, 5, ...

0, 1, 2, 3, 4, 5, ...

0, 1, 2, 3, 4, 5, ...

and a(n) is the hook sum: Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - R. J. Mathar, Jun 30 2013

From Klaus Purath, May 13 2021: (Start)

This sequence and A000326 provide all integers m such that 24*m + 1 is a square. The union of the two sequences is A001318.

If A is a sequence satisfying the recurrence t(n) = 3*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 2 or A(0) = -1, A(1) = n - 1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i + n^2 for i>0. (End)

a(n+1) is the number of Dyck paths of size (3,3n+2), i.e., the number of NE lattice paths from (0,0) to (3,3n+2) which stay above the line connecting these points. - Harry Richman, Jul 13 2021

Binomial transform of [0, 2, 3, 0, 0, 0, ...], being a(n) = 2*binomial(n,1) + 3*binomial(n,2). a(3) = 15 = [0, 2, 3, 0] dot [1, 3, 3, 1] = [0 + 6 + 9 + 0]. - Gary W. Adamson, Dec 17 2022

a(n) is the sum of longest side length of all nondegenerate integer-sided triangles with shortest side length n and middle side length (n + 1), n > 0. - Torlach Rush, Feb 04 2024

Also, 3 times triangular numbers, a(n) = 3*A000217(n).

In the 24-bit RGB color cube, the number of color-lattice-points in r+g+b = n planes at n < 256 equals the triangular numbers. For n = 256, ..., 765 the number of legitimate color partitions is less than A000217(n) because {r,g,b} components cannot exceed 255. For n = 256, ..., 511, the number of non-color partitions are computable with A045943(n-255), while for n = 512, ..., 765, the number of color points in r+g+b planes equals A000217(765-n). - Labos Elemer, Jun 20 2005

If a 3-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-3) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007

a(n) is also the smallest number that may be written both as the sum of n-1 consecutive positive integers and n consecutive positive integers. - Claudio Meller, Oct 08 2010

For n >= 3, a(n) equals 4^(2+n)*Pi^(1 - n) times the coefficient of zeta(3) in the following integral with upper bound Pi/4 and lower bound 0: int x^(n+1) tan x dx. - John M. Campbell, Jul 17 2011

The difference a(n)-a(n-1) = 3*n, for n >= 1. - Stephen Balaban, Jul 25 2011 [Comment clarified by N. J. A. Sloane, Aug 01 2024]

Sequence found by reading the line from 0, in the direction 0, 3, ..., and the same line from 0, in the direction 0, 9, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. This is one of the orthogonal axes of the spiral; the other is A032528. - Omar E. Pol, Sep 08 2011

A005449(a(n)) = A000332(3n + 3) = C(3n + 3, 4), a second pentagonal number of triangular matchstick number index number. Additionally, a(n) - 2n is a pentagonal number (A000326). - Raphie Frank, Dec 31 2012

Sum of the numbers from n to 2n. - Wesley Ivan Hurt, Nov 24 2015

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 5376 or 17920 or 20160. - Philippe A.J.G. Chevalier, Dec 28 2015

Also the number of 4-cycles in the (n+4)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017

Number of terms less than 10^k, k=0,1,2,3,...: 1, 3, 8, 26, 82, 258, 816, 2582, 8165, 25820, 81650, 258199, 816497, 2581989, 8164966, ... - Muniru A Asiru, Jan 24 2018

Numbers of the form 3*m*(2*m + 1) for m = 0, -1, 1, -2, 2, -3, 3, ... - Bruno Berselli, Feb 26 2018

Partial sums of A008585. - Omar E. Pol, Jun 20 2018

Column 1 of A273464. (Number of ways to select a unit lozenge inside an isosceles triangle of side length n; all vertices on a hexagonal lattice.) - R. J. Mathar, Jul 10 2019

Total number of pips in the n-th suit of a double-n domino set. - Ivan N. Ianakiev, Aug 23 2020

Numbers of the form 3*m^2+2*m, m an integer.

3*a(n) + 1 is a perfect square.

a(n) mod 10 belongs to a periodic sequence: 0, 1, 5, 8, 6, 1, 3, 0, 6, 5, 5, 6, 0, 3, 1, 6, 8, 5, 1, 0. - Mohamed Bouhamida, Sep 04 2009

A089801 is the characteristic function. - R. J. Mathar, Oct 07 2011

Exponents of powers of q in one form of the quintuple product identity. (-x^-2 + 1) * q^0 + (x^-3 - x) * q^1 + (-x^-5 + x^3) * q^5 + (x^-6 - x^4) * q^8 + ... = Sum_{n>=0} q^(3*n^2 + 2*n) * (x^(3*n) - x^(-3*n - 2)) = Product_{k>0} (1 - x * q^(2*k - 1)) * (1 - x^-1 * q^(2*k - 1)) * (1 - q^(2*k)) * (1 - x^2 * q^(4*k)) * (1 - x^-2 * q^(4*k - 4)). - Michael Somos, Dec 21 2011

The offset 0 would also be valid here, all other entries of generalized k-gonal numbers have offset 0 (see cross references). - Omar E. Pol, Jan 12 2013

Also, x values of the Diophantine equation x(x+3)+(x+1)(x+2) = (x+y)^2+(x-y)^2. - Bruno Berselli, Mar 29 2013

Numbers n such that Sum_{i=1..n} 2*i*(n-i)/n is an integer (the addend is the harmonic mean of i and n-i). - Wesley Ivan Hurt, Sep 14 2014

Equivalently, integers of the form m*(m+2)/3 (nonnegative values of m are listed in A032766). - Bruno Berselli, Jul 18 2016

Exponents of q in the expansion of Sum_{n >= 0} ( q^n * Product_{k = 1..n} (1 - q^(2*k-1)) ) = 1 + q - q^5 - q^8 + q^16 + q^21 - - + + .... - Peter Bala, Dec 03 2020

Exponents of q in the expansion of Product_{n >= 1} (1 - q^(6*n))*(1 + q^(6*n-1))*(1 + q^(6*n-5)) = 1 + q + q^5 + q^8 + q^16 + q^21 + .... - Peter Bala, Dec 09 2020

Exponents of q in the expansion of Product_{n >= 1} (1 - q^n)^2*(1 - q^(4*n))^2 /(1 - q^(2*n)) = 1 - 2*q + 4*q^5 - 5*q^8 + 7*q^16 - + ... (a consequence of the quintuple product identity). The series coefficients are a signed version of A001651. - Peter Bala, Feb 16 2021

From Peter Bala, Nov 26 2024: (Start)

Apart from the first two terms, the exponents of q in the expansion of Sum_{n >= 1} q^(3*n+2) * (Product_{k = 2..n} 1 - q^(2*k-1)) = q^5 + q^8 - q^16 - q^21 + + - - ... (in Andrews, equation 8, replace q with q^2 and set x = q).

Exponents of q^2 in the expansion of Sum_{n >= 0} q^n / (Product_{k = 1..n+1 } 1 + q^(2*k-1)) = 1 + (q^2)^1 - (q^2)^5 - (q^2)^8 + (q^2)^16 + (q^2)^21 - - + + ... (Chen, equation 22). (End)

This is the limiting probability that a large random binary matrix is nonsingular (cf. A002884).

This constant is very close to 2^(13/24) * sqrt(Pi/log(2)) / exp(Pi^2/(6*log(2))) = 0.288788095086602421278899775042039398383022429351580356839... - Vaclav Kotesovec, Aug 21 2018

This constant is irrational (see Penn link). - Paolo Xausa, Dec 09 2024

Suppose that S={-n,...,0,...,n} and that f(w,x,y,n) is a function, where w,x,y are in S. The number of ordered triples (w,x,y) satisfying f(w,x,y,n)=0, regarded as a function of n, is a sequence t of nonnegative integers. Sequences such as t/4 may also be integer sequences for all except certain initial values of n. In the following guide, such sequences are indicated in the related sequences column and may be included in the corresponding Mathematica programs.

...

sequence... f(w,x,y,n) ..... related sequences

A211415 ... w^2+x*y-1 ...... t+2, t/4, (t/4-1)/4

A211422 ... w^2+x*y ........ (t-1)/8, A120486

A211423 ... w^2+2x*y ....... (t-1)/4

A211424 ... w^2+3x*y ....... (t-1)/4

A211425 ... w^2+4x*y ....... (t-1)/4

A211426 ... 2w^2+x*y ....... (t-1)/4

A211427 ... 3w^2+x*y ....... (t-1)/4

A211428 ... 2w^2+3x*y ...... (t-1)/4

A211429 ... w^3+x*y ........ (t-1)/4

A211430 ... w^2+x+y ........ (t-1)/2

A211431 ... w^3+(x+y)^2 .... (t-1)/2

A211432 ... w^2-x^2-y^2 .... (t-1)/8

A003215 ... w+x+y .......... (t-1)/2, A045943

A202253 ... w+2x+3y ........ (t-1)/2, A143978

A211433 ... w+2x+4y ........ (t-1)/2

A211434 ... w+2x+5y ........ (t-1)/4

A211435 ... w+4x+5y ........ (t-1)/2

A211436 ... 2w+3x+4y ....... (t-1)/2

A211435 ... 2w+3x+5y ....... (t-1)/2

A211438 ... w+2x+2y ....... (t-1)/2, A118277

A001844 ... w+x+2y ......... (t-1)/4, A000217

A211439 ... w+3x+3y ........ (t-1)/2

A211440 ... 2w+3x+3y ....... (t-1)/2

A028896 ... w+x+y-1 ........ t/6, A000217

A211441 ... w+x+y-2 ........ t/3, A028387

A182074 ... w^2+x*y-n ...... t/4, A028387

A000384 ... w+x+y-n

A000217 ... w+x+y-2n

A211437 ... w*x*y-n ........ t/4, A007425

A211480 ... w+2x+3y-1

A211481 ... w+2x+3y-n

A211482 ... w*x+w*y+x*y-w*x*y

A211483 ... (n+w)^2-x-y

A182112 ... (n+w)^2-x-y-w

...

For the following sequences, S={1,...,n}, rather than

{-n,...,0,...n}. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.

A132188 ... w^2-x*y

A211506 ... w^2-x*y-n

A211507 ... w^2-x*y+n

A211508 ... w^2+x*y-n

A211509 ... w^2+x*y-2n

A211510 ... w^2-x*y+2n

A211511 ... w^2-2x*y ....... t/2

A211512 ... w^2-3x*y ....... t/2

A211513 ... 2w^2-x*y ....... t/2

A211514 ... 3w^2-x*y ....... t/2

A211515 ... w^3-x*y

A211516 ... w^2-x-y

A211517 ... w^3-(x+y)^2

A063468 ... w^2-x^2-y^2 .... t/2

A000217 ... w+x-y

A001399 ... w-2x-3y

A211519 ... w-2x+3y

A008810 ... w+2x-3y

A001399 ... w-2x-3y

A008642 ... w-2x-4y

A211520 ... w-2x+4y

A211521 ... w+2x-4y

A000115 ... w-2x-5y

A211522 ... w-2x+5y

A211523 ... w+2x-5y

A211524 ... w-3x-5y

A211533 ... w-3x+5y

A211523 ... w+3x-5y

A211535 ... w-4x-5y

A211536 ... w-4x+5y

A008812 ... w+4x-5y

A055998 ... w+x+y-2n

A074148 ... 2w+x+y-2n

A211538 ... 2w+2x+y-2n

A211539 ... 2w+2x-y-2n

A211540 ... 2w-3x-4y

A211541 ... 2w-3x+4y

A211542 ... 2w+3x-4y

A211543 ... 2w-3x-5y

A211544 ... 2w-3x+5y

A008812 ... 2w+3x-5y

A008805 ... w-2x-2y (repeated triangular numbers)

A001318 ... w-2x+2y

A000982 ... w+x-2y

A211534 ... w-3x-3y

A211546 ... w-3x+3y (triply repeated triangular numbers)

A211547 ... 2w-3x-3y (triply repeated squares)

A082667 ... 2w-3x+3y

A055998 ... w-x-y+2

A001399 ... w-2x-3y+1

A108579 ... w-2x-3y+n

...

Next, S={-n,...-1,1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated inequality. If f(w,x,y,n) is linear in w,x,y,n, then the sequence is a linear recurrence sequence.

A211545 ... w+x+y>0; recurrence degree: 4

A211612 ... w+x+y>=0

A211613 ... w+x+y>1

A211614 ... w+x+y>2

A211615 ... |w+x+y|<=1

A211616 ... |w+x+y|<=2

A211617 ... 2w+x+y>0; recurrence degree: 5

A211618 ... 2w+x+y>1

A211619 ... 2w+x+y>2

A211620 ... |2w+x+y|<=1

A211621 ... w+2x+3y>0

A211622 ... w+2x+3y>1

A211623 ... |w+2x+3y|<=1

A211624 ... w+2x+2y>0; recurrence degree: 6

A211625 ... w+3x+3y>0; recurrence degree: 8

A211626 ... w+4x+4y>0; recurrence degree: 10

A211627 ... w+5x+5y>0; recurrence degree: 12

A211628 ... 3w+x+y>0; recurrence degree: 6

A211629 ... 4w+x+y>0; recurrence degree: 7

A211630 ... 5w+x+y>0; recurrence degree: 8

A211631 ... w^2>x^2+y^2; all terms divisible by 8

A211632 ... 2w^2>x^2+y^2; all terms divisible by 8

A211633 ... w^2>2x^2+2y^2; all terms divisible by 8

...

Next, S={1,...,n}, and the sequence counts the cases (w,x,y) satisfying the indicated relation.

A211634 ... w^2<=x^2+y^2

A211635 ... w^2A211790

A211636 ... w^2>=x^2+y^2

A211637 ... w^2>x^2+y^2

A211638 ... w^2+x^2+y^2

A211639 ... w^2+x^2+y^2<=n

A211640 ... w^2+x^2+y^2>n

A211641 ... w^2+x^2+y^2>=n

A211642 ... w^2+x^2+y^2<2n

A211643 ... w^2+x^2+y^2<=2n

A211644 ... w^2+x^2+y^2>2n

A211645 ... w^2+x^2+y^2>=2n

A211646 ... w^2+x^2+y^2<3n

A211647 ... w^2+x^2+y^2<=3n

A063691 ... w^2+x^2+y^2=n

A211649 ... w^2+x^2+y^2=2n

A211648 ... w^2+x^2+y^2=3n

A211650 ... w^3A211790

A211651 ... w^3>x^3+y^3; see Comments at A211790

A211652 ... w^4A211790

A211653 ... w^4>x^4+y^4; see Comments at A211790