A035287 -id:A035287 - cp's OEIS frontend

A005408 The odd numbers: a(n) = 2*n + 1.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 0

N. J. A. Sloane

Leibniz's series: Pi/4 = Sum_{n>=0} (-1)^n/(2n+1) (cf. A072172).
Beginning of the ordering of the natural numbers used in Sharkovski's theorem - see the Cielsielski-Pogoda paper.
The Sharkovski ordering begins with the odd numbers >= 3, then twice these numbers, then 4 times them, then 8 times them, etc., ending with the powers of 2 in decreasing order, ending with 2^0 = 1.
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(6).
Also continued fraction for coth(1) (A073747 is decimal expansion). - Rick L. Shepherd, Aug 07 2002
a(1) = 1; a(n) is the smallest number such that a(n) + a(i) is composite for all i = 1 to n-1. - Amarnath Murthy, Jul 14 2003
Smallest number greater than n, not a multiple of n, but containing it in binary representation. - Reinhard Zumkeller, Oct 06 2003
Numbers n such that phi(2n) = phi(n), where phi is Euler's totient (A000010). - Lekraj Beedassy, Aug 27 2004
Pi*sqrt(2)/4 = Sum_{n>=0} (-1)^floor(n/2)/(2n+1) = 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 ... [since periodic f(x)=x over -Pi < x < Pi = 2(sin(x)/1 - sin(2x)/2 + sin(3x)/3 - ...) using x = Pi/4 (Maor)]. - Gerald McGarvey, Feb 04 2005
For n > 1, numbers having 2 as an anti-divisor. - Alexandre Wajnberg, Oct 02 2005
a(n) = shortest side a of all integer-sided triangles with sides a <= b <= c and inradius n >= 1.
First differences of squares (A000290). - Lekraj Beedassy, Jul 15 2006
The odd numbers are the solution to the simplest recursion arising when assuming that the algorithm "merge sort" could merge in constant unit time, i.e., T(1):= 1, T(n):= T(floor(n/2)) + T(ceiling(n/2)) + 1. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 14 2006
2n-5 counts the permutations in S_n which have zero occurrences of the pattern 312 and one occurrence of the pattern 123. - David Hoek (david.hok(AT)telia.com), Feb 28 2007
For n > 0: number of divisors of (n-1)th power of any squarefree semiprime: a(n) = A000005(A001248(k)^(n-1)); a(n) = A000005(A000302(n-1)) = A000005(A001019(n-1)) = A000005(A009969(n-1)) = A000005(A087752(n-1)). - Reinhard Zumkeller, Mar 04 2007
For n > 2, a(n-1) is the least integer not the sum of < n n-gonal numbers (0 allowed). - Jonathan Sondow, Jul 01 2007
A134451(a(n)) = abs(A134452(a(n))) = 1; union of A134453 and A134454. - Reinhard Zumkeller, Oct 27 2007
Numbers n such that sigma(2n) = 3*sigma(n). - Farideh Firoozbakht, Feb 26 2008
a(n) = A139391(A016825(n)) = A006370(A016825(n)). - Reinhard Zumkeller, Apr 17 2008
Number of divisors of 4^(n-1) for n > 0. - J. Lowell, Aug 30 2008
Equals INVERT transform of A078050 (signed - cf. comments); and row sums of triangle A144106. - Gary W. Adamson, Sep 11 2008
Odd numbers(n) = 2*n+1 = square pyramidal number(3*n+1) / triangular number(3*n+1). - Pierre CAMI, Sep 27 2008
A000035(a(n))=1, A059841(a(n))=0. - Reinhard Zumkeller, Sep 29 2008
Multiplicative closure of A065091. - Reinhard Zumkeller, Oct 14 2008
a(n) is also the maximum number of triangles that n+2 points in the same plane can determine. 3 points determine max 1 triangle; 4 points can give 3 triangles; 5 points can give 5; 6 points can give 7 etc. - Carmine Suriano, Jun 08 2009
Binomial transform of A130706, inverse binomial transform of A001787(without the initial 0). - Philippe Deléham, Sep 17 2009
Also the 3-rough numbers: positive integers that have no prime factors less than 3. - Michael B. Porter, Oct 08 2009
Or n without 2 as prime factor. - Juri-Stepan Gerasimov, Nov 19 2009
Given an L(2,1) labeling l of a graph G, let k be the maximum label assigned by l. The minimum k possible over all L(2,1) labelings of G is denoted by lambda(G). For n > 0, this sequence gives lambda(K_{n+1}) where K_{n+1} is the complete graph on n+1 vertices. - K.V.Iyer, Dec 19 2009
A176271 = odd numbers seen as a triangle read by rows: a(n) = A176271(A002024(n+1), A002260(n+1)). - Reinhard Zumkeller, Apr 13 2010
For n >= 1, a(n-1) = numbers k such that arithmetic mean of the first k positive integers is an integer. A040001(a(n-1)) = 1. See A145051 and A040001. - Jaroslav Krizek, May 28 2010
Union of A179084 and A179085. - Reinhard Zumkeller, Jun 28 2010
For n>0, continued fraction [1,1,n] = (n+1)/a(n); e.g., [1,1,7] = 8/15. - Gary W. Adamson, Jul 15 2010
Numbers that are the sum of two sequential integers. - Dominick Cancilla, Aug 09 2010
Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h and n in A000027), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 4). Also a(n)^2 - 1 == 0 (mod 8). - Bruno Berselli, Nov 17 2010
A004767 = a(a(n)). - Reinhard Zumkeller, Jun 27 2011
A001227(a(n)) = A000005(a(n)); A048272(a(n)) < 0. - Reinhard Zumkeller, Jan 21 2012
a(n) is the minimum number of tosses of a fair coin needed so that the probability of more than n heads is at least 1/2. In fact, Sum_{k=n+1..2n+1} Pr(k heads|2n+1 tosses) = 1/2. - Dennis P. Walsh, Apr 04 2012
A007814(a(n)) = 0; A037227(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2012
1/N (i.e., 1/1, 1/2, 1/3, ...) = Sum_{j=1,3,5,...,infinity} k^j, where k is the infinite set of constants 1/exp.ArcSinh(N/2) = convergents to barover(N). The convergent to barover(1) or [1,1,1,...] = 1/phi = 0.6180339..., whereas c.f. barover(2) converges to 0.414213..., and so on. Thus, with k = 1/phi we obtain 1 = k^1 + k^3 + k^5 + ..., and with k = 0.414213... = (sqrt(2) - 1) we get 1/2 = k^1 + k^3 + k^5 + .... Likewise, with the convergent to barover(3) = 0.302775... = k, we get 1/3 = k^1 + k^3 + k^5 + ..., etc. - Gary W. Adamson, Jul 01 2012
Conjecture on primes with one coach (A216371) relating to the odd integers: iff an integer is in A216371 (primes with one coach either of the form 4q-1 or 4q+1, (q > 0)); the top row of its coach is composed of a permutation of the first q odd integers. Example: prime 19 (q = 5), has 5 terms in each row of its coach: 19: [1, 9, 5, 7, 3] ... [1, 1, 1, 2, 4]. This is interpreted: (19 - 1) = (2^1 * 9), (19 - 9) = (2^1 * 5), (19 - 5) = (2^1 - 7), (19 - 7) = (2^2 * 3), (19 - 3) = (2^4 * 1). - Gary W. Adamson, Sep 09 2012
A005408 is the numerator 2n-1 of the term (1/m^2 - 1/n^2) = (2n-1)/(mn)^2, n = m+1, m > 0 in the Rydberg formula, while A035287 is the denominator (mn)^2. So the quotient a(A005408)/a(A035287) simulates the Hydrogen spectral series of all hydrogen-like elements. - Freimut Marschner, Aug 10 2013
This sequence has unique factorization. The primitive elements are the odd primes (A065091). (Each term of the sequence can be expressed as a product of terms of the sequence. Primitive elements have only the trivial factorization. If the products of terms of the sequence are always in the sequence, and there is a unique factorization of each element into primitive elements, we say that the sequence has unique factorization. So, e.g., the composite numbers do not have unique factorization, because for example 36 = 4*9 = 6*6 has two distinct factorizations.) - Franklin T. Adams-Watters, Sep 28 2013
These are also numbers k such that (k^k+1)/(k+1) is an integer. - Derek Orr, May 22 2014
a(n-1) gives the number of distinct sums in the direct sum {1,2,3,..,n} + {1,2,3,..,n}. For example, {1} + {1} has only one possible sum so a(0) = 1. {1,2} + {1,2} has three distinct possible sums {2,3,4} so a(1) = 3. {1,2,3} + {1,2,3} has 5 distinct possible sums {2,3,4,5,6} so a(2) = 5. - Derek Orr, Nov 22 2014
The number of partitions of 4*n into at most 2 parts. - Colin Barker, Mar 31 2015
a(n) is representable as a sum of two but no fewer consecutive nonnegative integers, e.g., 1 = 0 + 1, 3 = 1 + 2, 5 = 2 + 3, etc. (see A138591). - Martin Renner, Mar 14 2016
Unique solution a( ) of the complementary equation a(n) = a(n-1)^2 - a(n-2)*b(n-1), where a(0) = 1, a(1) = 3, and a( ) and b( ) are increasing complementary sequences. - Clark Kimberling, Nov 21 2017
Also the number of maximal and maximum cliques in the n-centipede graph. - Eric W. Weisstein, Dec 01 2017
Lexicographically earliest sequence of distinct positive integers such that the average of any number of consecutive terms is always an integer. (For opposite property see A042963.) - Ivan Neretin, Dec 21 2017
Maximum number of non-intersecting line segments between vertices of a convex (n+2)-gon. - Christoph B. Kassir, Oct 21 2022
a(n) is the number of parking functions of size n+1 avoiding the patterns 123, 132, and 231. - Lara Pudwell, Apr 10 2023

			G.f. = q + 3*q^3 + 5*q^5 + 7*q^7 + 9*q^9 + 11*q^11 + 13*q^13 + 15*q^15 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 28.
T. Dantzig, The Language of Science, 4th Edition (1954) page 276.
H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 73.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.1 Terminology, p. 264.
D. Hök, Parvisa mönster i permutationer [Swedish], (2007).
E. Maor, Trigonometric Delights, Princeton University Press, NJ, 1998, pp. 203-205.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
D. Applegate and J. C. Lagarias, The 3x+1 semigroup, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146-159; see also the arXiv version, arXiv:math/0411140 [math.NT], 2004-2005.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, J. Integer Sequ., Vol. 8 (2005), Article 05.4.5.
Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão and Graça Tomaz, Combinatorial Identities Associated with a Multidimensional Polynomial Sequence, J. Int. Seq., Vol. 21 (2018), Article 18.7.4.
Hongwei Chen, Evaluations of Some Variant Euler Sums, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.3.
K. Ciesielski and Z. Pogoda, On ordering the natural numbers, or the Sharkovski theorem, Amer. Math. Monthly, 115 (No. 2, 2008), 158-165.
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, arXiv:1601.01673 [math.NT], 2016. See p. 35.
T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, theorem 2.5, k=4.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 935
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Bridges.
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Franck Ramaharo, Enumerating the states of the twist knot, arXiv:1712.06543 [math.CO], 2017.
Michael Somos, Rational Function Multiplicative Coefficients
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, The modular forms database
Leo Tavares, Illustration: Triangular Sides
Eric Weisstein's World of Mathematics, Centipede Graph
Eric Weisstein's World of Mathematics, Davenport-Schinzel Sequence
Eric Weisstein's World of Mathematics, Gnomonic Number
Eric Weisstein's World of Mathematics, Inverse Cotangent,
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cotangent
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Tangent
Eric Weisstein's World of Mathematics, Inverse Tangent
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Nexus Number
Eric Weisstein's World of Mathematics, Odd Number
Eric Weisstein's World of Mathematics, Pythagorean Triple
Chai Wah Wu, Can machine learning identify interesting mathematics? An exploration using empirically observed laws, arXiv:1805.07431 [cs.LG], 2018.
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000027, A005843, A065091.
See A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf. A128200, A000290, A078050, A144106, A109613, A167875.
Cf. A001651 (n=1 or 2 mod 3), A047209 (n=1 or 4 mod 5).
Cf. A003558, A216371, A179480 (relating to the Coach theorem).
Cf. A000754 (boustrophedon transform).

List([0..100],n->2*n+1); # Muniru A Asiru, Oct 16 2018

a005408 n = (+ 1) . (* 2)
a005408_list = [1, 3 ..]  -- Reinhard Zumkeller, Feb 11 2012, Jun 28 2011

Magma
```
[ 2*n+1 : n in [0..100]];
```

A005408 := n->2*n+1;
A005408:=(1+z)/(z-1)^2; # Simon Plouffe in his 1992 dissertation

Table[2 n - 1, {n, 1, 50}] (* Stefan Steinerberger, Apr 01 2006 *)
Range[1, 131, 2] (* Harvey P. Dale, Apr 26 2011 *)
2 Range[0, 20] + 1 (* Eric W. Weisstein, Dec 01 2017 *)
LinearRecurrence[{2, -1}, {1, 3}, 20] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[(1 + x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

makelist(2*n+1, n, 0, 30); /* Martin Ettl, Dec 11 2012 */

PARI
```
{a(n) = 2*n + 1}
```

first(n) = Vec((1 + x)/(1 - x)^2 + O(x^n)) \\ Iain Fox, Dec 29 2017

a=lambda n: 2*n+1 # Indranil Ghosh, Jan 04 2017

[2*n+1 for n in range(100)] # G. C. Greubel, Nov 28 2018

a(n) = 2*n + 1. a(-1 - n) = -a(n). a(n+1) = a(n) + 2.
G.f.: (1 + x) / (1 - x)^2.
E.g.f.: (1 + 2*x) * exp(x).
G.f. with interpolated zeros: (x^3+x)/((1-x)^2 * (1+x)^2); e.g.f. with interpolated zeros: x*(exp(x)+exp(-x))/2. - Geoffrey Critzer, Aug 25 2012
a(n) = L(n,-2)*(-1)^n, where L is defined as in A108299. - Reinhard Zumkeller, Jun 01 2005
Euler transform of length 2 sequence [3, -1]. - Michael Somos, Mar 30 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v * (1 + 2*u) * (1 - 2*u + 16*v) - (u - 4*v)^2 * (1 + 2*u + 2*u^2). - Michael Somos, Mar 30 2007
a(n) = b(2*n + 1) where b(n) = n if n is odd is multiplicative. [This seems to say that A000027 is multiplicative? - R. J. Mathar, Sep 23 2011]
From Hieronymus Fischer, May 25 2007: (Start)
a(n) = (n+1)^2 - n^2.
G.f. g(x) = Sum_{k>=0} x^floor(sqrt(k)) = Sum_{k>=0} x^A000196(k). (End)
a(0) = 1, a(1) = 3, a(n) = 2*a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
a(n) = A000330(A016777(n))/A000217(A016777(n)). - Pierre CAMI, Sep 27 2008
a(n) = A034856(n+1) - A000217(n) = A005843(n) + A000124(n) - A000217(n) = A005843(n) + 1. - Jaroslav Krizek, Sep 05 2009
a(n) = (n - 1) + n (sum of two sequential integers). - Dominick Cancilla, Aug 09 2010
a(n) = 4*A000217(n)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. - Bruno Berselli, Nov 17 2010
n*a(2n+1)^2+1 = (n+1)*a(2n)^2; e.g., 3*15^2+1 = 4*13^2. - Charlie Marion, Dec 31 2010
arctanh(x) = Sum_{n>=0} x^(2n+1)/a(n). - R. J. Mathar, Sep 23 2011
a(n) = det(f(i-j+1))A113311(n);%20for%20n%20%3C%200%20we%20have%20f(n)=0.%20-%20_Mircea%20Merca">{1<=i,j<=n}, where f(n) = A113311(n); for n < 0 we have f(n)=0. - _Mircea Merca, Jun 23 2012
G.f.: Q(0), where Q(k) = 1 + 2*(k+1)*x/( 1 - 1/(1 + 2*(k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 11 2013
a(n) = floor(sqrt(2*A000384(n+1))). - Ivan N. Ianakiev, Jun 17 2013
a(n) = 3*A000330(n)/A000217(n), n > 0. - Ivan N. Ianakiev, Jul 12 2013
a(n) = Product_{k=1..2*n} 2*sin(Pi*k/(2*n+1)) = Product_{k=1..n} (2*sin(Pi*k/(2*n+1)))^2, n >= 0 (undefined product = 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - Wolfdieter Lang, Oct 10 2013
Noting that as n -> infinity, sqrt(n^2 + n) -> n + 1/2, let f(n) = n + 1/2 - sqrt(n^2 + n). Then for n > 0, a(n) = round(1/f(n))/4. - Richard R. Forberg, Feb 16 2014
a(n) = Sum_{k=0..n+1} binomial(2*n+1,2*k)*4^(k)*bernoulli(2*k). - Vladimir Kruchinin, Feb 24 2015
a(n) = Sum_{k=0..n} binomial(6*n+3, 6*k)*Bernoulli(6*k). - Michel Marcus, Jan 11 2016
a(n) = A000225(n+1) - A005803(n+1). - Miquel Cerda, Nov 25 2016
O.g.f.: Sum_{n >= 1} phi(2*n-1)*x^(n-1)/(1 - x^(2*n-1)), where phi(n) is the Euler totient function A000010. - Peter Bala, Mar 22 2019
Sum_{n>=0} 1/a(n)^2 = Pi^2/8 = A111003. - Bernard Schott, Dec 10 2020
Sum_{n >= 1} (-1)^n/(a(n)*a(n+1)) = Pi/4 - 1/2 = 1/(3 + (1*3)/(4 + (3*5)/(4 + ... + (4*n^2 - 1)/(4 + ... )))). Cf. A016754. - Peter Bala, Mar 28 2024
a(n) = A055112(n)/oblong(n) = A193218(n+1)/Hex number(n). Compare to the Sep 27 2008 comment by Pierre CAMI. - Klaus Purath, Apr 23 2024
a(k*m) = k*a(m) - (k-1). - Ya-Ping Lu, Jun 25 2024
a(n) = A000217(a(n))/n for n > 0. - Stefano Spezia, Feb 15 2025

Incorrect comment and example removed by Joerg Arndt, Mar 11 2010

Peripheral comments deleted by N. J. A. Sloane, May 09 2022

A000537 Sum of first n cubes; or n-th triangular number squared.

Original entry on oeis.org

0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, 6084, 8281, 11025, 14400, 18496, 23409, 29241, 36100, 44100, 53361, 64009, 76176, 90000, 105625, 123201, 142884, 164836, 189225, 216225, 246016, 278784, 314721, 354025, 396900, 443556, 494209, 549081

Offset: 0

N. J. A. Sloane

Number of parallelograms in an n X n rhombus. - Matti De Craene (Matti.DeCraene(AT)rug.ac.be), May 14 2000
Or, number of orthogonal rectangles in an n X n checkerboard, or rectangles in an n X n array of squares. - Jud McCranie, Feb 28 2003. Compare A085582.
Also number of 2-dimensional cage assemblies (cf. A059827, A059860).
The n-th triangular number T(n) = Sum_{r=1..n} r = n(n+1)/2 satisfies the relations: (i) T(n) + T(n-1) = n^2 and (ii) T(n) - T(n-1) = n by definition, so that n^2*n = n^3 = {T(n)}^2 - {T(n-1)}^2 and by summing on n we have Sum_{ r = 1..n } r^3 = {T(n)}^2 = (1+2+3+...+n)^2 = (n*(n+1)/2)^2. - Lekraj Beedassy, May 14 2004
Number of 4-tuples of integers from {0,1,...,n}, without repetition, whose last component is strictly bigger than the others. Number of 4-tuples of integers from {1,...,n}, with repetition, whose last component is greater than or equal to the others.
Number of ordered pairs of two-element subsets of {0,1,...,n} without repetition.
Number of ordered pairs of 2-element multisubsets of {1,...,n} with repetition.
1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2.
a(n) is the number of parameters needed in general to know the Riemannian metric g of an n-dimensional Riemannian manifold (M,g), by knowing all its second derivatives; even though to know the curvature tensor R requires (due to symmetries) (n^2)*(n^2-1)/12 parameters, a smaller number (and a 4-dimensional pyramidal number). - Jonathan Vos Post, May 05 2006
Also number of hexagons with vertices in an hexagonal grid with n points in each side. - Ignacio Larrosa Cañestro, Oct 15 2006
Number of permutations of n distinct letters (ABCD...) each of which appears twice with 4 and n-4 fixed points. - Zerinvary Lajos, Nov 09 2006
With offset 1 = binomial transform of [1, 8, 19, 18, 6, ...]. - Gary W. Adamson, Dec 03 2008
The sequence is related to A000330 by a(n) = n*A000330(n) - Sum_{i=0..n-1} A000330(i): this is the case d=1 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Apr 26 2010, Mar 01 2012
From Wolfdieter Lang, Jan 11 2013: (Start)
For sums of powers of positive integers S(k,n) := Sum_{j=1..n}j^k one has the recurrence S(k,n) = (n+1)*S(k-1,n) - Sum_{l=1..n} S(k-1,l), n >= 1, k >= 1.
This was used for k=4 by Ibn al-Haytham in an attempt to compute the volume of the interior of a paraboloid. See the Strick reference where the trick he used is shown, and the W. Lang link.
This trick generalizes immediately to arbitrary powers k. For k=3: a(n) = (n+1)*A000330(n) - Sum_{l=1..n} A000330(l), which coincides with the formula given in the previous comment by Berselli. (End)
Regarding to the previous contribution, see also Matem@ticamente in Links field and comments on this recurrences in similar sequences (partial sums of n-th powers). - Bruno Berselli, Jun 24 2013
A rectangular prism with sides A000217(n), A000217(n+1), and A000217(n+2) has surface area 6*a(n+1). - J. M. Bergot, Aug 07 2013, edited with corrected indices by Antti Karttunen, Aug 09 2013
A formula for the r-th successive summation of k^3, for k = 1 to n, is (6*n^2+r*(6*n+r-1)*(n+r)!)/((r+3)!*(n-1)!), (H. W. Gould). - Gary Detlefs, Jan 02 2014
Note that this sequence and its formula were known to (and possibly discovered by) Nicomachus, predating Ibn al-Haytham by 800 years. - Charles R Greathouse IV, Apr 23 2014
a(n) is the number of ways to paint the sides of a nonsquare rectangle using at most n colors. Cf. A039623. - Geoffrey Critzer, Jun 18 2014
For n > 0: A256188(a(n)) = A000217(n) and A256188(m) != A000217(n) for m < a(n), i.e., positions of first occurrences of triangular numbers in A256188. - Reinhard Zumkeller, Mar 26 2015
There is no cube in this sequence except 0 and 1. - Altug Alkan, Jul 02 2016
Also the number of chordless cycles in the complete bipartite graph K_{n+1,n+1}. - Eric W. Weisstein, Jan 02 2018
a(n) is the sum of the elements in the multiplication table [0..n] X [0..n]. - Michel Marcus, May 06 2021

			G.f. = x + 9*x^2 + 36*x^3 + 100*x^4 + 225*x^5 + 441*x^6 + ... - _Michael Somos_, Aug 29 2022

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 62, eq. (6.3) for k=3.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 110ff.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, pp. 36, 58.
Clifford Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Oxford University Press, 2001, p. 325.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. K. Strick, Geschichten aus der Mathematik II, Spektrum Spezial 3/11, p. 13.
D. Wells, You Are A Mathematician, "Counting rectangles in a rectangle", Problem 8H, pp. 240; 254, Penguin Books 1995.

T. D. Noe, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Luciano Ancora, Sum of cubes of the first "n" natural numbers
Luciano Ancora, The Square Pyramidal Number and other figurate numbers
M. Azaola and F. Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., 27 (2002), 29-48 (see Prop. 4.2(b)).
Marcel Berger, Encounter with a Geometer, Part II, Notices of the American Mathematical Society, Vol. 47, No. 3, (March 2000), pp. 326-340. [About the work of Mikhael Gromov.]
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
blackpenredpen, Math for fun, how many rectangles?, Youtube video (2018).
Bikash Chakraborty, Proof Without Words: Sums of Powers of Natural numbers, arXiv:2012.11539 [math.HO], 2020.
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
Shel Kaphan, How to see that the difference between two successive squared triangular numbers is a cube
Sameen Ahmed Khan, Sums of the powers of reciprocals of polygonal numbers, Int'l J. of Appl. Math. (2020) Vol. 33, No. 2, 265-282.
Seon-Hong Kim and Kenneth B. Stolarsky, Translations and Extensions of the Nicomachean Identity, J. Int. Seq. (2024), Vol. 27, Issue 6, Art. No. 24.6.3. See p. 1.
Wolfdieter Lang, Ibn al-Haytham's trick.
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, JIS 12 (2009) 09.5.5.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
Henri Picciotto, Sum of Cubes, Proof without words.
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Chordless Cycle
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Faulhaber's Formula
Wikipedia, Faulhaber's formula
Wikipedia, Squared triangular number
G. Xiao, Sigma Server, Operate on "n^3"
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Convolution of A000217 and A008458.
Cf. A000330, A000538, A006003.
Row sums of triangles A094414 and A094415.
Second column of triangle A008459.
Row 3 of array A103438.
Cf. A000578, A002415, A024166, A101102, A101094, A101097.
Cf. A236770 (see crossrefs).
Cf. A000290, A253169, A256188.
Cf. A000217, A000292, A000332, A000566, A035287, A039623, A053382, A053383, A059376, A059827, A059860, A085582, A127777, A176271.

List([0..40],n->(n*(n+1)/2)^2); # Muniru A Asiru, Dec 05 2018

a000537 = a000290 . a000217  -- Reinhard Zumkeller, Mar 26 2015

[(n*(n+1)/2)^2: n in [0..50]]; // Wesley Ivan Hurt, Jun 06 2014

a:= n-> (n*(n+1)/2)^2:
seq(a(n), n=0..40);

Accumulate[Range[0, 50]^3] (* Harvey P. Dale, Mar 01 2011 *)
f[n_] := n^2 (n + 1)^2/4; Array[f, 39, 0] (* Robert G. Wilson v, Nov 16 2012 *)
Table[CycleIndex[{{1, 2, 3, 4}, {3, 2, 1, 4}, {1, 4, 3, 2}, {3, 4, 1, 2}}, s] /. Table[s[i] -> n, {i, 1, 2}], {n, 0, 30}] (* Geoffrey Critzer, Jun 18 2014 *)
Accumulate @ Range[0, 50]^2 (* Waldemar Puszkarz, Jan 24 2015 *)
Binomial[Range[20], 2]^2 (* Eric W. Weisstein, Jan 02 2018 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 9, 36, 100}, 20] (* Eric W. Weisstein, Jan 02 2018 *)
CoefficientList[Series[-((x (1 + 4 x + x^2))/(-1 + x)^5), {x, 0, 20}], x] (* Eric W. Weisstein, Jan 02 2018 *)

PARI
```
a(n)=(n*(n+1)/2)^2
```

def A000537(n): return (n*(n+1)>>1)**2 # Chai Wah Wu, Oct 20 2023

a(n) = (n*(n+1)/2)^2 = A000217(n)^2 = Sum_{k=1..n} A000578(k), that is, 1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2.
G.f.: (x+4*x^2+x^3)/(1-x)^5. - Simon Plouffe in his 1992 dissertation
a(n) = Sum ( Sum ( 1 + Sum (6*n) ) ), rephrasing the formula in A000578. - Xavier Acloque, Jan 21 2003
a(n) = Sum_{i=1..n} Sum_{j=1..n} i*j, row sums of A127777. - Alexander Adamchuk, Oct 24 2004
a(n) = A035287(n)/4. - Zerinvary Lajos, May 09 2007
This sequence could be obtained from the general formula n*(n+1)*(n+2)*(n+3)*...*(n+k)*(n*(n+k) + (k-1)*k/6)/((k+3)!/6) at k=1. - Alexander R. Povolotsky, May 17 2008
G.f.: x*F(3,3;1;x). - Paul Barry, Sep 18 2008
Sum_{k > 0} 1/a(k) = (4/3)*(Pi^2-9). - Jaume Oliver Lafont, Sep 20 2009
a(n) = Sum_{1 <= k <= m <= n} A176271(m,k). - Reinhard Zumkeller, Apr 13 2010
a(n) = Sum_{i=1..n} J_3(i)*floor(n/i), where J_ 3 is A059376. - Enrique Pérez Herrero, Feb 26 2012
a(n) = Sum_{i=1..n} Sum_{j=1..n} Sum_{k=1..n} min(i,j,k). - Enrique Pérez Herrero, Feb 26 2013 [corrected by Ridouane Oudra, Mar 05 2025]
a(n) = 6*C(n+2,4) + C(n+1,2) = 6*A000332(n+2) + A000217(n), (Knuth). - Gary Detlefs, Jan 02 2014
a(n) = -Sum_{j=1..3} j*Stirling1(n+1,n+1-j)*Stirling2(n+3-j,n). - Mircea Merca, Jan 25 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*(3-4*log(2)). - Vaclav Kotesovec, Feb 13 2015
a(n)*((s-2)*(s-3)/2) = P(3, P(s, n+1)) - P(s, P(3, n+1)), where P(s, m) = ((s-2)*m^2-(s-4)*m)/2 is the m-th s-gonal number. For s=7, 10*a(n) = A000217(A000566(n+1)) - A000566(A000217(n+1)). - Bruno Berselli, Aug 04 2015
From Ilya Gutkovskiy, Jul 03 2016: (Start)
E.g.f.: x*(4 + 14*x + 8*x^2 + x^3)*exp(x)/4.
Dirichlet g.f.: (zeta(s-4) + 2*zeta(s-3) + zeta(s-2))/4. (End)
a(n) = (Bernoulli(4, n+1) - Bernoulli(4, 1))/4, n >= 0, with the Bernoulli polynomial B(4, x) from row n=4 of A053382/A053383. See, e.g., the Ash-Gross reference, p. 62, eq. (6.3) for k=3. - Wolfdieter Lang, Mar 12 2017
a(n) = A000217((n+1)^2) - A000217(n+1)^2. - Bruno Berselli, Aug 31 2017
a(n) = n*binomial(n+2, 3) + binomial(n+2, 4) + binomial(n+1, 4). - Tony Foster III, Nov 14 2017
Another identity: ..., a(3) = (1/2)*(1*(2+4+6)+3*(4+6)+5*6) = 36, a(4) = (1/2)*(1*(2+4+6+8)+3*(4+6+8)+5*(6+8)+7*(8)) = 100, a(5) = (1/2)*(1*(2+4+6+8+10)+3*(4+6+8+10)+5*(6+8+10)+7*(8+10)+9*(10)) = 225, ... - J. M. Bergot, Aug 27 2022
Comment from Michael Somos, Aug 28 2022: (Start)
The previous comment expresses a(n) as the sum of all of the n X n multiplication table array entries.
For example, for n = 4:
     1  2  3  4
     2  4  6  8
     3  6  9 12
     4  8 12 16
This array sum can be split up as follows:
   +---+---------------+
   | 0 | 1   2   3   4 |    (0+1)*(1+2+3+4)
   |   +---+-----------+
   | 0 | 2 | 4   6   8 |    (1+2)*(2+3+4)
   |   |   +---+-------+
   | 0 | 3 | 6 | 9  12 |    (2+3)*(3+4)
   |   |   |   +---+---+
   | 0 | 4 | 8 |12 |16 |    (3+4)*(4)
   +---+---+---+---+---+
This kind of row+column sums was used by Ramanujan and others for summing Lambert series. (End)
a(n) = 6*A000332(n+4) - 12*A000292(n+1) + 7*A000217(n+1) - n - 1. - Adam Mohamed, Sep 05 2024

Edited by M. F. Hasler, May 02 2015

A163102 a(n) = n^2*(n+1)^2/2.

Original entry on oeis.org

0, 2, 18, 72, 200, 450, 882, 1568, 2592, 4050, 6050, 8712, 12168, 16562, 22050, 28800, 36992, 46818, 58482, 72200, 88200, 106722, 128018, 152352, 180000, 211250, 246402, 285768, 329672, 378450, 432450, 492032, 557568, 629442, 708050, 793800, 887112, 988418

Offset: 0

Omar E. Pol, Jul 24 2009

Row sums of triangle A163282.
Also, the number of nonattacking placements of 2 rooks on an (n+1) X (n+1) board. - Thomas Zaslavsky, Jun 26 2010
If P_{k}(n) is the n-th k-gonal number, then a(n) = P_{s}(n+1)*P_{t}(n+1) - P_{s+1}(n+1)*P_{t-1}(n+1) for s=t+1. - Bruno Berselli, Sep 05 2014
Subsequence of A000982, see formula. - David James Sycamore, Jul 31 2018
Number of edges in the (n+1) X (n+1) rook complement graph. - Freddy Barrera, May 02 2019
Number of paths from (0,0) to (n+2,n+2) consisting of exactly three forward horizontal steps and three upward vertical steps. - Greg Dresden and Snezhana Tuneska, Aug 24 2023

Seth Chaiken, Christopher R. H. Hanusa, and Thomas Zaslavsky, A q-queens problem, in preparation. - Thomas Zaslavsky, Jun 26 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Seth Chaiken, Christopher R. H. Hanusa, and Thomas Zaslavsky, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016-2020.
A. Umar, Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations, J. Int. Seq., Vol. 14 (2011), Article 11.7.5.
Eric Weisstein's World of Mathematics, Rook Complement Graph.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Column k=2 of A144084.
Cf. A000217, A000537, A000982, A035287.
Cf. A006002, A099903, A163274, A163275, A163276, A163277, A163282.
Cf. A060300, A254371.

List([0..40],n->(n*(n+1))^2/2); # Muniru A Asiru, Aug 02 2018

[n^2*(n+1)^2/2: n in [0..40]]; // Vincenzo Librandi, Mar 26 2012

CoefficientList[Series[2*x*(1+4*x+x^2)/(1-x)^5,{x,0,40}],x] (* Vincenzo Librandi, Mar 26 2012 *)

a(n)=n^2*(n+1)^2/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*A000537(n) = A035287(n+1)/2. - Omar E. Pol, Nov 29 2011
G.f.: 2*x*(1+4*x+x^2)/(1-x)^5. - R. J. Mathar, Nov 30 2011
Let t(n) = A000217(n). Then a(n) = (t(n-1)*(t(n)+t(n+1)) + t(n)*(t(n-1)+t(n+1)) + t(n+1)*(t(n-1)+t(n)))/3. - J. M. Bergot, Jun 21 2012
a(n) = A000982(n*(n+1)). - David James Sycamore, Jul 31 2018
From Amiram Eldar, Nov 02 2021: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi^2/3 - 6.
Sum_{n>=1} (-1)^(n+1)/a(n) = 6 - 8*log(2). (End)
Another identity: ..., a(4) = 200 = 1*(2+4+6+8) + 3*(4+6+8) + 5*(6+8) + 7*(8), a(5) = 450 = 1*(2+4+6+8+10) + 3*(4+6+8+10) + 5*(6+8+10) + 7*(8+10) + 9*(10) = 30+84+120+126+90, and so on. - J. M. Bergot, Aug 25 2022
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: x*(2 + x)*(2 + 6*x + x^2)*exp(x)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A254371(n)/4 = A060300(n)/8. (End)

A162990 Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).

Original entry on oeis.org

4, 36, 9, 576, 144, 64, 14400, 3600, 1600, 900, 518400, 129600, 57600, 32400, 20736, 25401600, 6350400, 2822400, 1587600, 1016064, 705600, 1625702400, 406425600, 180633600, 101606400, 65028096, 45158400, 33177600, 131681894400

Offset: 1

Johannes W. Meijer, Jul 21 2009

The hypergeometric function 3F2([1,n+1,n+1],[n+2,n+2],z) = (n+1)^2*Li2(z)/z^(n+1) - MN(z;n)/(n!^2*z^n) for n >= 1, with Li2(z) the dilogarithm. The polynomial coefficients of MN(z;n) lead to the triangle given above.
We observe that 3F2([1,1,1],[2,2],z) = Li2(z)/z and that 3F2([1,0,0],[1,1],z) = 1.
The generating function for the EG1[3,n] coefficients of the EG1 matrix, see A162005, is GFEG1(z;m=2) = 1/(1-z)*(3*zeta(3)/2-2*z*log(2)* 3F2([1,1,1],[2,2],z) + sum((2^(1-2*n)* factorial(2*n-1)*z^(n+1)*3F2([1,n+1,n+1],[n+2,n+2],z))/(factorial(n+1)^2), n=1..infinity)).
The zeros of the MN(z;n) polynomials for larger values of n get ever closer to the unit circle and resemble the full moon, hence we propose to call the MN(z;n) the moon polynomials.

			The first few rows of the triangle are:
  [4]
  [36, 9]
  [576, 144, 64]
  [14400, 3600, 1600, 900]
The first few MN(z;n) polynomials are:
  MN(z;n=1) = 4
  MN(z;n=2) = 36 + 9*z
  MN(z;n=3) = 576 + 144*z + 64*z^2
  MN(z;n=4) = 14400 + 3600*z + 1600*z^2 + 900*z^3

Lewin, L., Polylogarithms and Associated Functions. New York, North-Holland, 1981.

Paolo Xausa, Table of n, a(n) for n = 1..5050 (rows 1..100 of the triangle, flattened).
Eric Weisstein's World of Mathematics, Dilogarithm.

A162995 is a scaled version of this triangle.
A001819(n)*(n+1)^2 equals the row sums for n>=1.
A162991 and A162992 equal the first and second right hand columns.
A001048, A052747, A052759, A052778, A052794 are related to the square root of the first five right hand columns.
A001044, A162993 and A162994 equal the first, second and third left hand columns.
A000142, A001710, A002301, A133799, A129923, A001715 are related to the square root of the first six left hand columns.
A027451(n+1) equals the denominators of M(z, n)/(n!)^2.
A129202(n)/A129203(n) = (n+1)^2*Li2(z=1)/(Pi^2) = (n+1)^2/6.
Cf. A002378 and A035287.

a := proc(n, m): ((n+1)!/m)^2 end: seq(seq(a(n, m), m=1..n), n=1..7); # Johannes W. Meijer, revised Nov 29 2012

Table[((n+1)!/m)^2, {n, 10}, {m, n}] (* Paolo Xausa, Mar 30 2024 *)

a(n,m) = ((n+1)!/m)^2 for n >= 1 and 1 <= m <= n.

A060300 a(n) = (2n(n+1))^2.

Original entry on oeis.org

0, 16, 144, 576, 1600, 3600, 7056, 12544, 20736, 32400, 48400, 69696, 97344, 132496, 176400, 230400, 295936, 374544, 467856, 577600, 705600, 853776, 1024144, 1218816, 1440000, 1690000, 1971216, 2286144, 2637376, 3027600, 3459600, 3936256, 4460544, 5035536, 5664400

Offset: 0

Jason Earls, Mar 25 2001

Arises from middle column 4^2, 12^2, 24^2, ... of following triangle: :
                        3^2 +  4^2 = 5^2
                10^2 + 11^2 + 12^2 = 13^2 + 14^2
         21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2
  36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2
...

C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, pp. 90-92.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000537, A002378, A035287, A046092, A163102, A254371.

[(2*n*(n+1))^2: n in [0..30]]; // Vincenzo Librandi, Nov 18 2016

CoefficientList[Series[16 x (1 + 4 x + x^2) / (1 - x)^5, {x, 0, 33}], x] (* Vincenzo Librandi, Nov 18 2016 *)
Table[(2n(n+1))^2,{n,0,30}] (* Harvey P. Dale, Jan 19 2019 *)

a(n) = { (2*n*(n + 1))^2 } \\ Harry J. Smith, Jul 03 2009

G.f.: 16*x*(1+4*x+x^2)/(1-x)^5. - Colin Barker, Apr 22 2012
a(n) = 4*A035287(n+1) = 4*A002378(n)^2. - Michel Marcus, May 24 2016
a(n) = 16*A000537(n) = 16*(n*(n+1)/2)^2 = 16*A000217(n)^2 = A046092(n)^2. - Bruce J. Nicholson, Jun 05 2017
a(n) = Integral_{x=1..2*n+1} (x^3-x) dx. - César Aguilera, Jun 27 2020
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: 4*x*(2 + x)*(2 + 6*x + x^2)*exp(x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 2*A254371(n) = 8*A163102(n). (End)

Name corrected by Harry J. Smith, Jul 03 2009

A221524 T(n,k)=Number of 0..k arrays of length n with each element differing from at least one neighbor by 2 or more.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 6, 2, 0, 0, 12, 10, 4, 0, 0, 20, 30, 36, 6, 0, 0, 30, 68, 144, 94, 10, 0, 0, 42, 130, 400, 536, 274, 16, 0, 0, 56, 222, 900, 1940, 2172, 768, 26, 0, 0, 72, 350, 1764, 5368, 9982, 8544, 2182, 42, 0, 0, 90, 520, 3136, 12458, 33380, 50400, 33960, 6170, 68, 0, 0

Offset: 1

R. H. Hardin Jan 19 2013

Table starts
.0...0......0.......0.........0..........0...........0...........0............0
.0...2......6......12........20.........30..........42..........56...........72
.0...2.....10......30........68........130.........222.........350..........520
.0...4.....36.....144.......400........900........1764........3136.........5184
.0...6.....94.....536......1940.......5368.......12458.......25544........47776
.0..10....274....2172......9982......33380.......90684......212812.......447962
.0..16....768....8544.....50400.....205080......654864.....1763328......4184064
.0..26...2182...33960....256018....1264378.....4738970....14629962.....39113752
.0..42...6170..134480...1297924....7787228....34274630...121342546....365574840
.0..68..17476..533248...6584320...47975704...247928860..1006508448...3416978176
.0.110..49470.2113456..33394958..295543282..1793345580..8348594292..31937713030
.0.178.140066.8377808.169387004.1820672982.12971955294.69248649436.298515152986

			Some solutions for n=6 k=4
..1....0....4....4....3....3....0....4....2....0....4....0....0....1....3....0
..3....3....0....0....0....1....2....0....4....3....2....4....2....3....1....3
..4....0....3....2....4....1....4....2....4....1....4....1....3....0....2....0
..1....0....4....4....1....4....2....3....2....2....2....1....0....0....4....4
..4....4....2....1....0....3....1....0....4....4....4....4....0....2....1....3
..2....1....4....4....2....1....4....3....0....2....2....2....3....4....3....1

R. H. Hardin, Table of n, a(n) for n = 1..1518

Column 2 is A006355
Row 2 is A002378(n-1)
Row 3 is A034262(n-1)
Row 4 is A035287

Empirical for column k:
k=2: a(n) = a(n-1) +a(n-2)
k=3: a(n) = a(n-1) +4*a(n-2) +3*a(n-3) +a(n-4)
k=4: a(n) = 2*a(n-1) +6*a(n-2) +6*a(n-3) +4*a(n-4) +4*a(n-6)
k=5: a(n) = 2*a(n-1) +11*a(n-2) +20*a(n-3) +17*a(n-4) -3*a(n-5) +a(n-6)
k=6: a(n) = 3*a(n-1) +14*a(n-2) +29*a(n-3) +28*a(n-4) +a(n-5) +27*a(n-6) +8*a(n-7) +2*a(n-8)
k=7: a(n) = 3*a(n-1) +21*a(n-2) +58*a(n-3) +79*a(n-4) +32*a(n-5) +23*a(n-6) +4*a(n-7) +8*a(n-8)
Empirical for row n:
n=2: a(n) = n^2 - n
n=3: a(n) = n^3 - 3*n^2 + 4*n - 2
n=4: a(n) = n^4 - 2*n^3 + n^2
n=5: a(n) = n^5 - n^4 - 10*n^3 + 38*n^2 - 60*n + 40 for n>2
n=6: a(n) = n^6 - 20*n^4 + 83*n^3 - 182*n^2 + 236*n - 148 for n>3
n=7: a(n) = n^7 + n^6 - 29*n^5 + 109*n^4 - 204*n^3 + 202*n^2 - 80*n for n>2

A145426 Decimal expansion of Sum_{k>=0} (k!/(k+2)!)^2.

Original entry on oeis.org

2, 8, 9, 8, 6, 8, 1, 3, 3, 6, 9, 6, 4, 5, 2, 8, 7, 2, 9, 4, 4, 8, 3, 0, 3, 3, 3, 2, 9, 2, 0, 5, 0, 3, 7, 8, 4, 3, 7, 8, 9, 9, 8, 0, 2, 4, 1, 3, 5, 9, 6, 8, 7, 5, 4, 7, 1, 1, 1, 6, 4, 5, 8, 7, 4, 0, 0, 1, 4, 9, 4, 0, 8, 0, 6, 4, 0, 1, 7, 4, 7, 6, 6, 7, 2, 5, 7, 8, 0, 1, 2, 3, 9

Offset: 0

R. J. Mathar, Feb 08 2009

			0.28986813369645287294483...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.31.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.19 Vallée's Constant, p. 161.

R. J. Mathar, Tightly circumscribed regular polygons, arXiv:1301.6293 [math.MG], 2013, value (A15).

Cf. A002388 (Pi^2), A002378 (oblong numbers), A035287, A348670.

Maple
```
evalf(1/3*Pi^2-3) ;
```

RealDigits[Pi^2/3 - 3, 10, 120][[1]] (* Amiram Eldar, Jun 17 2023 *)

Pi^2/3-3 \\ Seiichi Manyama, Dec 09 2021

sumnumrat(1/(x^4 + 2*x^3 + x^2), 1) \\ Charles R Greathouse IV, Jan 20 2022

Equals A002388/3-3 = Sum_{n>=1} 1/A002378(n)^2 = Sum_{n>=2} 1/A035287(n).

A254371 Sum of cubes of the first n even numbers (A016743).

Original entry on oeis.org

0, 8, 72, 288, 800, 1800, 3528, 6272, 10368, 16200, 24200, 34848, 48672, 66248, 88200, 115200, 147968, 187272, 233928, 288800, 352800, 426888, 512072, 609408, 720000, 845000, 985608, 1143072, 1318688, 1513800, 1729800, 1968128, 2230272, 2517768, 2832200, 3175200

Offset: 0

Luciano Ancora, Mar 16 2015

Property: for n >= 2, each (a(n), a(n)+1, a(n)+2) is a triple of consecutive terms that are the sum of two nonzero squares; precisely: a(n) = (n*(n + 1))^2 + (n*(n + 1))^2, a(n)+1 = (n^2+2n)^2 + (n^2-1)^2 and a(n)+2 = (n^2+n+1)^2 + (n^2+n-1)^2 (see Diophante link). - Bernard Schott, Oct 05 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Luciano Ancora, The Square Pyramidal Number and other figurate numbers, ch. 3.
Diophante, Une miniature avec trois entiers consécutifs (in French).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537 (sum of first n cubes); A002593 (sum of first n odd cubes).
Cf. A060300 (2*a(n)).
First bisection of A105636; second bisection of A212892.
Cf. A000217, A000415, A002378, A035287, A163102.

List([0..35],n->2*(n*(n+1))^2); # Muniru A Asiru, Oct 24 2018

[2*n^2*(n+1)^2: n in [0..40]]; // Bruno Berselli, Mar 23 2015

A254371:=n->2*n^2*(n + 1)^2: seq(A254371(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[2 n^2 (n+1)^2, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 72, 288, 800}, 40]
Accumulate[Range[0,80,2]^3] (* Harvey P. Dale, Jun 26 2017 *)

a(n)=sum(i=0, n, 8*i^3); \\ Michael B. Porter, Mar 16 2015

G.f.: 8*x*(1 + 4*x + x^2)/(1 - x)^5.
a(n) = 2*n^2*(n + 1)^2.
a(n) = 2*A035287(n+1) = 2*A002378(n)^2 = 8*A000217(n)^2. - Bruce J. Nicholson, Apr 23 2017
a(n) = 8*A000537(n). - Michel Marcus, Apr 23 2017
From Amiram Eldar, Aug 25 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 - 3/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - 2*log(2). (End)
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: 2*x*(2 + x)*(2 + 6*x + x^2)*exp(x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 4*A163102(n) = A060300(n)/2. (End)

A248619 a(n) = (n*(n+1))^4.

Original entry on oeis.org

0, 16, 1296, 20736, 160000, 810000, 3111696, 9834496, 26873856, 65610000, 146410000, 303595776, 592240896, 1097199376, 1944810000, 3317760000, 5473632256, 8767700496, 13680577296, 20851360000, 31116960000, 45558341136, 65554433296, 92844527616, 129600000000

Offset: 0

Eugene Chong, Oct 09 2014

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A016744, A059977; A002378: n*(n+1); A035287: n^2 *(n-1)^2; A060459: n^3*(n+1)^3.

Cf. A327773.

[(n*(n+1))^4: n in [0..30]]; // Vincenzo Librandi, Oct 16 2014

Maple
```
[ seq(n^4*(n+1)^4, n = 0..100) ];
```

Table[(n (n + 1))^4, {n, 0, 70}] (* or *) CoefficientList[Series[16 x (1 + 72 x + 603 x^2 + 1168 x^3 + 603 x^4 + 72 x^5 + x^6)/(1 - x)^9, {x, 0, 30}], x] (* Vincenzo Librandi, Oct 16 2014 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,16,1296,20736,160000,810000,3111696,9834496,26873856},30] (* Harvey P. Dale, Sep 09 2016 *)

a(n) = A002378(n)^4 = A016744(A000217(n)).
a(n) = 16*A059977(n) for n>0.
G.f.: 16*x*(1 + 72*x + 603*x^2 + 1168*x^3 + 603*x^4 + 72*x^5 + x^6)/(1 - x)^9. - Vincenzo Librandi, Oct 16 2014
Sum_{n>=1} 1/a(n) = A327773 = -35 + 10*Pi^2/3 + Pi^4/45. - Vaclav Kotesovec, Sep 25 2019

Terms a(76) and beyond corrected by Andrew Howroyd, Feb 20 2018

A358517 Decimal expansion of 4*log(2) - 11/4.

Original entry on oeis.org

0, 2, 2, 5, 8, 8, 7, 2, 2, 2, 3, 9, 7, 8, 1, 2, 3, 7, 6, 6, 8, 9, 2, 8, 4, 8, 5, 8, 3, 2, 7, 0, 6, 2, 7, 2, 3, 0, 2, 0, 0, 0, 5, 3, 7, 4, 4, 1, 0, 2, 1, 0, 1, 6, 4, 8, 2, 7, 2, 0, 0, 3, 7, 9, 7, 3, 5, 7, 4, 4, 8, 7, 8, 7, 8, 7, 7, 8, 8, 6, 2, 4, 2, 3, 4, 5, 3, 3, 0, 7, 9, 8, 5, 6, 7

Offset: 0

Claude H. R. Dequatre, Nov 20 2022

			0.0225887222397812376689284858327062723020005374...

Michael Ian Shamos, A catalog of the real numbers (2011), p. 63.
Index entries for transcendental numbers

Cf. A016639, A035287, A242023, A242024, A358516, A358519.

Cf. A002162.

Join[{0}, RealDigits[4*Log[2] - 11/4, 10, 120][[1]]] (* Amiram Eldar, Nov 21 2022 *)

PARI
```
4*log(2) - 11/4
```

Equals Sum_{k>=1} (-1)^(k+1) * k/((k + 1)*(k + 2)*(k + 3)) [Shamos].
Equals Sum_{k>=1} (-1)^(k+1)/((k+1)^2*(k+2)^2).
Equals Sum_{k>=3} (-1)^(k+1)/A035287(k). - Amiram Eldar, Nov 21 2022

cp's OEIS Frontend

A005408 The odd numbers: a(n) = 2*n + 1.

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A000537 Sum of first n cubes; or n-th triangular number squared.

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A163102 a(n) = n^2*(n+1)^2/2.

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A162990 Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).

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A060300 a(n) = (2*n*(n+1))^2.

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A060300 a(n) = (2n(n+1))^2.