A123865 -id:A123865 - cp's OEIS frontend

A000583 Fourth powers: a(n) = n^4.

0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 20736, 28561, 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1048576, 1185921

Offset: 0

N. J. A. Sloane

Figurate numbers based on 4-dimensional regular convex polytope called the 4-measure polytope, 4-hypercube or tesseract with Schlaefli symbol {4,3,3}. - Michael J. Welch (mjw1(AT)ntlworld.com), Apr 01 2004
Totally multiplicative sequence with a(p) = p^4 for prime p. - Jaroslav Krizek, Nov 01 2009
The binomial transform yields A058649. The inverse binomial transforms yields the (finite) 0, 1, 14, 36, 24, the 4th row in A019538 and A131689. - R. J. Mathar, Jan 16 2013
Generate Pythagorean triangles with parameters a and b to get sides of lengths x = b^2-a^2, y = 2*a*b, and z = a^2 + b^2. In particular use a=n-1 and b=n for a triangle with sides (x1,y1,z1) and a=n and b=n+1 for another triangle with sides (x2,y2,z2). Then x1*x2 + y1*y2 + z1*z2 = 8*a(n). - J. M. Bergot, Jul 22 2013
For n > 0, a(n) is the largest integer k such that k^4 + n is a multiple of k + n. Also, for n > 0, a(n) is the largest integer k such that k^2 + n^2 is a multiple of k + n^2. - Derek Orr, Sep 04 2014
Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 08 2017
a(n+2)/2 is the area of a trapezoid with vertices at (T(n), T(n+1)), (T(n+1), T(n)), (T(n+1), T(n+2)), and (T(n+2), T(n+1)) with T(n)=A000292(n) for n >= 0. - J. M. Bergot, Feb 16 2018

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 64.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 255; 2nd. ed., p. 269. Worpitzky's identity (6.37).
Dov Juzuk, Curiosa 56: An interesting observation, Scripta Mathematica 6 (1939), 218.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Page 47.

T. D. Noe, Table of n, a(n) for n = 0..1000
Henry Bottomley, Illustration of initial terms
Henry Bottomley, Some Smarandache-type multiplicative sequences
Ralph Greenberg, Math for Poets.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
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.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
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
Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42.
Eric Weisstein's World of Mathematics, Biquadratic Number.
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Index entries for sequences related to Benford's law

Cf. A000538, A005917 (first differences), A000332, A014820, A092181, A092182, A092183.
Cf. A004831, A002646.
Cf. A002593, A260810. - Bruno Berselli, Jul 31 2015
Cf. A002415, A000290, A006008, A132366, A039623, A139584, A071270, A047928, A187756.
Cf. A062392, A231303, A267315.

a000583 = (^ 4)
a000583_list = scanl (+) 0 a005917_list
-- Reinhard Zumkeller, Nov 13 2014, Nov 11 2012

[n^4 : n in [0..50]]; // Wesley Ivan Hurt, Sep 05 2014

A000583 := n->n^4: seq(A000583(n), n=0..50);
A000583:=-(z+1)*(z**2+10*z+1)/(z-1)**5; # Simon Plouffe in his 1992 dissertation; gives sequence without initial zero
with (combinat):seq(fibonacci(3, n^2)-1, n=0..33); # Zerinvary Lajos, May 25 2008

Range[0,100]^4 (* Vladimir Joseph Stephan Orlovsky, Mar 14 2011 *)

makelist(n^4,n,0,30); /* Martin Ettl, Nov 12 2012 */

A000583(n) = n^4 \\ Michael B. Porter, Nov 09 2009

def a(n): return n**4
print([a(n) for n in range(34)]) # Michael S. Branicky, Nov 10 2022

a(n) = A123865(n)+1 = A002523(n)-1.
Multiplicative with a(p^e) = p^(4e). - David W. Wilson, Aug 01 2001
G.f.: x*(1 + 11*x + 11*x^2 + x^3)/(1 - x)^5. More generally, g.f. for n^m is Euler(m, x)/(1-x)^(m+1), where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292).
Dirichlet generating function: zeta(s-4). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: (x + 7*x^2 + 6*x^3 + x^4)*e^x. More generally, the general form for the e.g.f. for n^m is phi_m(x)*e^x, where phi_m is the exponential polynomial of order n. - Franklin T. Adams-Watters, Sep 11 2005
Sum_{k>0} 1/a(k) = Pi^4/90 = A013662. - Jaume Oliver Lafont, Sep 20 2009
a(n) = C(n+3,4) + 11*C(n+2,4) + 11*C(n+1,4) + C(n,4). [Worpitzky's identity for powers of 4. See, e.g., Graham et al., eq. (6.37). - Wolfdieter Lang, Jul 17 2019]
a(n) = n*A177342(n) - Sum_{i=1..n-1} A177342(i) - (n - 1), with n > 1. - Bruno Berselli, May 07 2010
a(n) + a(n+1) + 1 = 2*A002061(n+1)^2. - Charlie Marion, Jun 13 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 24. - Ant King, Sep 23 2013
From Amiram Eldar, Jan 20 2021: (Start)
Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/720 (A267315).
Product_{n>=2} (1 - 1/a(n)) = sinh(Pi)/(4*Pi). (End)

A005563 a(n) = n*(n+2) = (n+1)^2 - 1.

Original entry on oeis.org

0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323, 360, 399, 440, 483, 528, 575, 624, 675, 728, 783, 840, 899, 960, 1023, 1088, 1155, 1224, 1295, 1368, 1443, 1520, 1599, 1680, 1763, 1848, 1935, 2024, 2115, 2208, 2303, 2400, 2499, 2600

Offset: 0

N. J. A. Sloane

Erdős conjectured that n^2 - 1 = k! has a solution if and only if n is 5, 11 or 71 (when k is 4, 5 or 7).
Second-order linear recurrences y(m) = 2y(m-1) + a(n)*y(m-2), y(0) = y(1) = 1, have closed form solutions involving only powers of integers. - Len Smiley, Dec 08 2001
Number of edges in the join of two cycle graphs, both of order n, C_n * C_n. - Roberto E. Martinez II, Jan 07 2002
Let k be a positive integer, M_n be the n X n matrix m_(i,j) = k^abs(i-j) then det(M_n) = (-1)^(n-1)*a(k-1)^(n-1). - Benoit Cloitre, May 28 2002
Also numbers k such that 4*k + 4 is a square. - Cino Hilliard, Dec 18 2003
For each term k, the function sqrt(x^2 + 1), starting with 1, produces an integer after k iterations. - Gerald McGarvey, Aug 19 2004
a(n) mod 3 = 0 if and only if n mod 3 > 0: a(A008585(n)) = 2; a(A001651(n)) = 0; a(n) mod 3 = 2*(1-A079978(n)). - Reinhard Zumkeller, Oct 16 2006
a(n) is the number of divisors of a(n+1) that are not greater than n. - Reinhard Zumkeller, Apr 09 2007
Nonnegative X values of solutions to the equation X^3 + X^2 = Y^2. To find Y values: b(n) = n(n+1)(n+2). - Mohamed Bouhamida, Nov 06 2007
Sequence allows us to find X values of the equation: X + (X + 1)^2 + (X + 2)^3 = Y^2. To prove that X = n^2 + 2n: Y^2 = X + (X + 1)^2 + (X + 2)^3 = X^3 + 7*X^2 + 15X + 9 = (X + 1)(X^2 + 6X + 9) = (X + 1)*(X + 3)^2 it means: (X + 1) must be a perfect square, so X = k^2 - 1 with k>=1. we can put: k = n + 1, which gives: X = n^2 + 2n and Y = (n + 1)(n^2 + 2n + 3). - Mohamed Bouhamida, Nov 12 2007
From R. K. Guy, Feb 01 2008: (Start)
Toads and Frogs puzzle:
This is also the number of moves that it takes n frogs to swap places with n toads on a strip of 2n + 1 squares (or positions, or lily pads) where a move is a single slide or jump, illustrated for n = 2, a(n) = 8 by
   T T - F F
   T - T F F
   T F T - F
   T F T F -
   T F - F T
   - F T F T
   F - T F T
   F F T - T
   F F - T T
I was alerted to this by the Holton article, but on consulting Singmaster's sources, I find that the puzzle goes back at least to 1867.
Probably the first to publish the number of moves for n of each animal was Edouard Lucas in 1883. (End)
a(n+1) = terms of rank 0, 1, 3, 6, 10 = A000217 of A120072 (3, 8, 5, 15). - Paul Curtz, Oct 28 2008
Row 3 of array A163280, n >= 1. - Omar E. Pol, Aug 08 2009
Final digit belongs to a periodic sequence: 0, 3, 8, 5, 4, 5, 8, 3, 0, 9. - Mohamed Bouhamida, Sep 04 2009 [Comment edited by N. J. A. Sloane, Sep 24 2009]
Let f(x) be a polynomial in x. Then f(x + n*f(x)) is congruent to 0 (mod f(x)); here n belongs to N. There is nothing interesting in the quotients f(x + n*f(x))/f(x) when x belongs to Z. However, when x is irrational these quotients consist of two parts, a) rational integers and b) integer multiples of x. The present sequence represents the non-integer part when the polynomial is x^2 + x + 1 and x = sqrt(2), f(x+n*f(x))/f(x) = A056108(n) + a(n)*sqrt(2). - A.K. Devaraj, Sep 18 2009
For n >= 1, a(n) is the number for which 1/a(n) = 0.0101... (A000035) in base (n+1). - Rick L. Shepherd, Sep 27 2009
For n > 0, continued fraction [n, 1, n] = (n+1)/a(n); e.g., [6, 1, 6] = 7/48. - Gary W. Adamson, Jul 15 2010
Starting (3, 8, 15, ...) = binomial transform of [3, 5, 2, 0, 0, 0, ...]; e.g., a(3) = 15 = (1*3 + 2*5 +1*2) = (3 + 10 + 2). - Gary W. Adamson, Jul 30 2010
a(n) is essentially the case 0 of the polygonal numbers. The polygonal numbers are defined as P_k(n) = Sum_{i=1..n} ((k-2)*i-(k-3)). Thus P_0(n) = 2*n-n^2 and a(n) = -P_0(n+2). See also A067998 and for the case k=1 A080956. - Peter Luschny, Jul 08 2011
a(n) is the maximal determinant of a 2 X 2 matrix with integer elements from {1, ..., n+1}, so the maximum determinant of a 2x2 matrix with integer elements from {1, ..., 5} = 5^2 - 1 = a(4) = 24. - Aldo González Lorenzo, Oct 12 2011
Using four consecutive triangular numbers t1, t2, t3 and t4, plot the points (0, 0), (t1, t2), and (t3, t4) to create a triangle. Twice the area of this triangle are the numbers in this sequence beginning with n = 1 to give 8. - J. M. Bergot, May 03 2012
Given a particle with spin S = n/2 (always a half-integer value), the quantum-mechanical expectation value of the square of the magnitude of its spin vector evaluates to  = S(S+1) = n(n+2)/4, i.e., one quarter of a(n) with n = 2S. This plays an important role in the theory of magnetism and magnetic resonance. - Stanislav Sykora, May 26 2012
Twice the harmonic mean [H(x, y) = (2*x*y)/(x + y)] of consecutive triangular numbers A000217(n) and A000217(n+1). - Raphie Frank, Sep 28 2012
Number m such that floor(sqrt(m)) = floor(m/floor(sqrt(m))) - 2 for m > 0. - Takumi Sato, Oct 10 2012
The solutions of equation 1/(i - sqrt(j)) = i + sqrt(j), when i = (n+1), j = a(n). For n = 1, 2 + sqrt(3) = 3.732050.. = A019973. For n = 2, 3 + sqrt(8) = 5.828427... = A156035. - Kival Ngaokrajang, Sep 07 2013
The integers in the closed form solution of a(n) = 2*a(n-1) + a(m-2)*a(n-2), n >= 2, a(0) = 0, a(1) = 1 mentioned by Len Smiley, Dec 08 2001, are m and -m + 2 where m >= 3 is a positive integer. - Felix P. Muga II, Mar 18 2014
Let m >= 3 be a positive integer. If a(n) = 2*a(n-1) + a(m-2) * a(n-2), n >= 2, a(0) = 0, a(1) = 1, then lim_{n->oo} a(n+1)/a(n) = m. - Felix P. Muga II, Mar 18 2014
For n >= 4 the Szeged index of the wheel graph W_n (with n + 1 vertices). In the Sarma et al. reference, Theorem 2.7 is incorrect. - Emeric Deutsch, Aug 07 2014
If P_{k}(n) is the n-th k-gonal number, then a(n) = t*P_{s}(n+2) - s*P_{t}(n+2) for s=t+1. - Bruno Berselli, Sep 04 2014
For n >= 1, a(n) is the dimension of the simple Lie algebra A_n. - Wolfdieter Lang, Oct 21 2015
Finding all positive integers (n, k) such that n^2 - 1 = k! is known as Brocard's problem, (see A085692). - David Covert, Jan 15 2016
For n > 0, a(n) mod (n+1) = a(n) / (n+1) = n. - Torlach Rush, Apr 04 2016
Conjecture: When using the Sieve of Eratosthenes and sieving (n+1..a(n)), with divisors (1..n) and n>0, there will be no more than a(n-1) composite numbers. - Fred Daniel Kline, Apr 08 2016
a(n) mod 8 is periodic with period 4 repeating (0,3,0,7), that is a(n) mod 8 = 5/2 - (5/2) cos(n*Pi) - sin(n*Pi/2) + sin(3*n*Pi/2). - Andres Cicuttin, Jun 02 2016
Also for n > 0, a(n) is the number of times that n-1 occurs among the first (n+1)! terms of A055881. - R. J. Cano, Dec 21 2016
The second diagonal of composites (the only prime is number 3) from the right on the Klauber triangle (see Kival Ngaokrajang link), which is formed by taking the positive integers and taking the first 1, the next 3, the following 5, and so on, each centered below the last. - Charles Kusniec, Jul 03 2017
Also the number of independent vertex sets in the n-barbell graph. - Eric W. Weisstein, Aug 16 2017
Interleaving of A000466 and A033996. - Bruce J. Nicholson, Nov 08 2019
a(n) is the number of degrees of freedom in a triangular cell for a Raviart-Thomas or Nédélec first kind finite element space of order n. - Matthew Scroggs, Apr 22 2020
From Muge Olucoglu, Jan 19 2021: (Start)
For n > 1, a(n-2) is the maximum number of elements in the second stage of the Quine-McCluskey algorithm whose minterms are not covered by the functions of n bits. At n=3, we have a(3-2) = a(1) = 1*(1+2) = 3 and f(A,B,C) = sigma(0,1,2,5,6,7).
.
          0 1 2   5 6 7
        +---------------
  *(0,1)| X X
   (0,2)| X   X
   (1,5)|   X     X
  *(2,6)|     X     X
  *(5,7)|         X   X
   (6,7)|           X X
.
*: represents the elements that are covered. (End)
1/a(n) is the ratio of the sum of the first k odd numbers and the sum of the next n*k odd numbers. - Melvin Peralta, Jul 15 2021
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {1, 2n}]. - Magus K. Chu, Sep 09 2022
Number of diagonals parallel to an edge in a regular (2*n+4)-gon (cf. A367204). - Paolo Xausa, Nov 21 2023
For n >= 1, also the number of minimum cyclic edge cuts in the (n+2)-trapezohedron graph. - Eric W. Weisstein, Nov 21 2024
For n >= 1, a(n) is the sum of the interior angles of a polygon with n+2 sides, in radians, multiplied by (n+2)/Pi. - Stuart E Anderson, Aug 06 2025

			G.f. = 3*x + 8*x^2 + 15*x^3 + 24*x^4 + 35*x^5 + 48*x^6 + 63*x^7 + 80*x^8 + ...

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see index under Toads and Frogs Puzzle.
Martin Gardner, Perplexing Puzzles and Tantalizing Teasers, p. 21 (for "The Dime and Penny Switcheroo").
R. K. Guy, Unsolved Problems in Theory of Numbers, Section D25.
Derek Holton, Math in School, 37 #1 (Jan 2008) 20-22.
Edouard Lucas, Récréations Mathématiques, Gauthier-Villars, Vol. 2 (1883) 141-143.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See p. 22.
R. K. Guy, Letter to N. J. A. Sloane, May 1990.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Kival Ngaokrajang, Klauber triangle
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.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
K. V. S. Sarma and I. V. N. Uma, On Szeged index of standard graphs, International J. of Math. Archive, 3(8), 2012, 3129-3135. - _Emeric Deutsch_, Aug 07 2014
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Leo Tavares, Illustration: Bounded Squares.
Leo Tavares, Illustration: Trapezoids.
Leo Tavares, Illustration: Trapagons.
Eric Weisstein's World of Mathematics, Barbell Graph.
Eric Weisstein's World of Mathematics, Cyclic Edge Cut.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Eric Weisstein's World of Mathematics, Trapezohedral Graph.
Wikipedia, Quine-McCluskey Algorithm.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

A column of triangle A102537.
a(n+1), n>=2, first column of triangle A120070.
Cf. A000035, A000217, A000290, A000466, A001651, A002378, A005563.
Cf. A007531, A008585, A008833, A013468, A016742, A019973, A028560.
Cf. A033996, A046092, A053186, A054000, A055881, A056108, A062196.
Cf. A068310, A067725, A067728, A067998, A069817, A079978, A080956.
Cf. A085692, A120072, A123865, A123866, A123867, A123868, A134582.
Cf. A140091, A140681, A143053, A144396, A156035, A163280, A212331.
Cf. A253607, A367204.

a005563 n = n * (n + 2)
a005563_list = zipWith (*) [0..] [2..]  -- Reinhard Zumkeller, Dec 16 2012

[n*(n+2): n in [0..60]]; // G. C. Greubel, Mar 29 2024

Table[n^2 - 1, {n, 42}] (* Zerinvary Lajos, Mar 21 2007 *)
ListCorrelate[{1, 2}, Range[-1, 50], {1, -1}, 0, Plus, Times] (* Harvey P. Dale, Aug 29 2015 *)
Range[20]^2 - 1 (* Eric W. Weisstein, Aug 16 2017 *)
Table[n (n + 2), {n, 20}] (* Eric W. Weisstein, Nov 21 2024 *)
CoefficientList[Series[(-3 + x)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 21 2024 *)
LinearRecurrence[{3, -3, 1}, {3, 8, 15}, 20] (* Eric W. Weisstein, Nov 21 2024 *)

makelist(n*(n+2), n, 0, 56); /* Martin Ettl, Oct 15 2012 */

a(n)=n*(n+2) \\ Charles R Greathouse IV, Dec 22 2011

concat(0, Vec(x*(3-x)/(1-x)^3 + O(x^90))) \\ Altug Alkan, Oct 22 2015

[n*(n+2) for n in range(61)] # G. C. Greubel, Mar 29 2024

G.f.: x*(3-x)/(1-x)^3. - Simon Plouffe in his 1992 dissertation
a(n) = A000290(n+1) - 1.
A002378(a(n)) = A002378(n)*A002378(n+1); e.g., A002378(15)=240=12*20. - Charlie Marion, Dec 29 2003
a(n) = A067725(n)/3. - Zerinvary Lajos, Mar 06 2007
a(n) = Sum_{k=1..n} A144396(k). - Zerinvary Lajos, May 11 2007
a(n) = A134582(n+1)/4. - Zerinvary Lajos, Feb 01 2008
A143053(a(n)) = A000290(n+1), for n > 0. - Reinhard Zumkeller, Jul 20 2008
a(n) = Real((n+1+i)^2). - Gerald Hillier, Oct 12 2008
A053186(a(n)) = 2*n. - Reinhard Zumkeller, May 20 2009
a(n) = (n! + (n+1)!)/(n-1)!, n > 0. - Gary Detlefs, Aug 10 2009
a(n) = floor(n^5/(n^3+1)) with offset 1 (a(1)=0). - Gary Detlefs, Feb 11 2010
a(n) = a(n-1) + 2*n + 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
Sum_{n>=1} 1/a(n) = 3/4. - Mohammad K. Azarian, Dec 29 2010
a(n) = 2/(Integral_{x=0..Pi/2} (sin(x))^(n-1)*(cos(x))^3), for n > 0. - Francesco Daddi, Aug 02 2011
a(n) = A002378(n) + floor(sqrt(A002378(n))); pronic number + its root. - Fred Daniel Kline, Sep 16 2011
a(n-1) = A008833(n) * A068310(n) for n > 1. - Reinhard Zumkeller, Nov 26 2011
G.f.: U(0) where U(k) = -1 + (k+1)^2/(1 - x/(x + (k+1)^2/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
a(n) = 15*C(n+4,3)*C(n+4,5)/(C(n+4,2)*C(n+4,4)). - Gary Detlefs, Aug 05 2013
a(n) = (n+2)!/((n-1)! + n!), n > 0. - Ivan N. Ianakiev, Nov 11 2013
a(n) = 3*C(n+1,2) - C(n,2) for n >= 0. - Felix P. Muga II, Mar 11 2014
a(n) = (A016742(n+1) - 4)/4 for n >= 0. - Felix P. Muga II, Mar 11 2014
a(-2 - n) = a(n) for all n in Z. - Michael Somos, Aug 07 2014
A253607(a(n)) = 1. - Reinhard Zumkeller, Jan 05 2015
E.g.f.: x*(x + 3)*exp(x). - Ilya Gutkovskiy, Jun 03 2016
For n >= 1, a(n^2 + n - 2) = a(n-1) * a(n). - Miko Labalan, Oct 15 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/4. - Amiram Eldar, Nov 04 2020
From Amiram Eldar, Feb 17 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = 2.
Product_{n>=1} (1 - 1/a(n)) = -sqrt(2)*sin(sqrt(2)*Pi)/Pi. (End)
a(n) = A000290(n+2) - n*2. See Bounded Squares illustration. - Leo Tavares, Oct 05 2021
From Leo Tavares, Oct 10 2021: (Start)
a(n) = A008585(n) + 2*A000217(n-1). See Trapezoids illustration.
2*A005563 = A054000(n+1). See Trapagons illustration.
a(n) = 2*A000217(n) + n. (End)
a(n) = (n+2)!!/(n-2)!! for n > 1. - Jacob Szlachetka, Jan 02 2022

Partially edited by Joerg Arndt, Mar 11 2010

More terms from N. J. A. Sloane, Aug 01 2010

A143325 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1) which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 8, 6, 0, 1, 4, 15, 24, 15, 0, 1, 5, 24, 60, 80, 27, 0, 1, 6, 35, 120, 255, 232, 63, 0, 1, 7, 48, 210, 624, 1005, 728, 120, 0, 1, 8, 63, 336, 1295, 3096, 4095, 2160, 252, 0, 1, 9, 80, 504, 2400, 7735, 15624, 16320, 6552, 495, 0, 1, 10, 99

Offset: 1

Alois P. Heinz, Aug 07 2008

Column k is Dirichlet convolution of mu(n) with k^(n-1). The coefficients of the polynomial of row n are given by the n-th row of triangle A054525; for example row 4 has polynomial -k+k^3.

			T(4,2)=6, because 6 words of length 4 over 2-letter alphabet {a,b} are primitive and earlier than others derived by cyclic shifts of the alphabet: aaab, aaba, aabb, abaa, abba, abbb; note that aaaa and abab are not primitive and words beginning with b can be derived by shifts of the alphabet from words in the list; secondly note that the words in the list need not be Lyndon words, for example aaba can be derived from aaab by a cyclic rotation of the positions.
Table begins:
  1,   1,    1,     1,     1,      1,      1,       1, ...
  0,   1,    2,     3,     4,      5,      6,       7, ...
  0,   3,    8,    15,    24,     35,     48,      63, ...
  0,   6,   24,    60,   120,    210,    336,     504, ...
  0,  15,   80,   255,   624,   1295,   2400,    4095, ...
  0,  27,  232,  1005,  3096,   7735,  16752,   32697, ...
  0,  63,  728,  4095, 15624,  46655, 117648,  262143, ...
  0, 120, 2160, 16320, 78000, 279720, 823200, 2096640, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to Lyndon words

Columns k=1-10 give: A063524, A000740, A034741, A295505, A295506, A320071, A320072, A320073, A320074, A320075.
Rows n=1-5, 7 give: A000012, A001477, A005563, A007531, A123865, A123866.
Main diagonal gives A075147.
Cf. A074650, A143324, A008683, A054525.

with(numtheory):
f1:= proc(n) option remember;
       unapply(k^(n-1)-add(f1(d)(k), d=divisors(n)minus{n}), k)
     end;
T:= (n,k)-> f1(n)(k);
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

t[n_, k_] := Sum[k^(d-1)*MoebiusMu[n/d], {d, Divisors[n]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 21 2014, from first formula *)

T(n,k) = Sum_{d|n} k^(d-1) * mu(n/d).

T(n,k) = k^(n-1) - Sum_{d

T(n,k) = A074650(n,k) * n/k.

T(n,k) = A143324(n,k) / k.

A033455 Convolution of nonzero squares A000290 with themselves.

Original entry on oeis.org

1, 8, 34, 104, 259, 560, 1092, 1968, 3333, 5368, 8294, 12376, 17927, 25312, 34952, 47328, 62985, 82536, 106666, 136136, 171787, 214544, 265420, 325520, 396045, 478296, 573678, 683704, 809999, 954304, 1118480, 1304512, 1514513, 1750728, 2015538, 2311464

Offset: 1

N. J. A. Sloane

Total area of all square regions from an n X n grid. E.g., at n = 3, there are nine individual squares, four 2 X 2's and one 3 X 3, total area 9 + 16 + 9 = 34, hence a(3) = 34. - Jon Perry, Jul 29 2003
If X is an n-set and Y and Z disjoint 2-subsets of X then a(n) is equal to the number of 7-subsets of X intersecting both Y and Z. - Milan Janjic, Aug 26 2007
Every fourth term is odd. However, there are no primes in the sequence. - Zak Seidov, Feb 28 2011
-120*a(n) is the real part of (n + n*i)*(n + 2 + n*i)*(n + (n + 2)i)*(n + 2+(n + 2)*i)*(n + 1 + (n + 1)*i), where i = sqrt(-1). - Jon Perry, Feb 05 2014
The previous formula rephrases the factorization of the 5th-order polynomial a(n) = (n+1)*((n+1)^4-1) = (n+1)*A123864(n+1) based on the factorization in A123865. - R. J. Mathar, Feb 08 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Abderrahim Arabi, Hacène Belbachir, Jean-Philippe Dubernard, Plateau Polycubes and Lateral Area, arXiv:1811.05707 [math.CO], 2018. See Column 1 Table 1 p. 8.
Milan Janjic, Two Enumerative Functions
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 2. - _N. J. A. Sloane_, Mar 23 2014
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000290, A001249, A123865, A219086.

List([1..40], n-> ((n+1)^5 -(n+1))/30); # G. C. Greubel, Jul 05 2019

[(n^5-n)/30: n in [2..41]]; // Vincenzo Librandi, Mar 24 2014

Table[(n^5 - n)/30, {n,2,41}] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2011 *)
CoefficientList[Series[(1+x)^2/(1-x)^6, {x,0,40}], x] (* Vincenzo Librandi, Mar 24 2014 *)

vector(40, n, ((n+1)^5 -n-1)/30) \\ G. C. Greubel, Jul 05 2019

[((n+1)^5 -n-1)/30 for n in (1..40)] # G. C. Greubel, Jul 05 2019

a(n-1) = n*(n^4 - 1)/30 = A061167(n)/30. - Henry Bottomley, Apr 18 2001
G.f.: x*(1+x)^2/(1-x)^6. - Philippe Deléham, Feb 21 2012
a(n) = Sum_{k=1..n+1} k^2*(n+1-k)^2. - Kolosov Petro, Feb 07 2019
E.g.f.: x*(30 +90*x +65*x^2 +15*x^3 +x^4)*exp(x)/30. - G. C. Greubel, Jul 05 2019

More terms from Vincenzo Librandi, Mar 24 2014

A073103 Number of solutions to x^4 == 1 (mod n).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 8, 8, 4, 2, 2, 8, 4, 2, 2, 8, 4, 4, 2, 4, 4, 8, 2, 8, 4, 4, 8, 4, 4, 2, 8, 16, 4, 4, 2, 4, 8, 2, 2, 16, 2, 4, 8, 8, 4, 2, 8, 8, 4, 4, 2, 16, 4, 2, 4, 8, 16, 4, 2, 8, 4, 8, 2, 8, 4, 4, 8, 4, 4, 8, 2, 32, 2, 4, 2, 8, 16, 2, 8, 8, 4, 8, 8, 4, 4, 2, 8, 16, 4, 2, 4, 8

Offset: 1

Benoit Cloitre, Aug 19 2002

a(n) = 2*A060594(n) for n = 5, 10, 13, 15, 16, 17, 20, 25, 26, 29, .... This subsequence, which contains all the primes of form 4k+1, seems to be asymptotic to 2n.

Shadow transform of A123865. - Michel Marcus, Jun 06 2013

T. D. Noe, Table of n, a(n) for n = 1..1000
Steven R. Finch, Quartic and Octic Characters Modulo n, arXiv:0907.4894 [math.NT], 2009.
Steven Finch, Greg Martin and Pascal Sebah, Roots of unity and nullity modulo n, Proc. Amer. Math. Soc., Vol. 138, No. 8 (2010), pp. 2729-2743.
Lorenz Halbeisen and Norbert Hungerbühler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150. (ps, pdf); see Definition 7 for the shadow transform.
Lorenz Halbeisen and Norbert Hungerbühler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
N. J. A. Sloane, Transforms.

Cf. A060594, A060839.

a:= n-> add(`if`(irem(j^4-1, n)=0, 1, 0), j=0..n-1):
seq(a(n), n=1..120);  # Alois P. Heinz, Jun 06 2013
# alternative
A073103 := proc(n)
    local a,pf,p,r;
    a := 1 ;
    for pf in ifactors(n)[2] do
        p := op(1,pf);
        r := op(2,pf);
        if p = 2 then
            a := a*p^min(r-1,3) ;
        else
            if modp(p,4) = 1 then
                a := 4*a ;
            else
                a := 2*a ;
            end if;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Mar 02 2015

a[n_] := Sum[If[Mod[j^4-1, n] == 0, 1, 0], {j, 0, n-1}]; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 12 2015, after Alois P. Heinz *)
f[2, e_] := 2^Min[e-1, 3]; f[p_, e_] := If[Mod[p, 4] == 1, 4, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

PARI
```
a(n)=sum(i=1,n,if((i^4-1)%n,0,1))
```

a(n)=my(f=factor(n)); prod(i=1,#f~, if(f[i,1]==2, 2^min(f[i,2]-1,3), if(f[i,1]%4==1, 4, 2))) \\ Charles R Greathouse IV, Mar 02 2015

from math import prod
from sympy import primefactors
def A073103(n): return (1<>1) for p in primefactors(n>>s)) # Chai Wah Wu, Oct 26 2022

Sum_{k=1..n} a(k) seems to be asymptotic to C*n*Log(n) with C>1.4...(when Sum_{k=1..n} A060594(k) is asymptotic to C/2*n*Log(n)).
Multiplicative with a(p^e) = p^min(e-1, 3) if p = 2, 4 if p == 1 (mod 4), 2 if p == 3 (mod 4). - David W. Wilson, Jun 09 2005
In fact, Sum_{k=1..n} a(k) is asymptotic to c*n*log(n)^2 where 2*c=0.190876.... - Steven Finch, Aug 12 2009 [c = 7/(2*Pi^3) * Product_{p prime == 1 (mod 4)} (1 - 4/(p+1)^2) = 0.0954383605... (Finch et al., 2010). - Amiram Eldar, Mar 26 2021]

A219086 a(n) = floor((n + 1/2)^4).

Original entry on oeis.org

0, 5, 39, 150, 410, 915, 1785, 3164, 5220, 8145, 12155, 17490, 24414, 33215, 44205, 57720, 74120, 93789, 117135, 144590, 176610, 213675, 256289, 304980, 360300, 422825, 493155, 571914, 659750, 757335, 865365, 984560, 1115664

Offset: 0

Clark Kimberling, Jan 01 2013

a(n) is the number k such that {k^p} < 1/2 < {(k+1)^p}, where p = 1/4 and { } = fractional part.  Equivalently, the jump sequence of f(x) = x^(1/4), in the sense that these are the nonnegative integers k for which round(k^p) < round((k+1)^p).  For details and a guide to related sequences, see A219085.
-4*a(n) gives the real part of (n+n*i)*((n+1)+n*i)*(n+(n+1)*i)*((n+1)+(n+1)*i). The imaginary part is always zero. - Jon Perry, Feb 05 2014
Numbers k such that 16*k+1 is a fourth power. - Bruno Berselli, May 29 2018
The row sums of "Floyd's Triangle", which is a triangular array of natural numbers beginning with the number 1, produce the sequence A006003. A006003 can be bisected to get the Rhombic Dodecahedron Sequence A005917, whose n-th partial sum is n^4, and A317297, whose n-th partial sum is a(n). Interleave n^4 or A000583 back with {a(n)} to get A011863, whose first differences are A019298. Finally, A011863(n)-A011863(n-2) = A006003(n-1). - Bruce J. Nicholson, Dec 22 2019

			0^(1/4) = 0.000...; 1^(1/4) = 1.000...
5^(1/4) = 1.495...; 6^(1/4) = 1.565...
39^(1/4) = 2.499...; 40^(1/4) = 2.514...

Clark Kimberling, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A123865, A219085, A234459, A000217, A001844.

Cf. A317297, A006003, A005917, A000583, A011863, A019298.

A219086:=n->floor((n + (1/2))^4); seq(A219086(n), n=0..50); # Wesley Ivan Hurt, Apr 05 2014

Table[Floor[(n + 1/2)^4], {n, 0, 100}]
LinearRecurrence[{5,-10,10,-5,1},{0,5,39,150,410},40] (* Harvey P. Dale, Jan 15 2023 *)

a(n)=floor((n + 1/2)^4) \\ Charles R Greathouse IV, Apr 15 2014

G.f.: (5*x^3 + 14*x^2 + 5*x)/(1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = (2*n^4 + 4*n^3 + 3*n^2 + n)/2. - J. M. Bergot, Apr 05 2014
a(n) = Sum_{i=0..n} i*(4*i^2 + 1) = n*(n + 1)*(2*n^2 + 2*n + 1)/2. - Bruno Berselli, Feb 09 2017
a(n) = lcm((2*n + 1)^2 - 1, (2*n + 1)^2 + 1)/8 for n>=1. - Lechoslaw Ratajczak, Mar 26 2017
a(n) = A000217(n) * A001844(n). - Bruce J. Nicholson, May 14 2017
E.g.f.: (1/2)*exp(x)*x*(10 + 29*x + 16*x^2 + 2*x^3). - Stefano Spezia, Dec 27 2019
a(n) = ((2*n+1)^4 - 1)/16. - Jianing Song, Jan 03 2023
Sum_{n>=1} 1/a(n) = 6 - 2*Pi*tanh(Pi/2). - Amiram Eldar, Jan 08 2023

A162622 Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n^4 - 1.

Original entry on oeis.org

0, 1, 1, 2, 17, 32, 3, 83, 163, 243, 4, 259, 514, 769, 1024, 5, 629, 1253, 1877, 2501, 3125, 6, 1301, 2596, 3891, 5186, 6481, 7776, 7, 2407, 4807, 7207, 9607, 12007, 14407, 16807, 8, 4103, 8198, 12293, 16388, 20483, 24578, 28673, 32768, 9, 6569, 13129

Offset: 0

Omar E. Pol, Jul 15 2009

Note that the last term of the n-th row is the 5th power of n, A000584(n).

See also the triangles of A162623 and A162624.

			Triangle begins:
  0;
  1,    1;
  2,   17,    32;
  3,   83,   163,   243;
  4,  259,   514,   769,  1024;
  5,  629,  1253,  1877,  2501,  3125;
  6, 1301,  2596,  3891,  5186,  6481,  7776;
  7, 2407,  4807,  7207,  9607, 12007, 14407, 16807;
  8, 4103,  8198, 12293, 16388, 20483, 24578, 28673, 32768;
  9, 6569, 13129, 19689, 26249, 32809, 39369, 45929, 52489, 59049; etc.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000583, A000584, A123865, A159797, A162609, A162610, A162611, A162612, A162613, A162614, A162615, A162616, A162623, A162624.

/* Triangle: */ [[n+k*(n^4-1): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Dec 14 2012

A162622 := proc(n,k) n+k*(n^4-1) ; end proc: seq(seq( A162622(n,k),k=0..n),n=0..15) ; # R. J. Mathar, Feb 11 2010

Flatten[Table[NestList[#+n^4-1&,n,n],{n,0,9}]] (* Harvey P. Dale, Jun 23 2013 *)

Sum_{k=0..n} T(n,k) = n*(n+1)*(1+n^4)/2 (row sums). [R. J. Mathar, Jul 20 2009]

7th and later rows from R. J. Mathar, Feb 11 2010

A123866 a(n) = n^6 - 1.

Original entry on oeis.org

0, 63, 728, 4095, 15624, 46655, 117648, 262143, 531440, 999999, 1771560, 2985983, 4826808, 7529535, 11390624, 16777215, 24137568, 34012223, 47045880, 63999999, 85766120, 113379903, 148035888, 191102975, 244140624, 308915775, 387420488

Offset: 1

Reinhard Zumkeller, Oct 16 2006

a(n) mod 7 = 0 iff n mod 7 > 0: a(A008589(n))=6; a(A047304(n)) = 0; a(n) mod 7 = 6*(1-A082784(n)).

a(n) = A005563(n-1)*A059826(n) = A068601(n)*A001093(n). - Reinhard Zumkeller, Feb 02 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A024004, A001014, A005563, A123865, A123867, A123868.

List([1..30], n-> n^6 -1); # G. C. Greubel, Aug 08 2019

a123866 = (subtract 1) . (^ 6)  -- Reinhard Zumkeller, Mar 11 2014

[n^6 - 1: n in [1..30]]; // Vincenzo Librandi, May 01 2011

A123866:=n->n^6 - 1; seq(A123866(n), n=1..40); # Wesley Ivan Hurt, Feb 26 2014

Table[n^6-1,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0,63,728,4095, 15624,46655, 117648}, 30] (* Harvey P. Dale, Nov 18 2012 *)

A123866(n):=n^6 -1 $ makelist(A123866(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

a(n)=n^6-1 \\ Charles R Greathouse IV, Oct 07 2015

[n^6 -1 for n in (1..30)] # G. C. Greubel, Aug 08 2019

G.f.: x^2*(63 + 287*x + 322*x^2 + 42*x^3 + 7*x^4 - x^5)/(1-x)^7. - Colin Barker, May 08 2012
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(1)=0, a(2)=63, a(3)=728, a(4)=4095, a(5)=15624, a(6)=46655, a(7)=117648. - Harvey P. Dale, Nov 18 2012
Sum_{n>=2} 1/a(n) = 11/12 - Pi*sqrt(3)*tanh(Pi*sqrt(3)/2)/6. - Vaclav Kotesovec, Feb 14 2015
E.g.f.: 1 + (-1 + x + 31*x^2 + 90*x^3 + 65*x^4 + 15*x^5 + x^6)*exp(x). - G. C. Greubel, Aug 08 2019
Product_{n>=2} (1 + 1/a(n)) = 6*Pi^2*sech(sqrt(3)*Pi/2)^2. - Amiram Eldar, Jan 20 2021

Showing 1-10 of 19 results. Next

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A162624 Triangle read by rows in which row n lists n terms, starting with n^4 + n - 1, such that the difference between successive terms is equal to n^4 - 1 = A123865(n).

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A162623 Triangle read by rows in which row n lists n terms, starting with n, such that the difference between successive terms is equal to n^4 - 1 = A123865(n).

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A000583 Fourth powers: a(n) = n^4.

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A005563 a(n) = n*(n+2) = (n+1)^2 - 1.

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A143325 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1) which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

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A033455 Convolution of nonzero squares A000290 with themselves.

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