A001024 -id:A001024 - cp's OEIS frontend

A060218 Number of orbits of length n under the full 15-shift (whose periodic points are counted by A001024).

15, 105, 1120, 12600, 151872, 1897840, 24408480, 320355000, 4271484000, 57664963104, 786341441760, 10812193870800, 149707312950720, 2085208989609360, 29192926025339776, 410525522071875000, 5795654431511374080, 82105104444274758000, 1166756747396368729440, 16626283650369421872480

Offset: 1

Thomas Ward, Mar 21 2001

Number of Lyndon words (aperiodic necklaces) with n beads of 15 colors. - Andrew Howroyd, Dec 10 2017

			a(2)=105 since there are 225 points of period 2 in the full 15-shift and 15 fixed points, so there must be (225-15)/2 = 105 orbits of length 2.

Robert Israel, Table of n, a(n) for n = 1..851
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
T. Ward, Exactly realizable sequences

Column 15 of A074650.

Cf. A001024.

A060218:= func< n | (&+[MoebiusMu(d)*15^Floor(n/d): d in Divisors(n)])/n >;
[A060218(n): n in [1..40]]; // G. C. Greubel, Aug 01 2024

f:= n -> 1/n*add(numtheory:-mobius(d)*15^(n/d), d = numtheory:-divisors(n)):
map(f, [$1..30]); # Robert Israel, Oct 28 2018

A060218[n_]:= DivisorSum[n, MoebiusMu[#]*15^(n/#) &]/n;
Table[A060218[n], {n, 40}] (* G. C. Greubel, Aug 01 2024 *)

a001024(n) = 15^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001024(n/d)); \\ Michel Marcus, Sep 11 2017

def A060218(n): return sum(moebius(k)*15^(n//k) for k in (1..n) if (k).divides(n))/n
[A060218(n) for n in range(1,41)] # G. C. Greubel, Aug 01 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*A001024(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 15*x^k))/k. - Ilya Gutkovskiy, May 19 2019

More terms from Michel Marcus, Sep 11 2017

A000290 The squares: a(n) = n^2.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500

Offset: 0

N. J. A. Sloane

To test if a number is a square, see Cohen, p. 40. - N. J. A. Sloane, Jun 19 2011
Zero followed by partial sums of A005408 (odd numbers). - Jeremy Gardiner, Aug 13 2002
Begin with n, add the next number, subtract the previous number and so on ending with subtracting a 1: a(n) = n + (n+1) - (n-1) + (n+2) - (n-2) + (n+3) - (n-3) + ... + (2n-1) - 1 = n^2. - Amarnath Murthy, Mar 24 2004
Sum of two consecutive triangular numbers A000217. - Lekraj Beedassy, May 14 2004
Numbers with an odd number of divisors: {d(n^2) = A048691(n); for the first occurrence of 2n + 1 divisors, see A071571(n)}. - Lekraj Beedassy, Jun 30 2004
See also A000037.
First sequence ever computed by electronic computer, on EDSAC, May 06 1949 (see Renwick link). - Russ Cox, Apr 20 2006
Numbers k such that the imaginary quadratic field Q(sqrt(-k)) has four units. - Marc LeBrun, Apr 12 2006
For n > 0: number of divisors of (n-1)th power of any squarefree semiprime: a(n) = A000005(A006881(k)^(n-1)); a(n) = A000005(A000400(n-1)) = A000005(A011557(n-1)) = A000005(A001023(n-1)) = A000005(A001024(n-1)). - Reinhard Zumkeller, Mar 04 2007
If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-2) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
Numbers a such that a^1/2 + b^1/2 = c^1/2 and a^2 + b = c. - Cino Hilliard, Feb 07 2008 (this comment needs clarification, Joerg Arndt, Sep 12 2013)
Numbers k such that the geometric mean of the divisors of k is an integer. - Ctibor O. Zizka, Jun 26 2008
Equals row sums of triangle A143470. Example: 36 = sum of row 6 terms: (23 + 7 + 3 + 1 + 1 + 1). - Gary W. Adamson, Aug 17 2008
Equals row sums of triangles A143595 and A056944. - Gary W. Adamson, Aug 26 2008
Number of divisors of 6^(n-1) for n > 0. - J. Lowell, Aug 30 2008
Denominators of Lyman spectrum of hydrogen atom. Numerators are A005563. A000290-A005563 = A000012. - Paul Curtz, Nov 06 2008
a(n) is the number of all partitions of the sum 2^2 + 2^2 + ... + 2^2, (n-1) times, into powers of 2. - Valentin Bakoev, Mar 03 2009
a(n) is the maximal number of squares that can be 'on' in an n X n board so that all the squares turn 'off' after applying the operation: in any 2 X 2 sub-board, a square turns from 'on' to 'off' if the other three are off. - Srikanth K S, Jun 25 2009
Zero together with the numbers k such that 2 is the number of perfect partitions of k. - Juri-Stepan Gerasimov, Sep 26 2009
Totally multiplicative sequence with a(p) = p^2 for prime p. - Jaroslav Krizek, Nov 01 2009
Satisfies A(x)/A(x^2), A(x) = A173277: (1, 4, 13, 32, 74, ...). - Gary W. Adamson, Feb 14 2010
Positive members are the integers with an odd number of odd divisors and an even number of even divisors. See also A120349, A120359, A181792, A181793, A181795. - Matthew Vandermast, Nov 14 2010
Besides the first term, this sequence is the denominator of Pi^2/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... . - Mohammad K. Azarian, Nov 01 2011
Partial sums give A000330. - Omar E. Pol, Jan 12 2013
Drmota, Mauduit, and Rivat proved that the Thue-Morse sequence along the squares is normal; see A228039. - Jonathan Sondow, Sep 03 2013
a(n) can be decomposed into the sum of the four numbers [binomial(n, 1) + binomial(n, 2) + binomial(n-1, 1) + binomial(n-1, 2)] which form a "square" in Pascal's Triangle A007318, or the sum of the two numbers [binomial(n, 2) + binomial(n+1, 2)], or the difference of the two numbers [binomial(n+2, 3) - binomial(n, 3)]. - John Molokach, Sep 26 2013
In terms of triangular tiling, the number of equilateral triangles with side length 1 inside an equilateral triangle with side length n. - K. G. Stier, Oct 30 2013
Number of positive roots in the root systems of type B_n and C_n (when n > 1). - Tom Edgar, Nov 05 2013
Squares of squares (fourth powers) are also called biquadratic numbers: A000583. - M. F. Hasler, Dec 29 2013
For n > 0, a(n) is the largest integer k such that k^2 + n is a multiple of k + n. More generally, for m > 0 and n > 0, the largest integer k such that k^(2*m) + n is a multiple of k + n is given by k = n^(2*m). - Derek Orr, Sep 03 2014
For n > 0, a(n) is the number of compositions of n + 5 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016
a(n), for n >= 3, is also the number of all connected subtrees of a cycle graph, having n vertices. - Viktar Karatchenia, Mar 02 2016
On every sequence of natural continuous numbers with an even number of elements, the summatory of the second half of the sequence minus the summatory of the first half of the sequence is always a square. Example: Sequence from 61 to 70 has an even number of elements (10). Then 61 + 62 + 63 + 64 + 65 = 315; 66 + 67 + 68 + 69 + 70 = 340; 340 - 315 = 25. (n/2)^2 for n = number of elements. - César Aguilera, Jun 20 2016
On every sequence of natural continuous numbers from n^2 to (n+1)^2, the sum of the differences of pairs of elements of the two halves in every combination possible is always (n+1)^2. - César Aguilera, Jun 24 2016
Suppose two circles with radius 1 are tangent to each other as well as to a line not passing through the point of tangency. Create a third circle tangent to both circles as well as the line. If this process is continued, a(n) for n > 0 is the reciprocals of the radii of the circles, beginning with the largest circle. - Melvin Peralta, Aug 18 2016
Does not satisfy Benford's law [Ross, 2012]. - N. J. A. Sloane, Feb 08 2017
Numerators of the solution to the generalization of the Feynman triangle problem, with an offset of 2. If each vertex of a triangle is joined to the point (1/p) along the opposite side (measured say clockwise), then the area of the inner triangle formed by these lines is equal to (p - 2)^2/(p^2 - p + 1) times the area of the original triangle, p > 2. For example, when p = 3, the ratio of the areas is 1/7. The denominators of the ratio of the areas is given by A002061. [Cook & Wood, 2004] - Joe Marasco, Feb 20 2017
Equals row sums of triangle A004737, n >= 1. - Martin Michael Musatov, Nov 07 2017
Right-hand side of the binomial coefficient identity Sum_{k = 0..n} (-1)^(n+k+1)*binomial(n,k)*binomial(n + k,k)*(n - k) = n^2. - Peter Bala, Jan 12 2022
Conjecture: For n>0, min{k such that there exist subsets A,B of {0,1,2,...,a(n)-1} such that |A|=|B|=k and A+B contains {0,1,2,...,a(n)-1}} = n. - Michael Chu, Mar 09 2022
Number of 3-permutations of n elements avoiding the patterns 132, 213, 321. See Bonichon and Sun. - Michel Marcus, Aug 20 2022
Number of intercalates in cyclic Latin squares of order 2n (cyclic Latin squares of odd order do not have intercalates). - Eduard I. Vatutin, Feb 15 2024
a(n) is the number of ternary strings of length n with at most one 0, exactly one 1, and no restriction on the number of 2's.  For example, a(3)=9, consisting of the 6 permutations of the string 102 and the 3 permutations of the string 122. - Enrique Navarrete, Mar 12 2025

			For n = 8, a(8) = 8 * 15 - (1 + 3 + 5 + 7 + 9 + 11 + 13) - 7 = 8 * 15 - 49 - 7 = 64. - _Bruno Berselli_, May 04 2010
G.f. = x + 4*x^2 + 9*x^3 + 16*x^4 + 25*x^5 + 36*x^6 + 49*x^7 + 64*x^8 + 81*x^9 + ...
a(4) = 16. For n = 4 vertices, the cycle graph C4 is A-B-C-D-A. The subtrees are: 4 singles: A, B, C, D; 4 pairs: A-B, BC, C-D, A-D; 4 triples: A-B-C, B-C-D, C-D-A, D-A-B; 4 quads: A-B-C-D, B-C-D-A, C-D-A-B, D-A-B-C; 4 + 4 + 4 + 4 = 16. - _Viktar Karatchenia_, Mar 02 2016

G. L. Alexanderson et al., The William Lowell Putnam Mathematical Competition, Problems and Solutions: 1965-1984, "December 1967 Problem B4(a)", pp. 8(157) MAA Washington DC 1985.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Chapter XV, pp. 135-167.
R. P. Burn & A. Chetwynd, A Cascade Of Numbers, "The prison door problem" Problem 4 pp. 5-7; 79-80 Arnold London 1996.
H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1996, p. 40.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 31, 36, 38, 63.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), p. 6.
M. Gardner, Time Travel and Other Mathematical Bewilderments, Chapter 6 pp. 71-2, W. H. Freeman NY 1988.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), p. 982.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.1 Terminology and §8.6 Figurate Numbers, pp. 264, 290-291.
Alfred S. Posamentier, The Art of Problem Solving, Section 2.4 "The Long Cell Block" pp. 10-1; 12; 156-7 Corwin Press Thousand Oaks CA 1996.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 35, 52-53, 129-132, 244.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
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).
J. K. Strayer, Elementary Number Theory, Exercise Set 3.3 Problems 32, 33, p. 88, PWS Publishing Co. Boston MA 1996.
C. W. Trigg, Mathematical Quickies, "The Lucky Prisoners" Problem 141 pp. 40, 141, Dover NY 1985.
R. Vakil, A Mathematical Mosaic, "The Painted Lockers" pp. 127;134 Brendan Kelly Burlington Ontario 1996.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

Franklin T. Adams-Watters, The first 10000 squares: Table of n, n^2 for n = 0..10000
Valentin P. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.
Stefano Barbero, Umberto Cerruti, and Nadir Murru, Transforming Recurrent Sequences by Using the Binomial and Invert Operators, J. Int. Seq. 13 (2010) # 10.7.7, section 4.4.
Anicius Manlius Severinus Boethius, De institutione arithmetica libri duo, Book 2, sections 10-12.
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
Henry Bottomley, Some Smarandache-type multiplicative sequences.
R. J. Cook and G. V. Wood, Feynman's Triangle, Mathematical Gazette, 88:299-302 (2004).
John Derbyshire, Monkeys and Doors.
Michael Drmota, Christian Mauduit, and Joël Rivat, The Thue-Morse Sequence Along The Squares is Normal, Abstract, ÖMG-DMV Congress, 2013.
Ralph Greenberg, Math for Poets.
Guo-Niu Han, Enumeration of Standard Puzzles.
Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
Vi Hart, Doodling in Math Class: Connecting Dots (2012) [Video].
Nick Hobson, Python program to determine whether an integer is in sequence A000290.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 338.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Milan Janjic, Two Enumerative Functions.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
L. B. W. Jolley, Summation of Series, Dover, 1961.
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.
Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
J. H. McKay, The William Lowell Putnam Mathematical Competition, Problem B4(a), The American Mathematical Monthly, vol. 75, no. 7, 1968, pp. 732-739.
Matthew Parker, The first million squares (7-Zip compressed file).
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)].
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; arXiv:0911.4975 [math.NT], 2009.
Omar E. Pol, Illustration of initial terms of A000217, A000290, A000326, A000384, A000566, A000567.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
William S. Renwick, The start of the EDSAC log.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Kenneth A. Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42.
John Scholes, 28th Putnam 1967 Prob.B4(a).
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
N. J. A. Sloane, Illustration of initial terms of A000217, A000290, A000326.
Michael Somos, Rational Function Multiplicative Coefficients.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Dinoj Surendran, Chimbumu and Chickwama get out of jail.
Eric Weisstein's World of Mathematics, Square Number.
Eric Weisstein's World of Mathematics, Unit.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for "core" sequences.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for two-way infinite sequences.
Index to sequences related to polygonal numbers.
Index entries for sequences related to Benford's law.

Cf. A092205, A128200, A005408, A128201, A002522, A005563, A008865, A059100, A143051, A143470, A143595, A056944, A001157 (inverse Möbius transform), A001788 (binomial transform), A228039, A001105, A004159, A159918, A173277, A095794, A162395, A186646 (Pisano periods), A028338 (2nd diagonal).
A row or column of A132191.
This sequence is related to partitions of 2^n into powers of 2, as it is shown in A002577. So A002577 connects the squares and A000447. - Valentin Bakoev, Mar 03 2009
Boustrophedon transforms: A000697, A000745.
Cf. A342819.
Cf. A013661.

a000290 = (^ 2)
a000290_list = scanl (+) 0 [1,3..]  -- Reinhard Zumkeller, Apr 06 2012

Magma
```
[ n^2 : n in [0..1000]];
```

A000290 := n->n^2; seq(A000290(n), n=0..50);
A000290 := -(1+z)/(z-1)^3; # Simon Plouffe, in his 1992 dissertation, for sequence starting at a(1)

Array[#^2 &, 51, 0] (* Robert G. Wilson v, Aug 01 2014 *)
LinearRecurrence[{3, -3, 1}, {0, 1, 4}, 60] (* Vincenzo Librandi, Jul 24 2015 *)
CoefficientList[Series[-(x^2 + x)/(x - 1)^3, {x, 0, 50}], x] (* Robert G. Wilson v, Jul 23 2018 *)
Range[0, 99]^2 (* Alonso del Arte, Nov 21 2019 *)
Join[{0},Accumulate[Range[1,101,2]]] (* Harvey P. Dale, May 11 2025 *)

A000290(n):=n^2$ makelist(A000290(n),n,0,30); /* Martin Ettl, Oct 25 2012 */

PARI
```
{a(n) = n^2};
```

b000290(maxn)=for(n=0,maxn,print(n," ",n^2);) \\ Anatoly E. Voevudko, Nov 11 2015

Python
```
# See Hobson link
```

def A000290(n): return n**2 # Chai Wah Wu, Nov 13 2022

(0 to 59).map(n => n * n) // Alonso del Arte, Oct 07 2019

(define (A000290 n) (* n n)) ;; Antti Karttunen, Oct 06 2017

G.f.: x*(1 + x) / (1 - x)^3.
E.g.f.: exp(x)*(x + x^2).
Dirichlet g.f.: zeta(s-2).
a(n) = a(-n).
Multiplicative with a(p^e) = p^(2*e). - David W. Wilson, Aug 01 2001
Sum of all matrix elements M(i, j) = 2*i/(i+j) (i, j = 1..n). a(n) = Sum_{i = 1..n} Sum_{j = 1..n} 2*i/(i + j). - Alexander Adamchuk, Oct 24 2004
a(0) = 0, a(1) = 1, a(n) = 2*a(n-1) - a(n-2) + 2. - Miklos Kristof, Mar 09 2005
From Pierre CAMI, Oct 22 2006: (Start)
a(n) is the sum of the odd numbers from 1 to 2*n - 1.
a(0) = 0, a(1) = 1, then a(n) = a(n-1) + 2*n - 1. (End)
For n > 0: a(n) = A130064(n)*A130065(n). - Reinhard Zumkeller, May 05 2007
a(n) = Sum_{k = 1..n} A002024(n, k). - Reinhard Zumkeller, Jun 24 2007
Left edge of the triangle in A132111: a(n) = A132111(n, 0). - Reinhard Zumkeller, Aug 10 2007
Binomial transform of [1, 3, 2, 0, 0, 0, ...]. - Gary W. Adamson, Nov 21 2007
a(n) = binomial(n+1, 2) + binomial(n, 2).
This sequence could be derived from the following general formula (cf. A001286, A000330): n*(n+1)*...*(n+k)*(n + (n+1) + ... + (n+k))/((k+2)!*(k+1)/2) at k = 0. Indeed, using the formula for the sum of the arithmetic progression (n + (n+1) + ... + (n+k)) = (2*n + k)*(k + 1)/2 the general formula could be rewritten as: n*(n+1)*...*(n+k)*(2*n+k)/(k+2)! so for k = 0 above general formula degenerates to n*(2*n + 0)/(0 + 2) = n^2. - Alexander R. Povolotsky, May 18 2008
From a(4) recurrence formula a(n+3) = 3*a(n+2) - 3*a(n+1) + a(n) and a(1) = 1, a(2) = 4, a(3) = 9. - Artur Jasinski, Oct 21 2008
The recurrence a(n+3) = 3*a(n+2) - 3*a(n+1) + a(n) is satisfied by all k-gonal sequences from a(3), with a(0) = 0, a(1) = 1, a(2) = k. - Jaume Oliver Lafont, Nov 18 2008
a(n) = floor(n*(n+1)*(Sum_{i = 1..n} 1/(n*(n+1)))). - Ctibor O. Zizka, Mar 07 2009
Product_{i >= 2} 1 - 2/a(i) = -sin(A063448)/A063448. - R. J. Mathar, Mar 12 2009
a(n) = A002378(n-1) + n. - Jaroslav Krizek, Jun 14 2009
a(n) = n*A005408(n-1) - (Sum_{i = 1..n-2} A005408(i)) - (n-1) = n*A005408(n-1) - a(n-1) - (n-1). - Bruno Berselli, May 04 2010
a(n) == 1 (mod n+1). - Bruno Berselli, Jun 03 2010
a(n) = a(n-1) + a(n-2) - a(n-3) + 4, n > 2. - Gary Detlefs, Sep 07 2010
a(n+1) = Integral_{x >= 0} exp(-x)/( (Pn(x)*exp(-x)*Ei(x) - Qn(x))^2 +(Pi*exp(-x)*Pn(x))^2 ), with Pn the Laguerre polynomial of order n and Qn the secondary Laguerre polynomial defined by Qn(x) = Integral_{t >= 0} (Pn(x) - Pn(t))*exp(-t)/(x-t). - Groux Roland, Dec 08 2010
Euler transform of length-2 sequence [4, -1]. - Michael Somos, Feb 12 2011
A162395(n) = -(-1)^n * a(n). - Michael Somos, Mar 19 2011
a(n) = A004201(A000217(n)); A007606(a(n)) = A000384(n); A007607(a(n)) = A001105(n). - Reinhard Zumkeller, Feb 12 2011
Sum_{n >= 1} 1/a(n)^k = (2*Pi)^k*B_k/(2*k!) = zeta(2*k) with Bernoulli numbers B_k = -1, 1/6, 1/30, 1/42, ... for k >= 0. See A019673, A195055/10 etc. [Jolley eq 319].
Sum_{n>=1} (-1)^(n+1)/a(n)^k = 2^(k-1)*Pi^k*(1-1/2^(k-1))*B_k/k! [Jolley eq 320] with B_k as above.
A007968(a(n)) = 0. - Reinhard Zumkeller, Jun 18 2011
A071974(a(n)) = n; A071975(a(n)) = 1. - Reinhard Zumkeller, Jul 10 2011
a(n) = A199332(2*n - 1, n). - Reinhard Zumkeller, Nov 23 2011
For n >= 1, a(n) = Sum_{d|n} phi(d)*psi(d), where phi is A000010 and psi is A001615. - Enrique Pérez Herrero, Feb 29 2012
a(n) = A000217(n^2) - A000217(n^2 - 1), for n > 0. - Ivan N. Ianakiev, May 30 2012
a(n) = (A000217(n) + A000326(n))/2. - Omar E. Pol, Jan 11 2013
a(n) = A162610(n, n) = A209297(n, n) for n > 0. - Reinhard Zumkeller, Jan 19 2013
a(A000217(n)) = Sum_{i = 1..n} Sum_{j = 1..n} i*j, for n > 0. - Ivan N. Ianakiev, Apr 20 2013
a(n) = A133280(A000217(n)). - Ivan N. Ianakiev, Aug 13 2013
a(2*a(n)+2*n+1) = a(2*a(n)+2*n) + a(2*n+1). - Vladimir Shevelev, Jan 24 2014
a(n+1) = Sum_{t1+2*t2+...+n*tn = n} (-1)^(n+t1+t2+...+tn)*multinomial(t1+t2 +...+tn,t1,t2,...,tn)*4^(t1)*7^(t2)*8^(t3+...+tn). - Mircea Merca, Feb 27 2014
a(n) = floor(1/(1-cos(1/n)))/2 = floor(1/(1-n*sin(1/n)))/6, n > 0. - Clark Kimberling, Oct 08 2014
a(n) = ceiling(Sum_{k >= 1} log(k)/k^(1+1/n)) = -Zeta'[1+1/n]. Thus any exponent greater than 1 applied to k yields convergence. The fractional portion declines from A073002 = 0.93754... at n = 1 and converges slowly to 0.9271841545163232... for large n. - Richard R. Forberg, Dec 24 2014
a(n) = Sum_{j = 1..n} Sum_{i = 1..n} ceiling((i + j - n + 1)/3). - Wesley Ivan Hurt, Mar 12 2015
a(n) = Product_{j = 1..n-1} 2 - 2*cos(2*j*Pi/n). - Michel Marcus, Jul 24 2015
From Ilya Gutkovskiy, Jun 21 2016: (Start)
Product_{n >= 1} (1 + 1/a(n)) = sinh(Pi)/Pi = A156648.
Sum_{n >= 0} 1/a(n!) = BesselI(0, 2) = A070910. (End)
a(n) = A028338(n, n-1), n >= 1 (second diagonal). - Wolfdieter Lang, Jul 21 2017
For n >= 1, a(n) = Sum_{d|n} sigma_2(d)*mu(n/d) = Sum_{d|n} A001157(d)*A008683(n/d). - Ridouane Oudra, Apr 15 2021
a(n) = Sum_{i = 1..2*n-1} ceiling(n - i/2). - Stefano Spezia, Apr 16 2021
From Richard L. Ollerton, May 09 2021: (Start) For n >= 1,
a(n) = Sum_{k=1..n} psi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} psi(gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} sigma_2(n/gcd(n,k))*mu(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} sigma_2(gcd(n,k))*mu(n/gcd(n,k))/phi(n/gcd(n,k)). (End)
a(n) = (A005449(n) + A000326(n))/3. - Klaus Purath, May 13 2021
Let T(n) = A000217(n), then a(T(n)) + a(T(n+1)) = T(a(n+1)). - Charlie Marion, Jun 27 2022
a(n) = Sum_{k=1..n} sigma_1(k) + Sum_{i=1..n} (n mod i). - Vadim Kataev, Dec 07 2022
a(n^2) + a(n^2+1) + ... + a(n^2+n) + 4*A000537(n) = a(n^2+n+1) + ... + a(n^2+2n).  In general, if P(k,n) = the n-th k-gonal number, then P(2k,n^2) + P(2k,n^2+1) + ... + P(2k,n^2+n) + 4*(k-1)*A000537(n) = P(2k,n^2+n+1) + ... + P(2k,n^2+2n). - Charlie Marion, Apr 26 2024
Sum_{n>=1} 1/a(n) = A013661. - Alois P. Heinz, Oct 19 2024
a(n) = 1 + 3^3*((n-1)/(n+1))^2 + 5^3*((n-1)*(n-2)/((n+1)*(n+2)))^2 + 7^3*((n-1)*(n-2)*(n-3)/((n+1)*(n+2)*(n+3)))^2 + ... for n >= 1. - Peter Bala, Dec 09 2024

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

A000302 Powers of 4: a(n) = 4^n.

Original entry on oeis.org

1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, 262144, 1048576, 4194304, 16777216, 67108864, 268435456, 1073741824, 4294967296, 17179869184, 68719476736, 274877906944, 1099511627776, 4398046511104, 17592186044416, 70368744177664, 281474976710656

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 4), L(1, 4), P(1, 4), T(1, 4). Essentially same as Pisot sequences E(4, 16), L(4, 16), P(4, 16), T(4, 16). See A008776 for definitions of Pisot sequences.
The convolution square root of this sequence is A000984, the central binomial coefficients: C(2n,n). - T. D. Noe, Jun 11 2002
With P(n) being the number of integer partitions of n, p(i) as the number of parts of the i-th partition of n, d(i) as the number of different parts of the i-th partition of n, m(i, j) the multiplicity of the j-th part of the i-th partition of n, one has a(n) = Sum_{i = 1..P(n)} p(i)!/(Product_{j = 1..d(i)} m(i, j)!) * 2^(n-1). - Thomas Wieder, May 18 2005
Sums of rows of the triangle in A122366. - Reinhard Zumkeller, Aug 30 2006
Hankel transform of A076035. - Philippe Deléham, Feb 28 2009
Equals the Catalan sequence: (1, 1, 2, 5, 14, ...), convolved with A032443: (1, 3, 11, 42, ...). - Gary W. Adamson, May 15 2009
Sum of coefficients of expansion of (1 + x + x^2 + x^3)^n.
a(n) is number of compositions of natural numbers into n parts less than 4. For example, a(2) = 16 since there are 16 compositions of natural numbers into 2 parts less than 4.
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 4-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
Squares in A002984. - Reinhard Zumkeller, Dec 28 2011
Row sums of Pascal's triangle using the rule that going left increases the value by a factor of k = 3. For example, the first three rows are {1}, {3, 1}, and {9, 6, 1}. Using this rule gives row sums as (k+1)^n. - Jon Perry, Oct 11 2012
First differences of A002450. - Omar E. Pol, Feb 20 2013
Sum of all peak heights in Dyck paths of semilength n+1. - David Scambler, Apr 22 2013
Powers of 4 exceed powers of 2 by A020522 which is the m-th oblong number A002378(m), m being the n-th Mersenne number A000225(n); hence, we may write, a(n) = A000079(n) + A002378(A000225(n)). - Lekraj Beedassy, Jan 17 2014
a(n) is equal to 1 plus the sum for 0 < k < 2^n of the numerators and denominators of the reduced fractions k/2^n. - J. M. Bergot, Jul 13 2015
Binomial transform of A000244. - Tony Foster III, Oct 01 2016
From Ilya Gutkovskiy, Oct 01 2016: (Start)
Number of nodes at level n regular 4-ary tree.
Partial sums of A002001. (End)
Satisfies Benford's law [Berger-Hill, 2011]. - N. J. A. Sloane, Feb 08 2017
Also the number of connected dominating sets in the (n+1)-barbell graph. - Eric W. Weisstein, Jun 29 2017
Side length of the cells at level n in a pyramid scheme where a square grid is decomposed into overlapping 2 X 2 blocks (cf. Kropatsch, 1985). - Felix Fröhlich, Jul 04 2019
a(n-1) is the number of 3-compositions of n; see Hopkins & Ouvry reference. - Brian Hopkins, Aug 15 2020

H. W. Gould, Combinatorial Identities, 1972, eq. (1.93), p. 12.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, eq. (5.39), p. 187.
D. Phulara and L. W. Shapiro, Descendants in ordered trees with a marked vertex, Congressus Numerantium, 205 (2011), 121-128.
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).
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

T. D. Noe, Table of n, a(n) for n = 0..100
Arno Berger and Theodore P. Hill, Benford's law strikes back: no simple explanation in sight for mathematical gem, The Mathematical Intelligencer 33.1 (2011): 85-91.
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.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
G. Dresden and Y. Li, Periodic Weighted Sums of Binomial Coefficients, arXiv:2210.04322 [math.NT], 2022.
R. Duarte and A. G. de Oliveira, Short note on the convolution of binomial coefficients, arXiv preprint arXiv:1302.2100 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.7.6.
Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
Madeleine Goertz and Aaron Williams, The Quaternary Gray Code and How It Can Be Used to Solve Ziggurat and Other Ziggu Puzzles, arXiv:2411.19291 [math.CO], 2024. See p. 5.
R. K. Guy, Letter to N. J. A. Sloane
Brian Hopkins and Stéphane Ouvry, Combinatorics of Multicompositions, arXiv:2008.04937 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 8
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 269
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
Tanya Khovanova, Recursive Sequences
Craig Knecht, Number of tilings for a 6 sphinx tile repetitive unit.
Walter G. Kropatsch, A pyramid that grows by powers of 2, Pattern Recognition Letters, Vol. 3, No. 5 (1985), 315-322 [Subscription required].
Mircea Merca, A Note on Cosine Power Sums J. Integer Sequences, Vol. 15 (2012), Article 12.5.3.
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
Robert Price, Comments on A000302 concerning Elementary Cellular Automata, Feb 26 2016.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Robert Schneider, Partition zeta functions, Research in Number Theory, 2(1):9., 2016.
Paul K. Stockmeyer, The Pascal Rhombus and the Stealth Configuration, arXiv preprint arXiv:1504.04404 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Barbell Graph
Eric Weisstein's World of Mathematics, Cantor Dust
Eric Weisstein's World of Mathematics, Connected Dominating Set
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata
Index entries for "core" sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (4).
Index entries for sequences related to Benford's law

Cf. A024036, A052539, A032443, A000351 (Binomial transform).
Cf. A249307.
Cf. A083420.

a000302 = (4 ^)
a000302_list = iterate (* 4) 1  -- Reinhard Zumkeller, Apr 04 2012

A000302 := n->4^n;
for n from 0 to 10 do sum(2^(n-j)*binomial(n+j,j),j=0..n); od; # Peter C. Heinig (algorithms(AT)gmx.de), Apr 06 2007
A000302:=-1/(-1+4*z); # Simon Plouffe in his 1992 dissertation.

Table[4^n, {n, 0, 30}] (* Stefan Steinerberger, Apr 01 2006 *)
CoefficientList[Series[1/(1 - 4 x), {x, 0, 50}], x]  (* Vincenzo Librandi, May 29 2014 *)
NestList[4 # &, 1, 30] (* Harvey P. Dale, Mar 26 2015 *)
4^Range[0, 30] (* Eric W. Weisstein, Jun 29 2017 *)
LinearRecurrence[{4}, {1}, 31] (* Robert A. Russell, Nov 08 2018 *)

A000302(n):=4^n$ makelist(A000302(n),n,0,30); /* Martin Ettl, Oct 24 2012 */

A000302(n)=4^n \\ Michael B. Porter, Nov 06 2009

print([4**n for n in range(25)]) # Michael S. Branicky, Jan 04 2021

is_A000302 = lambda n: n.bit_count()==1 and n.bit_length()&1 # M. F. Hasler, Nov 25 2024

[4**n for n in range(0,25)] # Stefano Spezia, Jul 23 2025

(List.fill(20)(4: BigInt)).scanLeft(1: BigInt)( * ) // Alonso del Arte, Jun 22 2019

a(n) = 4^n.
a(0) = 1; a(n) = 4*a(n-1).
G.f.: 1/(1-4*x).
E.g.f.: exp(4*x).
a(n) = Sum_{k = 0..n} binomial(2k, k) * binomial(2(n - k), n - k). - Benoit Cloitre, Jan 26 2003 [See Graham et al., eq. (5.39), p. 187. - Wolfdieter Lang, Aug 16 2019]
1 = Sum_{n >= 1} 3/a(n) = 3/4 + 3/16 + 3/64 + 3/256 + 3/1024, ...; with partial sums: 3/4, 15/16, 63/64, 255/256, 1023/1024, ... - Gary W. Adamson, Jun 16 2003
a(n) = A001045(2*n) + A001045(2*n+1). - Paul Barry, Apr 27 2004
A000005(a(n)) = A005408(n+1). - Reinhard Zumkeller, Mar 04 2007
a(n) = Sum_{j = 0..n} 2^(n - j)*binomial(n + j, j). - Peter C. Heinig (algorithms(AT)gmx.de), Apr 06 2007
Hankel transform of A115967. - Philippe Deléham, Jun 22 2007
a(n) = 6*Stirling2(n+1, 4) + 6*Stirling2(n+1, 3) + 3*Stirling2(n+1, 2) + 1 = 2*Stirling2(2^n, 2^n - 1) + Stirling2(n+1, 2) + 1. - Ross La Haye, Jun 26 2008
a(n) = A159991(n)/A001024(n) = A047653(n) + A181765(n). A160700(a(n)) = A010685(n). - Reinhard Zumkeller, May 02 2009
a(n) = A188915(A006127(n)). - Reinhard Zumkeller, Apr 14 2011
a(n) = Sum_{k = 0..n} binomial(2*n+1, k). - Mircea Merca, Jun 25 2011
Sum_{n >= 1} Mobius(n)/a(n) = 0.1710822479183... - R. J. Mathar, Aug 12 2012
a(n) = Sum_{k = 0..n} binomial(2*k + x, k)*binomial(2*(n - k) - x, n - k) for every real number x. - Rui Duarte and António Guedes de Oliveira, Feb 16 2013
a(n) = 5*a(n - 1) - 4*a(n - 2). - Jean-Bernard François, Sep 12 2013
a(n) = (2*n+1) * binomial(2*n,n) * Sum_{j=0..n} (-1)^j/(2*j+1)*binomial(n,j). - Vaclav Kotesovec, Sep 15 2013
a(n) = A000217(2^n - 1) + A000217(2^n). - J. M. Bergot, Dec 28 2014
a(n) = (2^n)^2 = A000079(n)^2. - Doug Bell, Jun 23 2015
a(n) = A002063(n)/3 - A004171(n). - Zhandos Mambetaliyev, Nov 19 2016
a(n) = (1/2) * Product_{k = 0..n} (1 + (2*n + 1)/(2*k + 1)). - Peter Bala, Mar 06 2018
a(n) = A001045(n+1)*A001045(n+2) + A001045(n)^2. - Ezhilarasu Velayutham, Aug 30 2019
a(n) = denominator(zeta_star({2}(n + 1))/zeta(2*n + 2)) where zeta_star is the multiple zeta star values and ({2}_n) represents (2, ..., 2) where the multiplicity of 2 is n. - _Roudy El Haddad, Feb 22 2022
a(n) = 1 + 3*Sum_{k=0..n} binomial(2*n, n+k)*(k|9), where (k|9) is the Jacobi symbol. - Greg Dresden, Oct 11 2022
a(n) = Sum_{k = 0..n} binomial(2*n+1, 2*k) = Sum_{k = 0..n} binomial(2*n+1, 2*k+1). - Sela Fried, Mar 23 2023

Partially edited by Joerg Arndt, Mar 11 2010

A159991 Powers of 60: a(n) = 60^n.

Original entry on oeis.org

1, 60, 3600, 216000, 12960000, 777600000, 46656000000, 2799360000000, 167961600000000, 10077696000000000, 604661760000000000, 36279705600000000000, 2176782336000000000000, 130606940160000000000000, 7836416409600000000000000, 470184984576000000000000000

Offset: 0

Reinhard Zumkeller, May 01 2009

			G.f. = 1 + 60*x + 3600*x^2 + 216000*x^3 + 12960000*x^4 + 77600000*x^5 + ... - _Michael Somos_, Jan 01 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..150
Wikipedia, Sexagesimal
Index entries for linear recurrences with constant coefficients, signature (60).

Subsequence of A051037.

Cf. A159990, A159993, A159995, A088157, A091649, A125628, A091720, A091721, A091722, A060707, A070197.

List([0..20],n->60^n); # Muniru A Asiru, Nov 21 2018

[60^n: n in [0..20]]; // Vincenzo Librandi, May 02 2011

[60^n$n=0..20]; # Muniru A Asiru, Nov 21 2018

60^Range[0,15] (* Harvey P. Dale, Jun 02 2011 *)

A159991(n):=60^n$
makelist(A159991(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=60^n \\ Charles R Greathouse IV, May 02 2011

a(n)=60^n \\ Charles R Greathouse IV, Jun 19 2015

powers(60,8) \\ Charles R Greathouse IV, Jun 19 2015

for n in range(0,20): print(60**n, end=', ') # Stefano Spezia, Nov 21 2018

a(n) = A000400(n)*A011557(n) = A000351(n)*A001021(n) = A000302(n)*A001024(n) = A000244(n)*A009964(n). (Corrected by Robert B Fowler, Jan 25 2023)
From Muniru A Asiru, Nov 21 2018: (Start)
a(n) = 60^n.
a(n) = 60*a(n-1) for n > 0, a(0) = 1.
G.f.: 1/(1-60*x).
E.g.f: exp(60*x). (End)
a(n) = 1/a(-n) for all n in Z. - Michael Somos, Jan 01 2019

A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 9, 8, 1, 0, 1, 5, 16, 27, 16, 1, 0, 1, 6, 25, 64, 81, 32, 1, 0, 1, 7, 36, 125, 256, 243, 64, 1, 0, 1, 8, 49, 216, 625, 1024, 729, 128, 1, 0, 1, 9, 64, 343, 1296, 3125, 4096, 2187, 256, 1, 0, 1, 10, 81, 512, 2401, 7776, 15625, 16384, 6561, 512, 1, 0

Offset: 0

Marc LeBrun

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020

			Rows begin:
[1, 0,  0,   0,    0,     0,      0,      0, ...],
[1, 1,  1,   1,    1,     1,      1,      1, ...],
[1, 2,  4,   8,   16,    32,     64,    128, ...],
[1, 3,  9,  27,   81,   243,    729,   2187, ...],
[1, 4, 16,  64,  256,  1024,   4096,  16384, ...],
[1, 5, 25, 125,  625,  3125,  15625,  78125, ...],
[1, 6, 36, 216, 1296,  7776,  46656, 279936, ...],
[1, 7, 49, 343, 2401, 16807, 117649, 823543, ...], ...

T. D. Noe, Rows n = 0..50 of triangle, flattened

Rows 0-49 are A000007, A000012, A000079, A000244, A000302, A000351, A000400, A000420, A001018, A001019, A011557, A001020, A001021, A001022, A001023, A001024, A001025, A001026, A001027, A001029, A009964-A009992, A087752.
Columns 0-26 are A000012, A001477, A000290, A000578, A000583, A000584, A001014, A001015, A001016, A001017, A008454, A008455, A008456, A010801-A010813, A089081.
Main diagonal is A000312. Other diagonals include A000169, A007778, A000272, A008788. Antidiagonal sums are in A026898.
Cf. A099555.
Transpose is A004248. See A051128, A095884, A009999 for other versions.
Cf. A277504 (unoriented), A293500 (chiral).

[[(n-k)^k: k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 08 2018

Table[If[k == 0, 1, (n - k)^k], {n, 0, 11}, {k, 0, n}]//Flatten

T(n,k) = (n-k)^k \\ Charles R Greathouse IV, Feb 07 2017

E.g.f.: Sum T(n,k)*x^n*y^k/k! = 1/(1-x*exp(y)). - Paul D. Hanna, Oct 22 2004

E.g.f.: Sum T(n,k)*x^n/n!*y^k/k! = e^(x*e^y). - Franklin T. Adams-Watters, Jun 23 2006

More terms from David W. Wilson

Edited by Paul D. Hanna, Oct 22 2004

A329332 Table of powers of squarefree numbers, powers of A019565(n) in increasing order in row n. Square array A(n,k) n >= 0, k >= 0 read by descending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 9, 6, 1, 1, 16, 27, 36, 5, 1, 1, 32, 81, 216, 25, 10, 1, 1, 64, 243, 1296, 125, 100, 15, 1, 1, 128, 729, 7776, 625, 1000, 225, 30, 1, 1, 256, 2187, 46656, 3125, 10000, 3375, 900, 7, 1, 1, 512, 6561, 279936, 15625, 100000, 50625, 27000, 49, 14

Offset: 0

Peter Munn, Nov 10 2019

The A019565 row order gives the table neat relationships with A003961, A003987, A059897, A225546, A319075 and A329050. See the formula section.

Transposition of this table, that is reflection about its main diagonal, has subtle symmetries. For example, consider the unique factorization of a number into powers of distinct primes. This can be restated as factorization into numbers from rows 2^n (n >= 0) with no more than one from each row. Reflecting about the main diagonal, this factorization becomes factorization (of a related number) into numbers from columns 2^k (k >= 0) with no more than one from each column. This is also unique and is factorization into powers of squarefree numbers with distinct exponents that are powers of two. See the example section.

			Square array A(n,k) begins:
n\k |  0   1     2      3        4          5           6             7
----+------------------------------------------------------------------
   0|  1   1     1      1        1          1           1             1
   1|  1   2     4      8       16         32          64           128
   2|  1   3     9     27       81        243         729          2187
   3|  1   6    36    216     1296       7776       46656        279936
   4|  1   5    25    125      625       3125       15625         78125
   5|  1  10   100   1000    10000     100000     1000000      10000000
   6|  1  15   225   3375    50625     759375    11390625     170859375
   7|  1  30   900  27000   810000   24300000   729000000   21870000000
   8|  1   7    49    343     2401      16807      117649        823543
   9|  1  14   196   2744    38416     537824     7529536     105413504
  10|  1  21   441   9261   194481    4084101    85766121    1801088541
  11|  1  42  1764  74088  3111696  130691232  5489031744  230539333248
  12|  1  35  1225  42875  1500625   52521875  1838265625   64339296875
Reflection of factorization about the main diagonal: (Start)
The canonical (prime power) factorization of 864 is 2^5 * 3^3 = 32 * 27. Reflecting the factors about the main diagonal of the table gives us 10 * 36 = 10^1 * 6^2 = 360. This is the unique factorization of 360 into powers of squarefree numbers with distinct exponents that are powers of two.
Reflection about the main diagonal is given by the self-inverse function A225546(.). Clearly, all positive integers are in the domain of A225546, whether or not they appear in the table. It is valid to start from 360, observe that A225546(360) = 864, then use 864 to derive 360's factorization into appropriate powers of squarefree numbers as above.
(End)

The range of values is A072774.
Rows (abbreviated list): A000079(1), A000244(2), A000400(3), A000351(4), A011557(5), A001024(6), A009974(7), A000420(8), A001023(9), A009965(10), A001020(16), A001022(32), A001026(64).
A019565 is column 1, A334110 is column 2, and columns that are sorted in increasing order (some without the 1) are: A005117(1), A062503(2), A062838(3), A113849(4), A113850(5), A113851(6), A113852(7).
Other subtables: A182944, A319075, A329050.
Re-ordered subtable of A297845, A306697, A329329.
A000290, A003961, A003987, A059897 and A225546 are used to express relationships between terms of this sequence.
Cf. A285322.

A(n,k) = A019565(n)^k.
A(k,n) = A225546(A(n,k)).
A(n,2k) = A000290(A(n,k)) = A(n,k)^2.
A(2n,k) = A003961(A(n,k)).
A(n,2k+1) = A(n,2k) * A(n,1).
A(2n+1,k) = A(2n,k) * A(1,k).
A(A003987(n,m), k) = A059897(A(n,k), A(m,k)).
A(n, A003987(m,k)) = A059897(A(n,m), A(n,k)).
A(2^n,k) = A319075(k,n+1).
A(2^n, 2^k) = A329050(n,k).
A(n,k) = A297845(A(n,1), A(1,k)) = A306697(A(n,1), A(1,k)), = A329329(A(n,1), A(1,k)).
Sum_{n>=0} 1/A(n,k) = zeta(k)/zeta(2*k), for k >= 2. - Amiram Eldar, Dec 03 2022

A072978 Numbers of the form m*2^Omega(m), where m>1 is odd and Omega(m)=A001222(m), the number of prime factors of m.

Original entry on oeis.org

1, 6, 10, 14, 22, 26, 34, 36, 38, 46, 58, 60, 62, 74, 82, 84, 86, 94, 100, 106, 118, 122, 132, 134, 140, 142, 146, 156, 158, 166, 178, 194, 196, 202, 204, 206, 214, 216, 218, 220, 226, 228, 254, 260, 262, 274, 276, 278, 298, 302, 308, 314, 326, 334, 340, 346

Offset: 1

Reinhard Zumkeller, Aug 20 2002

(number of odd prime factors) = (number of even prime factors).
A000400, A011557, A001023, A001024, A009965, A009966 and A009975 are subsequences. - Reinhard Zumkeller, Jan 06 2008
Subsequence of A028260. - Reinhard Zumkeller, Sep 20 2008

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001222, A007814, A087436.

Cf. A000400, A011557, A001023, A001024, A009965, A009966, A009975, A028260.

Join[{1}, Select[Range[2, 500, 2], First[#] == Total[Rest[#]] & [FactorInteger[#][[All, 2]]] &]] (* Paolo Xausa, Feb 19 2025 *)

isok(k) = {my(v = valuation(k, 2)); bigomega(k >> v) == v;} \\ Amiram Eldar, May 15 2025

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A072978(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def h(x,n): return sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,1,3,1,n))
    def f(x): return int(n+x-primepi(x>>1)-sum(h(x>>m,m) for m in range(2,x.bit_length()+1))) if x>1 else 1
    return bisection(f,n,n) # Chai Wah Wu, Apr 10 2025

A007814(a(n)) = A087436(a(n)). - Reinhard Zumkeller, Jan 06 2008

A275069 Number A(n,k) of set partitions of [n] such that i-j is a multiple of k for all i,j belonging to the same block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 15, 1, 1, 1, 1, 1, 4, 52, 1, 1, 1, 1, 1, 2, 10, 203, 1, 1, 1, 1, 1, 1, 4, 25, 877, 1, 1, 1, 1, 1, 1, 2, 8, 75, 4140, 1, 1, 1, 1, 1, 1, 1, 4, 20, 225, 21147, 1, 1, 1, 1, 1, 1, 1, 2, 8, 50, 780, 115975, 1

Offset: 0

Alois P. Heinz, Jul 15 2016

			A(5,0) = 1: 1|2|3|4|5.
A(5,1) = 52 = A000110(5).
A(5,2) = 10: 135|24, 13|24|5, 135|2|4, 13|2|4|5, 15|24|3, 1|24|35, 1|24|3|5, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
A(5,3) = 4: 14|25|3, 14|2|3|5, 1|25|3|4, 1|2|3|4|5.
A(5,4) = 2: 15|2|3|4, 1|2|3|4|5.
Square array A(n,k) begins:
  1,      1,    1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,      1,    1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,      2,    1,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,      5,    2,   1,   1,  1,  1, 1, 1, 1, 1, ...
  1,     15,    4,   2,   1,  1,  1, 1, 1, 1, 1, ...
  1,     52,   10,   4,   2,  1,  1, 1, 1, 1, 1, ...
  1,    203,   25,   8,   4,  2,  1, 1, 1, 1, 1, ...
  1,    877,   75,  20,   8,  4,  2, 1, 1, 1, 1, ...
  1,   4140,  225,  50,  16,  8,  4, 2, 1, 1, 1, ...
  1,  21147,  780, 125,  40, 16,  8, 4, 2, 1, 1, ...
  1, 115975, 2704, 375, 100, 32, 16, 8, 4, 2, 1, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A000110, A124419, A275070, A275071, A275072, A275073, A275074, A275075, A275076, A275077.

A(k*n,n) for k=1-4 gives: A000012, A000079, A000351, A001024.

with(combinat):
A:= (n, k)-> mul(bell(floor((n+i)/k)), i=0..k-1):
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := Product[BellB[Floor[(n+i)/k]], {i, 0, k-1}]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 17 2017, translated from Maple *)

A(n,k) = Product_{i=0..k-1} A000110(floor((n+i)/k)).

A183827 T(n,k)=1/256 the number of (n+1)X(k+1) 0..3 arrays with no 2X2 subblock being a reflection across the shared element pair of any horizontal or vertical neighbor.

Original entry on oeis.org

1, 15, 15, 225, 813, 225, 3375, 43947, 43947, 3375, 50625, 2377341, 8538066, 2377341, 50625, 759375, 128578815, 1661285961, 1661285961, 128578815, 759375, 11390625, 6954559893, 323115202755, 1163575148283, 323115202755, 6954559893, 11390625

Offset: 1

R. H. Hardin Jan 07 2011

Table starts
...........1...............15.................225..................3375
..........15..............813...............43947...............2377341
.........225............43947.............8538066............1661285961
........3375..........2377341..........1661285961.........1163575148283
.......50625........128578815........323115202755.......814479338538099
......759375.......6954559893......62851591581012....570213177028325955
....11390625.....376152548283...12225396762980829.399185865558804067443
...170859375...20345104031589.2378005715920214025
..2562890625.1100412240703119
.38443359375

			Some solutions with upper left block zero for 4X3
..0..0..0....0..0..3....0..0..2....0..0..2....0..0..0....0..0..1....0..0..1
..0..0..1....0..0..3....0..0..2....0..0..1....0..0..2....0..0..3....0..0..1
..1..0..3....1..2..0....0..3..2....2..1..1....2..2..0....2..3..0....1..0..3
..3..3..0....2..2..2....0..2..3....3..1..1....1..3..3....2..2..1....1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..60

Column 1 is A001024(n-1)

A185567 T(n,k)=1/4 the number of nXk 0..3 arrays with no element equal both to the element above and to the element to its left.

Original entry on oeis.org

1, 4, 4, 16, 60, 16, 64, 900, 900, 64, 256, 13500, 50580, 13500, 256, 1024, 202500, 2842560, 2842560, 202500, 1024, 4096, 3037500, 159749820, 598507920, 159749820, 3037500, 4096, 16384, 45562500, 8977824540, 126017211780, 126017211780

Offset: 1

R. H. Hardin Jan 31 2011

Table starts
.....1.........4.............16..................64.....................256
.....4........60............900...............13500..................202500
....16.......900..........50580.............2842560...............159749820
....64.....13500........2842560...........598507920............126017211780
...256....202500......159749820........126017211780..........99407416968000
..1024...3037500.....8977824540......26533211918040.......78416544382742160
..4096..45562500...504547256880....5586628402633500....61858104930689319360
.16384.683437500.28355191537860.1176277376648694960.48796145962555619907720

			Some solutions for 4X3 with a(1,1)=0
..0..2..2....0..0..2....0..0..2....0..0..0....0..0..2....0..0..2....0..0..0
..0..2..0....2..0..2....2..2..0....2..0..2....0..2..0....2..2..3....2..2..1
..1..0..3....1..3..0....3..3..2....1..1..3....2..0..2....3..2..3....3..3..0
..1..2..3....1..0..1....1..3..1....0..3..0....3..3..0....3..3..2....1..3..1

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 2 is 4*A001024(n-1)

cp's OEIS Frontend

A060218 Number of orbits of length n under the full 15-shift (whose periodic points are counted by A001024).

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A000290 The squares: a(n) = n^2.

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A000302 Powers of 4: a(n) = 4^n.

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A159991 Powers of 60: a(n) = 60^n.

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A003992 Square array read by upwards antidiagonals: T(n,k) = n^k for n >= 0, k >= 0.

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