Displaying 1-10 of 14 results found.
Numerator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/ A045896.
+20
6
0, 1, 1, 1, 1, 5, 1, 7, 1, 3, 5, 11, 1, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 1, 25, 13, 9, 7, 29, 5, 31, 4, 11, 17, 35, 3, 37, 19, 13, 5, 41, 7, 43, 11, 15, 23, 47, 2, 49, 25, 17, 13, 53, 9, 55, 7, 19, 29, 59, 5, 61, 31, 21, 8, 65, 11, 67, 17, 23, 35, 71, 3, 73
FORMULA
c(n) = a(n+1) is multiplicative with c(2^e) = 2^(e-3) if e > 2 and 1 otherwise, c(3^e) = 3^(e-1), and c(p^e) = p^e if p >= 5. [corrected by Amiram Eldar, Nov 20 2022] Sum_{k=1..n} a(k) ~ (301/1152) * n^2. - Amiram Eldar, Nov 20 2022
EXAMPLE
0, 1/60, 1/60, 1/70, 1/84, 5/504, 1/120, 7/990, 1/165, 3/572,...
MATHEMATICA
b[n_] := n/((n + 1) (n + 2)); Numerator[-Differences[Array[b, 100]]]
(* or *)
f[p_, e_] := p^e; f[2, e_] := If[e < 3, 1, 2^(e - 3)]; f[3, e_] := 3^(e - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n - 1]; Array[a, 100] (* Amiram Eldar, Nov 20 2022 *)
Denominator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/ A045896.
+20
4
1, 60, 60, 70, 84, 504, 120, 990, 165, 572, 1092, 2730, 280, 4080, 2448, 1938, 855, 7980, 1540, 10626, 3036, 4600, 7800, 17550, 819, 21924, 12180, 8990, 7440, 32736, 5984, 39270, 5355, 15540, 25308, 54834, 4940, 63960, 34440
EXAMPLE
0, 1/60, 1/60, 1/70, 1/84, 5/504, 1/120, 7/990, 1/165, 3/572,...
MATHEMATICA
Denominator[#[[1]]-#[[2]]&/@(Partition[#[[1]]/(#[[2]]#[[3]])&/@Partition[ Range[50], 3, 1], 2, 1])] (* Harvey P. Dale, Nov 15 2014 *)
Hexagonal numbers: a(n) = n*(2*n-1).
(Formerly M4108 N1705)
+10
442
0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, 1891, 2016, 2145, 2278, 2415, 2556, 2701, 2850, 3003, 3160, 3321, 3486, 3655, 3828, 4005, 4186, 4371, 4560
COMMENTS
Number of edges in the join of two complete graphs, each of order n, K_n * K_n. - Roberto E. Martinez II, Jan 07 2002 The power series expansion of the entropy function H(x) = (1+x)log(1+x) + (1-x)log(1-x) has 1/a_i as the coefficient of x^(2i) (the odd terms being zero). - Tommaso Toffoli (tt(AT)bu.edu), May 06 2002
Sequence also gives the greatest semiperimeter of primitive Pythagorean triangles having inradius n-1. Such a triangle has consecutive longer sides, with short leg 2n-1, hypotenuse a(n) - (n-1) = A001844(n), and area (n-1)*a(n) = 6* A000330(n-1). - Lekraj Beedassy, Apr 23 2003 More generally, if p1 and p2 are two arbitrarily chosen distinct primes then a(n) is the number of divisors of (p1^2*p2)^(n-1) or equivalently of any member of A054753^(n-1). - Ant King, Aug 29 2011 Number of standard tableaux of shape (2n-1,1,1) (n>=1). - Emeric Deutsch, May 30 2004 It is well known that for n>0, A014105(n) [0,3,10,21,...] is the first of 2n+1 consecutive integers such that the sum of the squares of the first n+1 such integers is equal to the sum of the squares of the last n; e.g., 10^2 + 11^2 + 12^2 = 13^2 + 14^2. Less well known is that for n>1, a(n) [0,1,6,15,28,...] is the first of 2n consecutive integers such that sum of the squares of the first n such integers is equal to the sum of the squares of the last n-1 plus n^2; e.g., 15^2 + 16^2 + 17^2 = 19^2 + 20^2 + 3^2. - Charlie Marion, Dec 16 2006 Sequence found by reading the line from 0, in the direction 0, 6, ... and the line from 1, in the direction 1, 15, ..., in the square spiral whose vertices are the generalized hexagonal numbers A000217. - Omar E. Pol, Jan 09 2009 Let Hex(n)=hexagonal number, T(n)=triangular number, then Hex(n)=T(n)+3*T(n-1). - Vincenzo Librandi, Nov 10 2010 For n>=1, 1/a(n) = Sum_{k=0..2*n-1} ((-1)^(k+1)*binomial(2*n-1,k)*binomial(2*n-1+k,k)*H(k)/(k+1)) with H(k) harmonic number of order k.
The number of possible distinct colorings of any 2 colors chosen from n colors of a square divided into quadrants. - Paul Cleary, Dec 21 2010 For n>0, a(n-1) is the number of triples (w,x,y) with all terms in {0,...,n} and max(|w-x|,|x-y|) = |w-y|. - Clark Kimberling, Jun 12 2012 a(n) is the number of positions of one domino in an even pyramidal board with base 2n. - César Eliud Lozada, Sep 26 2012 Let a triangle have T(0,0) = 0 and T(r,c) = |r^2 - c^2|. The sum of the differences of the terms in row(n) and row(n-1) is a(n). - J. M. Bergot, Jun 17 2013 With T_(i+1,i)=a(i+1) and all other elements of the lower triangular matrix T zero, T is the infinitesimal generator for A176230, analogous to A132440 for the Pascal matrix. - Tom Copeland, Dec 11 2013 a(n) is the number of length 2n binary sequences that have exactly two 1's. a(2) = 6 because we have: {0,0,1,1}, {0,1,0,1}, {0,1,1,0}, {1,0,0,1}, {1,0,1,0}, {1,1,0,0}. The ordinary generating function with interpolated zeros is: (x^2 + 3*x^4)/(1-x^2)^3. - Geoffrey Critzer, Jan 02 2014 For n > 0, a(n) is the largest integer k such that k^2 + n^2 is a multiple of k + n. More generally, for m > 0 and n > 0, the largest integer k such that k^(2*m) + n^(2*m) is a multiple of k + n is given by k = 2*n^(2*m) - n. - Derek Orr, Sep 04 2014 Binomial transform of (0, 1, 4, 0, 0, 0, ...) and second partial sum of (0, 1, 4, 4, 4, ...). - Gary W. Adamson, Oct 05 2015 a(n) also gives the dimension of the simple Lie algebras D_n, for n >= 4. - Wolfdieter Lang, Oct 21 2015 For n > 0, a(n) equals the number of compositions of n+11 into n parts avoiding parts 2, 3, 4. - Milan Janjic, Jan 07 2016 Also the number of minimum dominating sets and maximal irredundant sets in the n-cocktail party graph. - Eric W. Weisstein, Jun 29 and Aug 17 2017 As Beedassy's formula shows, this Hexagonal number sequence is the odd bisection of the Triangle number sequence. Both of these sequences are figurative number sequences. For A000384, a(n) can be found by multiplying its triangle number by its hexagonal number. For example let's use the number 153. 153 is said to be the 17th triangle number but is also said to be the 9th hexagonal number. Triangle(17) Hexagonal(9). 17*9=153. Because the Hexagonal number sequence is a subset of the Triangle number sequence, the Hexagonal number sequence will always have both a triangle number and a hexagonal number. n* (2*n-1) because (2*n-1) renders the triangle number. - Bruce J. Nicholson, Nov 05 2017 Also numbers k with the property that in the symmetric representation of sigma(k) the smallest Dyck path has a central valley and the largest Dyck path has a central peak, n >= 1. Thus all hexagonal numbers > 0 have middle divisors. (Cf. A237593.) - Omar E. Pol, Aug 28 2018 Consider all Pythagorean triples (X, Y, Z=Y+1) ordered by increasing Z: a(n+1) gives the semiperimeter of related triangles; A005408, A046092 and A001844 give the X, Y and Z values. - Ralf Steiner, Feb 25 2020 It appears that these are the numbers k with the property that the smallest subpart in the symmetric representation of sigma(k) is 1. - Omar E. Pol, Aug 28 2021 The n-th hexagonal number equals the sum of the n consecutive integers with the same parity starting at 2*n-1; for example, 1, 2+4, 3+5+7, 4+6+8+10, etc. In general, the n-th 2k-gonal number is the sum of the n consecutive integers with the same parity starting at (k-2)*n - (k-3). When k = 1 and 2, this result generates the positive integers, A000027, and the squares, A000290, respectively. - Charlie Marion, Mar 02 2022 Conjecture: For n>0, min{k such that there exist subsets A,B of {0,1,2,...,a(n)} such that |A|=|B|=k and A+B={0,1,2,...,2*a(n)}} = 2*n. - Michael Chu, Mar 09 2022
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77-78. (In the integral formula on p. 77 a left bracket is missing for the cosine argument.)
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 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).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See pp. 122-123.
LINKS
Jonathan M. Borwein, Dirk Nuyens, Armin Straub and James Wan, Random Walk Integrals, The Ramanujan Journal, October 2011, 26:109. DOI: 10.1007/s11139-011-9325-y. Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
FORMULA
E.g.f.: exp(x)*(x+2x^2). - Paul Barry, Jun 09 2003 G.f.: x*(1+3*x)/(1-x)^3. - Simon Plouffe in his 1992 dissertation, dropping the initial zero a(n) = right term of M^n * [1,0,0], where M = the 3 X 3 matrix [1,0,0; 1,1,0; 1,4,1]. Example: a(5) = 45 since M^5 *[1,0,0] = [1,5,45]. - Gary W. Adamson, Dec 24 2006 Starting with offset 1, = binomial transform of [1, 5, 4, 0, 0, 0, ...]. Also, A004736 * [1, 4, 4, 4, ...]. - Gary W. Adamson, Oct 25 2007 a(n)^2 + (a(n)+1)^2 + ... + (a(n)+n-1)^2 = (a(n)+n+1)^2 + ... + (a(n)+2n-1)^2 + n^2; e.g., 6^2 + 7^2 = 9^2 + 2^2; 28^2 + 29^2 + 30^2 + 31^2 = 33^2 + 34^2 + 35^2 + 4^2. - Charlie Marion, Nov 10 2007 a(n) = binomial(n+1,2) + 3*binomial(n,2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=6. - Jaume Oliver Lafont, Dec 02 2008 a(n) = 2*a(n-1) - a(n-2) + 4. - Ant King, Aug 26 2011 a(n+1) = trinomial(2*n+1, 2) = trinomial(2*n+1, 4*n), for n >= 0, with the trinomial irregular triangle A027907. a(n+1) = (n+1)*(2*n+1) = (1/Pi)*Integral_{x=0..2} (1/sqrt(4 - x^2))*(x^2 - 1)^(2*n+1)*R(4*n-2, x) with the R polynomial coefficients given in A127672. [Comtet, p. 77, the integral formula for q=3, n -> 2*n+1, k = 2, rewritten with x = 2*cos(phi)]. - Wolfdieter Lang, Apr 19 2018 Product_{n>=2} (1 - 1/a(n)) = 2/3. - Amiram Eldar, Jan 21 2021 a(n) = floor(Sum_{k=(n-1)^2..n^2} sqrt(k)), for n >= 1. - Amrit Awasthi, Jun 13 2021
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 1, 6}, 50] (* Harvey P. Dale, Sep 10 2015 *) Join[{0}, Accumulate[Range[1, 312, 4]]] (* Harvey P. Dale, Mar 26 2016 *) (* For Mathematica 10.4+ *) Table[PolygonalNumber[RegularPolygon[6], n], {n, 0, 48}] (* Arkadiusz Wesolowski, Aug 27 2016 *) CoefficientList[Series[x*(1 + 3*x)/(1 - x)^3 , {x, 0, 100}], x] (* Stefano Spezia, Sep 02 2018 *)
PROG
(PARI) a(n)=n*(2*n-1)
(PARI) a(n) = binomial(2*n, 2) \\ Altug Alkan, Oct 06 2015 (Haskell)
a000384 n = n * (2 * n - 1)
a000384_list = scanl (+) 0 a016813_list
(Python) # Intended to compute the initial segment of the sequence, not isolated terms.
def aList():
x, y = 1, 1
yield 0
while True:
yield x
x, y = x + y + 4, y + 4
CROSSREFS
a(n)= A093561(n+1, 2), (4, 1)-Pascal column. Cf. A002939 (twice a(n): sums of Pythagorean triples (X, Y, Z=Y+1)).
a(n) = n if n odd, n/2 if n even.
+10
195
0, 1, 1, 3, 2, 5, 3, 7, 4, 9, 5, 11, 6, 13, 7, 15, 8, 17, 9, 19, 10, 21, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 16, 33, 17, 35, 18, 37, 19, 39, 20, 41, 21, 43, 22, 45, 23, 47, 24, 49, 25, 51, 26, 53, 27, 55, 28, 57, 29, 59, 30, 61, 31, 63, 32, 65, 33, 67, 34, 69, 35, 71, 36, 73, 37, 75, 38
COMMENTS
a(n) is the size of largest conjugacy class in D_2n, the dihedral group with 2n elements. - Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002
a(n+1) is the composition length of the n-th symmetric power of the natural representation of a finite subgroup of SL(2,C) of type D_4 (quaternion group). - Paul Boddington, Oct 23 2003 For n > 1, a(n) is the greatest common divisor of all permutations of {0, 1, ..., n} treated as base n + 1 integers. - David Scambler, Nov 08 2006 (see the Mathematics Stack Exchange link below). From Dimitrios Choussos (choussos(AT)yahoo.de), May 11 2009: (Start)
Sequence A075888 and the above sequence are fitting together. First 2 entries of this sequence have to be taken out.
In some cases two three or more sequenced entries of this sequence have to be added together to get the next entry of A075888. Example: Sequences begin with 1, 3, 2, 5, 3, 7, 4, 9 (4 + 9 = 13, the next entry in A075888). But it works out well up to primes around 50000 (haven't tested higher ones).
As A075888 gives a very regular graph. There seems to be a regularity in the primes. (End) Let M be an infinite lower triangular matrix with (1, 1, 1, 0, 0, 0, ...) in every column, shifted down twice. This sequence starting with 1 = M * (1, 2, 3, ...)
M =
1;
1, 0;
1, 1, 0;
0, 1, 0, 0;
0, 1, 1, 0, 0;
0, 0, 1, 0, 0, 0;
0, 0, 1, 1, 0, 0, 0;
...
A026741 = M * (1, 2, 3, ...); but A002487 = lim_{n->infinity} M^n, a left-shifted vector considered as a sequence. (End)
A particular case of sequence for which a(n+3) = (a(n+2) * a(n+1)+q)/a(n) for every n > n0. Here n0 = 1 and q = -1. - Richard Choulet, Mar 01 2010 For n >= 2, a(n+1) is the smallest m such that s_n(2*m*(n-1))/(n-1) is even, where s_b(c) is the sum of digits of c in base b. - Vladimir Shevelev, May 02 2011 For n >= 3, a(n) is the periodic of integer of spiral length ratio of spiral that have (n-1) circle centers. See illustration in links. - Kival Ngaokrajang, Dec 28 2013 This is the sequence of Lehmer numbers u_n(sqrt(R), Q) with the parameters R = 4 and Q = 1. It is a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all natural numbers n and m. Cf. A005013 and A108412. - Peter Bala, Apr 18 2014 The sequence of convergents of the 2-periodic continued fraction [0; 1, -4, 1, -4, ...] = 1/(1 - 1/(4 - 1/(1 - 1/(4 - ...)))) = 2 begins [0/1, 1/1, 4/3, 3/2, 8/5, 5/3, 12/7, ...]. The present sequence is the sequence of denominators; the sequence of numerators of the continued fraction convergents [0, 1, 4, 3, 8, 5, 12, ...] is A022998, also a strong divisibility sequence. - Peter Bala, May 19 2014 For n >= 3, (a(n-2)/a(n))*Pi = vertex angle of a regular n-gon. See illustration in links. - Kival Ngaokrajang, Jul 17 2014 For n > 1, the numerator of the harmonic mean of the first n triangular numbers. - Colin Barker, Nov 13 2014 The difference sequence is a permutation of the integers. - Clark Kimberling, Apr 19 2015 Given the function a(n, p) = n/p if n mod p = 0, else n, then a possible formula is: a(n, p) = n*(1 + (p-1)*((n^(p-1)) mod p))/p, p prime, (n^(p-1)) mod p = 1, n not divisible by p. (Fermat's Little Theorem). Examples: p = 2; a(n), p = 3; A051176(n), p = 5; A060791(n), p = 7; A106608(n). Conjecture: lcm(n, p) = p*a(n, p), gcd(n, p) = n/a(n, p).
(End)
Let r(n) = (a(n+1) + 1)/a(n+1) if n mod 2 = 1, a(n+1)/(a(n+1) + 2) otherwise; then lim_{k->oo} 2^(k+2) * Product_{n=0..k} r(n)^(k-n) = Pi. - Dimitris Valianatos, Mar 22 2021
REFERENCES
David Wells, Prime Numbers: The Most Mysterious Figures in Math. Hoboken, New Jersey: John Wiley & Sons (2005), p. 53.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, 2nd Ed. Penguin (1997), p. 79.
FORMULA
G.f.: x*(1 + x + x^2)/(1-x^2)^2. - Len Smiley, Apr 30 2001 a(n) = 2*a(n-2) - a*(n-4) for n >= 4.
Multiplicative with a(2^e) = 2^(e-1) and a(p^e) = p^e, p > 2. - Vladeta Jovovic, Apr 05 2002 a(n) = n / gcd(n, 2). a(n)/ A045896(n) = n/((n+1)*(n+2)). For n > 1, a(n) = GCD of the n-th and (n-1)th triangular numbers ( A000217). - Ross La Haye, Sep 13 2003 Euler transform of finite sequence [1, 2, -1]. - Michael Somos, Jun 15 2005 G.f.: x * (1 - x^3) / ((1 - x) * (1 - x^2)^2) = Sum_{k>0} k * (x^k - x^(2*k)). - Michael Somos, Jun 15 2005 a(n+3) + a(n+2) = 3 + a(n+1) + a(n). a(n+3) * a(n) = - 1 + a(n+2) * a(n+1). a(n) = -a(-n) for all n in Z. - Michael Somos, Jun 15 2005 For n > 1, a(n) is the numerator of the average of 1, 2, ..., n - 1; i.e., numerator of A000217(n-1)/(n-1), with corresponding denominators [1, 2, 1, 2, ...] ( A000034). - Rick L. Shepherd, Jun 05 2006 Dirichlet g.f.: zeta(s-1)*(1 - 1/2^s). (End)
a(n+1) = denominator(H(n, 1)), n >= 0, with H(n, 1) = 2*n/(n+1) the harmonic mean of n and 1. a(n+1) = A227042(n, 1). See the formula a(n) = n/gcd(n, 2) given above. - Wolfdieter Lang, Jul 04 2013 a(n) = numerator(1 - 2/(n+2)), n >= 0; a(n) = denominator(1 - 2/n), n >= 1. - Kival Ngaokrajang, Jul 17 2014 Euler transform of length 3 sequence [1, 2, -1]. - Michael Somos, Jan 20 2017 G.f.: x / (1 - x / (1 - 2*x / (1 + 7*x / (2 - 9*x / (7 - 4*x / (3 - 7*x / (2 + 3*x))))))). - Michael Somos, Jan 20 2017 a(n) = Sum_{d|n, n/d odd} phi(d), where phi(n) is the Euler totient function A000010. O.g.f.: Sum_{n >= 1} phi(n)*x^n/(1 - x^(2*n)). (End)
For k >= 0, a(k) = gcd(k + 1, k*(k + 1)/2).
If (k mod 4) = 0 or 2 then a(k) = (k + 1).
If (k mod 4) = 1 or 3 then a(k) = (k + 1)/2. (End)
EXAMPLE
G.f. = x + x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 3*x^6 + 7*x^7 + 4*x^8 + ...
MATHEMATICA
Numerator[Abs[Table[Det[DiagonalMatrix[Table[1/i^2 - 1, {i, 1, n - 1}]] + 1], {n, 20}]]] (* Alexander Adamchuk, Jun 02 2006 *) halfMax = 40; Riffle[Range[0, halfMax], Range[1, 2halfMax + 1, 2]] (* Harvey P. Dale, Mar 27 2011 *)
PROG
(Haskell)
import Data.List (transpose)
a026741 n = a026741_list !! n
a026741_list = concat $ transpose [[0..], [1, 3..]]
(Python)
AUTHOR
J. Carl Bellinger (carlb(AT)ctron.com)
0, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832, 8190, 8556, 8930
COMMENTS
a(n) is the number of edges in (n+1) X (n+1) square grid with all horizontal, vertical and diagonal segments filled in. - Asher Auel, Jan 12 2000 In other words, the edge count of the (n+1) X (n+1) king graph. - Eric W. Weisstein, Jun 20 2017 Write 0,1,2,... in clockwise spiral; sequence gives numbers on one of 4 diagonals. (See Example section.)
The identity (4*n+1)^2 - (4*n^2+2*n)*(2)^2 = 1 can be written as A016813(n)^2 - a(n)*2^2 = 1. - Vincenzo Librandi, Jul 20 2010 - Nov 25 2012 Starting with "6" = binomial transform of [6, 14, 8, 0, 0, 0, ...]. - Gary W. Adamson, Aug 27 2010 The hyper-Wiener index of the crown graph G(n) (n>=3). The crown graph G(n) is the graph with vertex set {x(1), x(2), ..., x(n), y(1), y(2), ..., y(n)} and edge set {(x(i), y(j)): 1 <= i,j <= n, i != j} (= the complete bipartite graph K(n,n) with horizontal edges removed). The Hosoya-Wiener polynomial of G(n) is n(n-1)(t+t^2)+nt^3. - Emeric Deutsch, Aug 29 2013
REFERENCES
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
LINKS
Eric Weisstein's World of Mathematics, Edge Count. Eric Weisstein's World of Mathematics, King Graph.
FORMULA
a(n) = 4*n^2 + 2*n.
Sum_{n>=1} 1/a(n) = 1 - log(2).
Sum_{n>=1} 1/a(n)^2 = 2*log(2) + Pi^2/6 - 3. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 + log(2)/2 - 1. - Amiram Eldar, Feb 22 2022
EXAMPLE
64--65--66--67--68--69--70--71--72
|
63 36--37--38--39--40--41--42
| | |
62 35 16--17--18--19--20 43
| | | | |
61 34 15 4---5---6 21 44
| | | | | | |
60 33 14 3 0 7 22 45
| | | | | | | |
59 32 13 2---1 8 23 46
| | | | | |
58 31 12--11--10---9 24 47
| | | |
57 30--29--28--27--26--25 48
| |
56--55--54--53--52--51--50--49
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 6, 20}, 40] (* Harvey P. Dale, Aug 11 2011 *)
CROSSREFS
Cf. A001477, A007395, A007494, A007742, A014105, A016813, A033954, A045896, A046092, A054000, A118729, A173511. Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, this sequence, A033996, A033988.
Numerators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)*(a(n,k) - a(n,k+1)), n >= 0, k >= 0.
+10
22
1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 3, 1, -1, 1, 1, 2, 1, -1, 0, 1, 1, 5, 2, -3, -1, 1, 1, 1, 3, 5, -1, -1, 1, 0, 1, 1, 7, 5, 0, -4, 1, 1, -1, 1, 1, 4, 7, 1, -1, -1, 1, -1, 0, 1, 1, 9, 28, 49, -29, -5, 8, 1, -5, 5, 1, 1, 5, 3, 8, -7, -9, 5, 7, -5, 5, 0, 1, 1, 11, 15, 27, -28, -343, 295, 200, -44, -1017, 691, -691
FORMULA
a(n,k) = numerator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)).
E.g.f.: A(x,t) = (x+log(1-t))/(1-t-exp(-x)) = (1+(1/2)*x+(1/6)*x^2/2!-(1/30)*x^4/4!+...)*1 + (1/2+(1/3)*x+(1/6)*x^2/2!+...)*t + (1/3+(1/4)*x+(3/20)*x^2/2!+...)*t^2 + .... (End)
EXAMPLE
Table begins:
1 1/2 1/3 1/4 1/5 1/6 1/7 ...
1/2 1/3 1/4 1/5 1/6 1/7 ...
1/6 1/6 3/20 2/15 5/42 ...
0 1/30 1/20 2/35 5/84 ...
-1/30 -1/30 -3/140 -1/105 ...
Antidiagonals of numerator(a(n,k)):
1;
1, 1;
1, 1, 1;
1, 1, 1, 0;
1, 1, 3, 1, -1;
1, 1, 2, 1, -1, 0;
1, 1, 5, 2, -3, -1, 1;
1, 1, 3, 5, -1, -1, 1, 0;
1, 1, 7, 5, 0, -4, 1, 1, -1;
1, 1, 4, 7, 1, -1, -1, 1, -1, 0;
1, 1, 9, 28, 49, -29, -5, 8, 1, -5, 5;
MAPLE
a:= proc(n, k) option remember;
`if`(n=0, 1/(k+1), (k+1)*(a(n-1, k)-a(n-1, k+1)))
end:
seq(seq(numer(a(n, d-n)), n=0..d), d=0..12); # Alois P. Heinz, Apr 17 2013
MATHEMATICA
nmax = 12; a[0, k_]:= 1/(k+1); a[n_, k_]:= a[n, k]= (k+1)(a[n-1, k]-a[n-1, k+1]); Numerator[Flatten[Table[a[n-k, k], {n, 0, nmax}, {k, n, 0, -1}]]] (* Jean-François Alcover, Nov 28 2011 *)
PROG
(Magma)
function a(n, k)
if n eq 0 then return 1/(k+1);
else return (k+1)*(a(n-1, k) - a(n-1, k+1));
end if;
end function;
A051714:= func< n, k | Numerator(a(n, k)) >;
(SageMath)
def a(n, k):
if (n==0): return 1/(k+1)
else: return (k+1)*(a(n-1, k) - a(n-1, k+1))
def A051714(n, k): return numerator(a(n, k))
Denominators of table a(n,k) read by antidiagonals: a(0,k) = 1/(k+1), a(n+1,k) = (k+1)(a(n,k)-a(n,k+1)), n >= 0, k >= 0.
+10
19
1, 2, 2, 3, 3, 6, 4, 4, 6, 1, 5, 5, 20, 30, 30, 6, 6, 15, 20, 30, 1, 7, 7, 42, 35, 140, 42, 42, 8, 8, 28, 84, 105, 28, 42, 1, 9, 9, 72, 84, 1, 105, 140, 30, 30, 10, 10, 45, 120, 140, 28, 105, 20, 30, 1, 11, 11, 110, 495, 3960, 924, 231, 165, 220, 66, 66, 12, 12, 66, 55, 495, 264, 308, 132, 165, 44, 66, 1
FORMULA
a(n,k) = denominator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)/(j+k+1)). - Fabián Pereyra, Jan 14 2023
EXAMPLE
Table begins:
1 1/2 1/3 1/4 1/5 1/6 1/7 ...
1/2 1/3 1/4 1/5 1/6 1/7 ...
1/6 1/6 3/20 2/15 5/42 ...
0 1/30 1/20 2/35 5/84 ...
-1/30 -1/30 -3/140 -1/105 ...
MAPLE
a:= proc(n, k) option remember;
`if`(n=0, 1/(k+1), (k+1)*(a(n-1, k)-a(n-1, k+1)))
end:
seq(seq(denom(a(n, d-n)), n=0..d), d=0..12); # Alois P. Heinz, Apr 17 2013
MATHEMATICA
nmax = 12; a[0, k_] := 1/(k+1); a[n_, k_] := a[n, k] = (k+1)(a[n-1, k]-a[n-1, k+1]); Denominator[ Flatten[ Table[ a[n-k, k], {n, 0, nmax}, {k, n, 0, -1}]]](* Jean-François Alcover, Nov 28 2011 *)
Denominator of c(n) = (n^2+n+2)/((n+1)*(n+2)*(n+3)).
+10
8
3, 6, 15, 60, 105, 21, 126, 360, 495, 330, 429, 1092, 1365, 420, 1020, 2448, 2907, 1710, 1995, 4620, 5313, 759, 3450, 7800, 8775, 4914, 5481, 12180, 13485, 3720, 8184, 17952, 19635, 10710, 11655, 25308, 27417, 3705, 15990, 34440, 37023, 19866, 21285, 45540
COMMENTS
All terms are multiples of 3.
Difference table of c(n):
1/3, 1/6, 2/15, 7/60, 2/21,...
-1/6, -1/30, -1/60, -1/84, -1/105,...
2/15, 1/60, 1/210, 1/420, 1/630,...
-7/60, -1/84, -1/420, -1/1260, -1/2520,... .
This is an autosequence of the second kind; the inverse binomial transform is the signed sequence. The main diagonal is the first upper diagonal multiplied by 2.
Denominators of the main diagonal: A051133(n+1). Denominators of the first upper diagonal; A000911(n). Based on the Akiyama-Tanigawa transform applied to 1/(n+1) which yields the Bernoulli numbers A164555(n)/ A027642(n). Are the numerators of the main diagonal (-1)^n? If yes, what is the value of 1/3 - 1/30 + 1/210,... or 1 - 1/10 + 1/70 - 1/420, ... , from A002802(n)? Is a(n+40) - a(n) divisible by 10?
Are the common divisors to A014206(n) and A007531(n+3) of period 16: repeat 2, 4, 4, 2, 2, 16, 4, 2, 2, 4, 4, 2, 2, 8, 4, 2? Reduce c(n) = f(n) = b(n)/a(n) = 1/3, 1/6, 2/15, 7/60, 11/105, 2/21, 11/126, 29/360, ... .
Consider the successively interleaved autosequences (also called eigensequences) of the second kind and of the first kind
1, 1/2, 1/3, 1/4, 1/5, 1/6, ...
0, 1/6, 1/6, 3/20, 2/15, 5/42, ...
1/3, 1/6, 2/15, 7/60, 11/105, 2/21, ...
0, 1/10, 1/10, 13/140, 3/35, 5/63, ...
1/5, 1/10, 3/35, 11/140, 23/315, 43/630, ...
0, 1/14, 1/14, 17/252, 4/63, ...
This array is Au1(m,n). Au1(0,0)=1, Au1(0,1)=1/2.
Au1(m+1,n) = 2*Au1(m,n+1) - Au1(m,n).
First row: see A003506, Leibniz's Harmonic Triangle. a(n) is the denominator of the third row f(n).
The first column is 1, 0, 1/3, 0, 1/5, 0, 1/7, 0, ... . Numerators: A093178(n+1). This incites, considering tan(1), to introduce before the first row Ta0(n) = 0, 1/2, 1/2, 5/12, 1/3, 4/15, 13/60, 151/840, ... .
FORMULA
The sum of the difference table main diagonal is 1/3 - 1/30 + 1/210 - ... = 10* A086466-4 = 4*(sqrt(5)*log(phi)-1) = 0.3040894... - Jean-François Alcover, Apr 22 2014 a(n) = (n+1)*(n+2)*(n+3)/gcd(4*n - 4, n^2 + n + 2), where gcd(4*n - 4, n^2 + n + 2) is periodic with period 16. - Robert Israel, Jul 17 2023
MAPLE
seq(denom((n^2+n+2)/((n+1)*(n+2)*(n+3))), n=0..1000);
MATHEMATICA
Denominator[Table[(n^2+n+2)/Times@@(n+{1, 2, 3}), {n, 0, 50}]] (* Harvey P. Dale, Mar 27 2015 *)
PROG
(PARI) for(n=0, 100, print1(denominator((n^2+n+2)/((n+1)*(n+2)*(n+3))), ", ")) \\ Colin Barker, Apr 18 2014
Row sums of the Eta triangle A160464
+10
6
-1, -9, -87, -2925, -75870, -2811375, -141027075, -18407924325, -1516052821500, -153801543183750, -18845978136851250, -2744283682352086875, -468435979952504313750, -92643070481933918821875
COMMENTS
It is conjectured that the row sums of the Eta triangle depend on five different sequences.
Two Maple algorithms are given. The first one gives the row sums according to the Eta triangle A160464 and the second one gives the row sums according to our conjecture.
FORMULA
Rowsums(n) = (-1) * A119951(n-1) * FF(n) for n >= 2. FF(n) = SF(n) * FF(n-1) for n >= 3 with FF(2) =1.
MAPLE
nmax:=15; c(2) := -1/3: for n from 3 to nmax do c(n):=(2*n-2)*c(n-1)/(2*n-1)-1/ ((n-1)*(2*n-1)) end do: for n from 2 to nmax do GCS(n-1) := ln(1/(2^(-(2*(n-1)-1-floor(ln(n-1)/ ln(2))))))/ln(2); p(n):=2^(-GCS(n-1))*(2*n-1)!; ETA(n, 1) := p(n)*c(n) end do: mmax:=nmax: for m from 2 to mmax do ETA(2, m) := 0 end do: for n from 3 to nmax do for m from 2 to mmax do q(n) := (1+(-1)^(n-3)*(floor(ln(n-1)/ln(2)) - floor(ln(n-2)/ln(2)))): ETA(n, m) := q(n)*(-ETA(n-1, m-1)+(n-1)^2*ETA(n-1, m)) end do end do: for n from 2 to nmax do s1(n):=0: for m from 1 to n-1 do s1(n) := s1(n) + ETA(n, m) end do end do: seq(s1(n), n=2..nmax);
# End first program.
nmax:=nmax; A160467 := proc(n): denom(4*(4^n-1)*bernoulli(2*n)/n) end: A043529 := proc(n): ceil(frac(log[2](n+1))+1) end proc: A000466 := proc(n): 4*n^2-1 end proc: A045896 := proc(n): denom((n)/((n+1)*(n+2))) end proc: A119951 := proc(n) : numer(sum(((2*k1)!/(k1!*(k1+1)!))/2^(2*(k1-1)), k1=1..n)) end proc: for n from 1 to nmax do SF(2*n+1):= A000466(n)/ A043529(n-1); SF(2*n+2) := A045896(n-1)/ A160467(n+1) end do: FF(2):=1: for n from 3 to nmax do FF(n) := SF(n) * FF(n-1) end do: for n from 2 to nmax do s2(n):= (-1)* A119951(n-1)*FF(n) end do: seq(s2(n), n=2..nmax); # End second program.
Period length of pairs (a,b) where a has period 2n-2 and b has period n.
+10
4
0, 2, 12, 12, 40, 30, 84, 56, 144, 90, 220, 132, 312, 182, 420, 240, 544, 306, 684, 380, 840, 462, 1012, 552, 1200, 650, 1404, 756, 1624, 870, 1860, 992, 2112, 1122, 2380, 1260, 2664, 1406, 2964, 1560
FORMULA
a(n) = n*(n-1) for n even.
a(n) = 2*n*(n-1) for n odd.
a(n) = lcm(2*n-2, n).
Sum_{n>=2} 1/a(n) = (log(2)+1)/2. (End)
MATHEMATICA
Table[ LCM[ 2*n-2, n ], {n, 40} ]
PROG
(PARI) for(n=1, 50, print1(lcm(2*n-2, n), ", ")) \\ G. C. Greubel, Jun 15 2018 (Magma) [Lcm(2*n-2, n): n in [1..50]]; // G. C. Greubel, Jun 15 2018
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