cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A046092 4 times triangular numbers: a(n) = 2*n*(n+1).

Original entry on oeis.org

0, 4, 12, 24, 40, 60, 84, 112, 144, 180, 220, 264, 312, 364, 420, 480, 544, 612, 684, 760, 840, 924, 1012, 1104, 1200, 1300, 1404, 1512, 1624, 1740, 1860, 1984, 2112, 2244, 2380, 2520, 2664, 2812, 2964, 3120, 3280, 3444, 3612, 3784, 3960, 4140, 4324
Offset: 0

Views

Author

N. J. A. Sloane and Eric W. Weisstein

Keywords

Comments

Consider all Pythagorean triples (X,Y,Z=Y+1) ordered by increasing Z; sequence gives Y values. X values are 1, 3, 5, 7, 9, ... (A005408), Z values are A001844.
In the triple (X, Y, Z) we have X^2=Y+Z. Actually, the triple is given by {x, (x^2 -+ 1)/2}, where x runs over the odd numbers (A005408) and x^2 over the odd squares (A016754). - Lekraj Beedassy, Jun 11 2004
a(n) is the number of edges in n X n square grid with all horizontal and vertical segments filled in. - Asher Auel, Jan 12 2000 [Corrected by Felix Huber, Apr 09 2024]
a(n) is the only number satisfying an inequality related to zeta(2) and zeta(3): Sum_{i>a(n)+1} 1/i^2 < Sum_{i>n} 1/i^3 < Sum_{i>a(n)} 1/i^2. - Benoit Cloitre, Nov 02 2001
Number of right triangles made from vertices of a regular n-gon when n is even. - Sen-Peng Eu, Apr 05 2001
Number of ways to change two non-identical letters in the word aabbccdd..., where there are n type of letters. - Zerinvary Lajos, Feb 15 2005
a(n) is the number of (n-1)-dimensional sides of an (n+1)-dimensional hypercube (e.g., squares have 4 corners, cubes have 12 edges, etc.). - Freek van Walderveen (freek_is(AT)vanwal.nl), Nov 11 2005
From Nikolaos Diamantis (nikos7am(AT)yahoo.com), May 23 2006: (Start)
Consider a triangle, a pentagon, a heptagon, ..., a k-gon where k is odd. We label a triangle with n=1, a pentagon with n=2, ..., a k-gon with n = floor(k/2). Imagine a player standing at each vertex of the k-gon.
Initially there are 2 frisbees, one held by each of two neighboring players. Every time they throw the frisbee to one of their two nearest neighbors with equal probability. Then a(n) gives the average number of steps needed so that the frisbees meet.
I verified this by simulating the processes with a computer program. For example, a(2) = 12 because in a pentagon that's the expected number of trials we need to perform. That is an exercise in Concrete Mathematics and it can be done using generating functions. (End)
A diagonal of A059056. - Zerinvary Lajos, Jun 18 2007
If X_1,...,X_n is a partition of a 2n-set X into 2-blocks then a(n-1) is equal to the number of 2-subsets of X containing none of X_i, (i=1,...,n). - Milan Janjic, Jul 16 2007
X values of solutions to the equation 2*X^3 + X^2 = Y^2. To find Y values: b(n) = 2n(n+1)(2n+1). - Mohamed Bouhamida, Nov 06 2007
Number of (n+1)-permutations of 3 objects u,v,w, with repetition allowed, containing n-1 u's. Example: a(1)=4 because we have vv, vw, wv and ww; a(2)=12 because we can place u in each of the previous four 2-permutations either in front, or in the middle, or at the end. - Zerinvary Lajos, Dec 27 2007
Sequence found by reading the line from 0, in the direction 0, 4, ... and the same line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, May 03 2008
a(n) is also the least weight of self-conjugate partitions having n different even parts. - Augustine O. Munagi, Dec 18 2008
From Peter Luschny, Jul 12 2009: (Start)
The general formula for alternating sums of powers of even integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,1)-(-1)^k P(n,2k+1))/2. Here n=2, thus
a(k) = |(P(2,1) - (-1)^k*P(2,2k+1))/2|. (End)
The sum of squares of n+1 consecutive numbers between a(n)-n and a(n) inclusive equals the sum of squares of n consecutive numbers following a(n). For example, for n = 2, a(2) = 12, and the corresponding equation is 10^2 + 11^2 + 12^2 = 13^2 + 14^2. - Tanya Khovanova, Jul 20 2009
Number of roots in the root system of type D_{n+1} (for n>2). - Tom Edgar, Nov 05 2013
Draw n ellipses in the plane (n>0), any 2 meeting in 4 points; sequence gives number of intersections of these ellipses (cf. A051890, A001844); a(n) = A051890(n+1) - 2 = A001844(n) - 1. - Jaroslav Krizek, Dec 27 2013
a(n) appears also as the second member of the quartet [p0(n), a(n), p2(n), p3(n)] of the square of [n, n+1, n+2, n+3] in the Clifford algebra Cl_2 for n >= 0. p0(n) = -A147973(n+3), p2(n) = A054000(n+1) and p3(n) = A139570(n). See a comment on A147973, also with a reference. - Wolfdieter Lang, Oct 15 2014
a(n) appears also as the third and fourth member of the quartet [p0(n), p0(n), a(n), a(n)] of the square of [n, n, n+1, n+1] in the Clifford algebra Cl_2 for n >= 0. p0(n) = A001105(n). - Wolfdieter Lang, Oct 16 2014
Consider two equal rectangles composed of unit squares. Then surround the 1st rectangle with 1-unit-wide layers to build larger rectangles, and surround the 2nd rectangle just to hide the previous layers. If r(n) and h(n) are the number of unit squares needed for n layers in the 1st case and the 2nd case, then for all rectangles, we have a(n) = r(n) - h(n) for n>=1. - Michel Marcus, Sep 28 2015
When greater than 4, a(n) is the perimeter of a Pythagorean triangle with an even short leg 2*n. - Agola Kisira Odero, Apr 26 2016
Also the number of minimum connected dominating sets in the (n+1)-cocktail party graph. - Eric W. Weisstein, Jun 29 2017
a(n+1) is the harmonic mean of A000384(n+2) and A014105(n+1). - Bob Andriesse, Apr 27 2019
Consider a circular cake from which wedges of equal center angle c are cut out in clockwise succession and turned around so that the bottom comes to the top. This goes on until the cake shows its initial surface again. An interesting case occurs if 360°/c is not an integer. Then, with n = floor(360°/c), the number of wedges which have to be cut out and turned equals a(n). (For the number of cutting line segments see A005408.) - According to Peter Winkler's book "Mathematical Mind-Benders", which presents the problem and its solution (see Winkler, pp. 111, 115) the problem seems to be of French origin but little is known about its history. - Manfred Boergens, Apr 05 2022
a(n-3) is the maximum irregularity over all maximal 2-degenerate graphs with n vertices. The extremal graphs are 2-stars (K_2 joined to n-2 independent vertices). (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023
Number of ways of placing a domino on a (n+1)X(n+1) board of squares. - R. J. Mathar, Apr 24 2024
The sequence terms are the exponents in the expansion of (1/(1 + x)) * Sum_{n >= 0} x^n * Product_{k = 1..n} (1 - x^(2*k-1))/(1 + x^(2*k+1)) = 1 - x^4 + x^12 - x^24 + x^40 - x^60 + - ... (Andrews and Berndt, Entry 9.3.3, p. 229). Cf. A153140. - Peter Bala, Feb 15 2025
Number of edges in an (n+1)-dimensional orthoplex. 2D orthoplexes (diamonds) have 4 edges, 3D orthoplexes (octahedrons) have 12 edges, 4D orthoplexes (16-cell) have 24 edges, and so on. - Aaron Franke, Mar 23 2025

Examples

			a(7)=112 because 112 = 2*7*(7+1).
The first few triples are (1,0,1), (3,4,5), (5,12,13), (7,24,25), ...
The first such partitions, corresponding to a(n)=1,2,3,4, are 2+2, 4+4+2+2, 6+6+4+4+2+2, 8+8+6+6+4+4+2+2. - _Augustine O. Munagi_, Dec 18 2008

References

  • George E. Andrews and Bruce C. Berndt, Ramanujan's Lost Notebook, Part I, Springer, 2005.
  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 3.
  • Albert H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 125, 1964.
  • Ronald L. Graham, D. E. Knuth and Oren Patashnik, Concrete Mathematics, Reading, Massachusetts: Addison-Wesley, 1994.
  • Peter Winkler, Mathematical Mind-Benders, Wellesley, Massachusetts: A K Peters, 2007.

Links

Crossrefs

Cf. A045943, A028895, A002943, A054000, A000330, A007290, A002378, A033996, A124080, A028896, A049598, A005563, A000217, A033586, A085250.
Main diagonal of array in A001477.
Equals A033996/2. Cf. A001844. - Augustine O. Munagi, Dec 18 2008
Cf. A078371, A141530 (see Librandi's comment in A078371).
Cf. A097080, A001845.
Cf. similar sequences listed in A299645.
Cf. A005408.
Cf. A016754.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).

Programs

Formula

a(n) = A100345(n+1, n-1) for n>0.
a(n) = 2*A002378(n) = 4*A000217(n). - Lekraj Beedassy, May 25 2004
a(n) = C(2n, 2) - n = 4*C(n, 2). - Zerinvary Lajos, Feb 15 2005
From Lekraj Beedassy, Jun 04 2006: (Start)
a(n) - a(n-1)=4*n.
Let k=a(n). Then a(n+1) = k + 2*(1 + sqrt(2k + 1)). (End)
Array read by rows: row n gives A033586(n), A085250(n+1). - Omar E. Pol, May 03 2008
O.g.f.:4*x/(1-x)^3; e.g.f.: exp(x)*(2*x^2+4*x). - Geoffrey Critzer, May 17 2009
From Stephen Crowley, Jul 26 2009: (Start)
a(n) = 1/int(-(x*n+x-1)*(step((-1+x*n)/n)-1)*n*step((x*n+x-1)/(n+1)),x=0..1) where step(x)=piecewise(x<0,0,0<=x,1) is the Heaviside step function.
Sum_{n>=1} 1/a(n) = 1/2. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=4, a(2)=12. - Harvey P. Dale, Jul 25 2011
For n > 0, a(n) = 1/(Integral_{x=0..Pi/2} (sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011
a(n) = A001844(n) - 1. - Omar E. Pol, Oct 03 2011
(a(n) - A000217(k))^2 = A000217(2n-k)*A000217(2n+1+k) - (A002378(n) - A000217(k)), for all k. See also A001105. - Charlie Marion, May 09 2013
From Ivan N. Ianakiev, Aug 30 2013: (Start)
a(n)*(2m+1)^2 + a(m) = a(n*(2m+1)+m), for any nonnegative integers n and m.
t(k)*a(n) + t(k-1)*a(n+1) = a((n+1)*(t(k)-t(k-1)-1)), where k>=2, n>=1, t(k)=A000217(k). (End)
a(n) = A245300(n,n). - Reinhard Zumkeller, Jul 17 2014
2*a(n)+1 = A016754(n) = A005408(n)^2, the odd squares. - M. F. Hasler, Oct 02 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) - 1/2 = A187832. - Ilya Gutkovskiy, Mar 16 2017
a(n) = lcm(2*n,2*n+2). - Enrique Navarrete, Aug 30 2017
a(n)*a(n+k) + k^2 = m^2 (a perfect square), n >= 1, k >= 0. - Ezhilarasu Velayutham, May 13 2019
From Amiram Eldar, Jan 29 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = cosh(Pi/2)/(Pi/2).
Product_{n>=1} (1 - 1/a(n)) = -2*cos(sqrt(3)*Pi/2)/Pi. (End)
a(n) = A016754(n) - A001844(n). - Leo Tavares, Sep 20 2022

A078371 a(n) = (2*n+5)*(2*n+1).

Original entry on oeis.org

5, 21, 45, 77, 117, 165, 221, 285, 357, 437, 525, 621, 725, 837, 957, 1085, 1221, 1365, 1517, 1677, 1845, 2021, 2205, 2397, 2597, 2805, 3021, 3245, 3477, 3717, 3965, 4221, 4485, 4757, 5037, 5325, 5621, 5925, 6237, 6557, 6885, 7221, 7565, 7917, 8277, 8645
Offset: 0

Views

Author

Wolfdieter Lang, Nov 29 2002

Keywords

Comments

This is the generic form of D in the (nontrivially) solvable Pell equation x^2 - D*y^2 = +4. See A077428 and A078355.
Consider all primitive Pythagorean triples (a,b,c) with c-a=8, sequence gives values of a. (Corresponding values for b are A017113(n), while c follows A078370(n).) - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 19 2004
From Vincenzo Librandi, Aug 08 2010: (Start)
The identity (4*n^3 + 18*n^2 + 24*n + 9)^2 - (4*n^2 + 12*n + 5)*(2*n^2 + 6*n + 4)^2 = 1 (see Ramasamy's paper in link) can be written as A141530(n+2)^2 - a(n)*A046092(n+1)^2 = 1.
a(n)^3 + 6*a(n)^2 + 9*a(n) + 4 is a square: in fact, a(n)^3 + 6*a(n)^2 + 9*a(n) + 4 = (a(n) + 1)^2*(a(n) + 4), where a(n) + 4 = (2*n+3)^2. (End)
Products of two positive odd integers with difference 4 (i.e., 1*5, 3*7, 5*9, 7*11, 9*13, ...). - Wesley Ivan Hurt, Nov 19 2013
Starting with stage 1, the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 675", based on the 5-celled von Neumann neighborhood. - Robert Price, May 21 2016
The continued fraction expansion of (sqrt(a(n))-1)/2 is [n; {1,2*n+1}] with periodic part of length 2: repeat {1,2*n+1}. - Ron Knott, May 11 2017
a(n) is the sum of 2*n+5 consecutive integers starting from n-1. - Bruno Berselli, Jan 16 2018
The continued fraction expansion of sqrt(a(n)) is [2n+2; {1, n, 2, n, 1, 4n+4}]. For n=0, this collapses to [2; {4}]. - Magus K. Chu, Aug 26 2022

Links

Crossrefs

Subsequence of A077425 (D values (not a square) for which Pell x^2 - D*y^2 = +4 is solvable in positive integers).
Cf. A017113, A046092, A061037, A077428, A078355, A078370.
Supersequence of A143206.

Programs

  • Magma
    [(2*n+5)*(2*n+1): n in [0..100]]; // G. C. Greubel, Sep 19 2018
  • Maple
    seq((2*n+5)*(2*n+1), n=0..48); # Emeric Deutsch, Feb 24 2005
  • Mathematica
    Table[(2 n + 5) (2 n + 1), {n, 0, 100}] (* Wesley Ivan Hurt, Nov 19 2013 *)
LinearRecurrence[{3,-3,1},{5,21,45},50] (* Harvey P. Dale, Oct 18 2020 *)
  • PARI
    lista(nn) = {for (n=0, nn, print1((2*n+1)*(2*n+5), ", "));} \\ Michel Marcus, Nov 21 2013

Formula

a(n) = 8*(binomial(n+2, 2)-1)+5, hence subsequence of A004770 (5 (mod 8) numbers).
G.f.: (5 + 6*x - 3*x^2)/(1-x)^3.
a(n) = A061037(2*n+1) = (2*n+3)^2 - 4. For A061037: a(2*n+1) = (2*n+1)*(2*n+5) = (2*n+3)^2-4. - Paul Curtz, Sep 24 2008
a(n) = 8*(n+1) + a(n-1) for n > 0, a(0)=5. - Vincenzo Librandi, Aug 08 2010
From Ilya Gutkovskiy, May 22 2016: (Start)
E.g.f.: (5 + 4*x*(4 + x))*exp(x).
Sum_{n>=0} 1/a(n) = 1/3. (End)
Sum_{n>=0} (-1)^n/a(n) = 1/6. - Amiram Eldar, Oct 08 2023

Extensions

More terms from Emeric Deutsch, Feb 24 2005

A154560 a(n) = (n+3)^2*n/2 + 1.

Original entry on oeis.org

1, 9, 26, 55, 99, 161, 244, 351, 485, 649, 846, 1079, 1351, 1665, 2024, 2431, 2889, 3401, 3970, 4599, 5291, 6049, 6876, 7775, 8749, 9801, 10934, 12151, 13455, 14849, 16336, 17919, 19601, 21385, 23274, 25271, 27379, 29601, 31940, 34399, 36981
Offset: 0

Views

Author

Klaus Brockhaus, Jan 12 2009

Keywords

Comments

8*a(n) is the y value of a solution (x, y) to the Diophantine equation 2*x^3+12*x^2 = y^2. The corresponding x value is A152811(n+1).

Examples

			a(5) = (5+3)^2*5/2+1 = 64*5/2+1 = 161.

Links

Crossrefs

Cf. A058794 (row 3 of A007754), A117560 (n*(n^2-1)/2-1), A144129 (4*n^3-3*n), A141530, A152811 (2*(n^2+2*n-2)).

Programs

  • Magma
    [(n+3)^2*n/2 + 1: n in [0..50]]; // Vincenzo Librandi, Sep 06 2011
  • PARI
    {for(n=0,40,print1((n+3)^2*n/2+1,","))}

Formula

G.f.: (1+5*x-4*x^2+x^3)/(1-x)^4.
a(n) = A058794(n)/2.
a(n) = A117560(n+2) - n - 1.
a(2*n) = A144129(n+1).
a(2*n-1) = A141530(n+1). a(n) = -a(-n-4). - Bruno Berselli, Sep 05 2011
a(n) = ((n+2-i)^3+(n+2+i)^3)/4, where i is the imaginary unit. - Nicolas Bělohoubek, Jul 03 2025

A157371 a(n) = (n+1)*(9-9*n+5*n^2-n^3).

Original entry on oeis.org

9, 8, 9, 0, -55, -216, -567, -1216, -2295, -3960, -6391, -9792, -14391, -20440, -28215, -38016, -50167, -65016, -82935, -104320, -129591, -159192, -193591, -233280, -278775, -330616, -389367, -455616, -529975, -613080, -705591, -808192, -921591, -1046520, -1183735, -1334016, -1498167
Offset: 0

Views

Author

Paul Curtz, Feb 28 2009

Keywords

Comments

This is the fourth in a family of sequences that appear in columns on pages 36 and 56 of the reference: (i) sequence n+1, A000029, (ii) sequence (n+1)*(1-n), A147998 and (iii) (n+1)*(5-5*n+2*n^2), A152064.
First differences along columns shown on page 56 of the reference are columns of what is shown on page 36 of the reference. Example: the third column of page 56, A152064, has first differences which constitute the third column p page 36, A140811.

References

  • Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil (1969).

Links

Programs

  • Magma
    [(n+1)*(9-9*n+5*n^2-n^3): n in [0..40] ]; // Vincenzo Librandi, Jul 14 2011
  • Mathematica
    LinearRecurrence[{5,-10,10,-5,1},{9,8,9,0,-55},40] (* or *) Table[(n+1)(9-9n+5n^2-n^3),{n,0,40}] (* or *) CoefficientList[ Series[ (55x^3- 59x^2+ 37x-9)/ (x-1)^5,{x,0,40}],x] (* Harvey P. Dale, Jul 13 2011 *)
  • PARI
    a(n)=(n+1)*(9-9*n+5*n^2-n^3) \\ Charles R Greathouse IV, Oct 16 2015

Formula

First differences: a(n+1)-a(n) = -A141530(n).
Fourth differences: a(n+4)-4*a(n+3)+6*a(n+2)-4*a(n+1)+a(n) = -24 = -A010863(n).
From Harvey P. Dale, Jul 13 2011: (Start)
a(0)=9, a(1)=8, a(2)=9, a(3)=0, a(4)=-55, a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5).
G.f.: (9-37*x+59*x^2-55*x^3)/(1-x)^5. (End)
E.g.f.: (9 - x + x^2 - 2*x^3 - x^4)*exp(x). - G. C. Greubel, Feb 02 2018

Extensions

Edited, extended by R. J. Mathar, Sep 25 2009

A157411 a(n) = 30*n^4 - 120*n^3 + 120*n^2 - 19.

Original entry on oeis.org

-19, 11, -19, 251, 1901, 6731, 17261, 36731, 69101, 119051, 191981, 294011, 431981, 613451, 846701, 1140731, 1505261, 1950731, 2488301, 3129851, 3887981, 4776011, 5807981, 6998651, 8363501, 9918731, 11681261, 13668731, 15899501, 18392651
Offset: 0

Views

Author

Paul Curtz, Feb 28 2009

Keywords

Comments

These are the numerators in column j=4 of the array in A140825 (reference p. 36).
The other columns in A140825 are represented by A000012, A005408, A140811 and A141530.
The link between these columns is given by the first differences: a(n+1) - a(n) = 30*A141530(n), where 30 = A027760(4) = A027760(3) = A027642(4) = A002445(2), then for j=3, A141530(n+1) - A141530(n) = A140070(2)*A140811(n).

References

  • P. Curtz, Integration numerique des systemes differentiels a conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil (1969).

Links

Programs

  • Magma
    [30*n^4 - 120*n^3 + 120*n^2 - 19: n in [0..50]]; // Vincenzo Librandi, Aug 07 2011
  • Mathematica
    Table[30n^4-120n^3+120n^2-19,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{-19,11,-19,251,1901},40] (* Harvey P. Dale, Mar 08 2015 *)
  • PARI
    a(n)=30*n^4-120*n^3+120*n^2-19 \\ Charles R Greathouse IV, Oct 16 2015

Formula

a(n)= 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: (-19 + 106*x - 264*x^2 + 646*x^3 + 251*x^4)/(1-x)^5.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 720. Fourth differences are constant, 720.

Extensions

Edited, one index corrected and extended by R. J. Mathar, Sep 17 2009

A165313 Triangle T(n,k) = A091137(k-1) read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 1, 2, 12, 24, 1, 2, 12, 24, 720, 1, 2, 12, 24, 720, 1440, 1, 2, 12, 24, 720, 1440, 60480, 1, 2, 12, 24, 720, 1440, 60480, 120960, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 1, 2, 12
Offset: 1

Views

Author

Paul Curtz, Sep 14 2009

Keywords

Comments

From a study of modified initialization formulas in Adams-Bashforth (1855-1883) multisteps method for numerical integration. On p.36, a(i,j) comes from (j!)*a(i,j) = Integral_{u=i,..,i+1} u*(u-1)*...*(u-j+1) du; see p.32.
Then, with i vertical, j horizontal, with unreduced fractions, partial array is:
0) 1, 1/2, -1/12, 1/24, -19/720, 27/1440, ... = 1/log(2)
1) 1, 3/2, 5/12, -1/24, 11/720, -11/1440, ... = 2/log(2)
2) 1, 5/2, 23/12, 9/24, -19/720, 11/1440, ... = 4/log(2)
3) 1, 7/2, 53/12, 55/24, 251/720, -27/1440, ... = 8/log(2)
4) 1, 9/2, 95/12, 161/24, 1901/720, 475/1440, ... = 16/log(2)
5) 1, 11/2, 149/12, 351/24, 6731/720, 4277/1440, ... = 32/log(2)
... [improved by Paul Curtz, Jul 13 2019]
First line: the reduced terms are A002206/A002207, logarithmic or Gregory numbers G(n). The difference between the second line and the first one is 0 together A002206/A002207. This is valid for the next lines. - Paul Curtz, Jul 13 2019
See A141417, A140825, A157982, horizontal numerators: A141047, vertical numerators: A000012, A005408, A140811, A141530, A157411. On p.56, coefficients are s(i,q) = (1/q!)* Integral_{u=-i-1,..,1} u*(u+1)*...*(u+q-1) du.
Unreduced fractions array is:
-1) 1, 1/2, 5/12, 9/24, 251/720, 475/1440, ... = A002657/A091137
0) 2, 0/2, 4/12, 8/24, 232/720, 448/1440, ... = A195287/A091137
1) 3, -3/2, 9/12, 9/24, 243/720, 459/1440, ...
2) 4, -8/2, 32/12, 0/24, 224/720, 448/1440, ...
3) 5, -15/2, 85/12, -55/24, 475/720, 475/1440, ...
...
(on p.56 up to 6)). See A147998. Vertical numerators: A000027, A147998, A152064, A157371, A165281.
From Paul Curtz, Jul 14 2019: (Start)
Difference table from the second line and the first one difference:
1, -1/2, -1/12, -1/24, -19/720, -27/1440, ...
-3/2, 5/12, 1/24, 11/720, 11/1440, ...
23/12, -9/24, -19/720, -11/1440, ...
-55/24, 251/720, 27/1440, ...
1901/720, -475/1440,
-4277/1440, ...
...
Compare the lines to those of the first array.
The verticals are the signed diagonals of the first array. (End)

Examples

			1;
1,2;
1,2,12;
1,2,12,24;
1,2,12,24,720;

References

  • P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Note 12, Arcueil, 1969.

Crossrefs

Cf. A090624, A091137.
Cf. A000012, A000079, A002657, A005408, A007525, A131920, A140811, A140825, A141047, A141417, A141530, A157411, A157982, A195287.
Cf. A002206, A002207.

Programs

  • Mathematica
    (* a = A091137 *) a[n_] := a[n] = Product[d, {d, Select[Divisors[n]+1, PrimeQ]}]*a[n-1]; a[0]=1; Table[Table[a[k-1], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Dec 18 2014 *)

A370349 a(n) is the number of integer triples (x,y,z) satisfying a system of linear inequalities and congruences specified in the comments.

Original entry on oeis.org

1, 6, 18, 39, 72, 120, 185, 270, 378, 511, 672, 864, 1089, 1350, 1650, 1991, 2376, 2808, 3289, 3822, 4410, 5055, 5760, 6528, 7361, 8262, 9234, 10279, 11400, 12600, 13881, 15246, 16698, 18239, 19872, 21600, 23425, 25350, 27378, 29511, 31752, 34104, 36569, 39150, 41850, 44671, 47616, 50688, 53889, 57222
Offset: 0

Views

Author

Jeffery Opoku, Feb 16 2024

Keywords

Comments

The inequalities are
n + x + y + z >= 0,
49*n + 13*x - 11*y - 23*z >= 0,
49*n - 11*x - 23*y + 13*z >= 0,
49*n - 23*x + 13*y - 11*z >= 0,
The congruences are
n + x + y + z == 0 (mod 12),
49*n + 13*x - 11*y - 23*z == 0 (mod 7).

Examples

			For n=0, the sole solution is (x,y,z) = (0,0,0) so a(0) = 1.
For n=1, the a(1)=6 solutions are (-1, -3, 3), (-2, 0, 1), (-3, 3, -1), (1, -2, 0), (0, 1, -2), (3, -1, -3).

Links

Crossrefs

Cf. A002620, A102283, A141530, A384127, A384295.

Programs

  • Mathematica
    n = Range[0, 500, 2];
Floor[(10 + 24*n + 18*n^2 + 4*n^3)/9]
  • Python
    def A370349(n): return ((n<<2)+10)*(n+1)**2//9 # Chai Wah Wu, Mar 11 2024

Formula

a(n) = floor((10 + 24*n + 18*n^2 + 4*n^3)/9).
a(n) = (A141530(n+2) - A102283(n))/9. - Stefano Spezia, Feb 17 2024

A157982 Triangle T(n,m) read by rows which contains the coefficients [x^m] of the polynomial generating the numerators of the column A140825(.,n).

Original entry on oeis.org

1, 1, 2, -1, 0, 6, 1, 0, -6, 4, -19, 0, 120, -120, 30, 27
Offset: 0

Views

Author

Paul Curtz, Mar 10 2009

Keywords

Comments

The first five polynomials describing the first five columns of A140825 are in A000012, A005408, A140811, A141530 and A157411.

Examples

			1;
1,2;     # 2n+1
-1,0,6;  # 6n^2-1
1,0,-6,4;  # 4n^3-6n^2+1, A141530
-19,0,120,-120,30;  # 30n^4-120n^3+120n^2-19, A157411

References

  • P Curtz Integration numerique des systemes differentiels a conditions initiales, C.C.S.A., Arcueil, 1969, p.36.

Crossrefs

Cf. A141417.
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