A140811 -id:A140811 - cp's OEIS frontend

A005843 The nonnegative even numbers: a(n) = 2n.

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 0

N. J. A. Sloane

-2, -4, -6, -8, -10, -12, -14, ... are the trivial zeros of the Riemann zeta function. - Vivek Suri (vsuri(AT)jhu.edu), Jan 24 2008
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 2-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
A134452(a(n)) = 0; A134451(a(n)) = 2 for n > 0. - Reinhard Zumkeller, Oct 27 2007
Omitting the initial zero gives the number of prime divisors with multiplicity of product of terms of n-th row of A077553. - Ray Chandler, Aug 21 2003
A059841(a(n))=1, A000035(a(n))=0. - Reinhard Zumkeller, Sep 29 2008
(APSO) Alternating partial sums of (a-b+c-d+e-f+g...) = (a+b+c+d+e+f+g...) - 2*(b+d+f...), it appears that APSO(A005843) = A052928 = A002378 - 2*(A116471), with A116471=2*A008794. - Eric Desbiaux, Oct 28 2008
A056753(a(n)) = 1. - Reinhard Zumkeller, Aug 23 2009
Twice the nonnegative numbers. - Juri-Stepan Gerasimov, Dec 12 2009
The number of hydrogen atoms in straight-chain (C(n)H(2n+2)), branched (C(n)H(2n+2), n > 3), and cyclic, n-carbon alkanes (C(n)H(2n), n > 2). - Paul Muljadi, Feb 18 2010
For n >= 1; a(n) = the smallest numbers m with the number of steps n of iterations of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = m. See A175126 and A175127. A175126(a(n)) = A175126(A175127(n)) = n. Example (a(4)=8): 8-2=6, 6-2=4, 4-2=2, 2-2=0; iterations has 4 steps and number 8 is the smallest number with such result. - Jaroslav Krizek, Feb 15 2010
For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is not integer. A040001(a(n)) > 1. See A145051 and A040001. - Jaroslav Krizek, May 28 2010
Union of A179082 and A179083. - Reinhard Zumkeller, Jun 28 2010
a(k) is the (Moore lower bound on and the) order of the (k,4)-cage: the smallest k-regular graph having girth four: the complete bipartite graph with k vertices in each part. - Jason Kimberley, Oct 30 2011
For n > 0: A048272(a(n)) <= 0. - Reinhard Zumkeller, Jan 21 2012
Let n be the number of pancakes that have to be divided equally between n+1 children. a(n) is the minimal number of radial cuts needed to accomplish the task. - Ivan N. Ianakiev, Sep 18 2013
For n > 0, a(n) is the largest number k such that (k!-n)/(k-n) is an integer. - Derek Orr, Jul 02 2014
a(n) when n > 2 is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
It appears that for n > 2, a(n) = A020482(n) + A002373(n), where all sequences are infinite. This is consistent with Goldbach's conjecture, which states that every even number > 2 can be expressed as the sum of two prime numbers. - Bob Selcoe, Mar 08 2015
Number of partitions of 4n into exactly 2 parts. - Colin Barker, Mar 23 2015
Number of neighbors in von Neumann neighborhood. - Dmitry Zaitsev, Nov 30 2015
Unique solution b( ) of the complementary equation a(n) = a(n-1)^2 - a(n-2)*b(n-1), where a(0) = 1, a(1) = 3, and a( ) and b( ) are increasing complementary sequences. - Clark Kimberling, Nov 21 2017
Also the maximum number of non-attacking bishops on an (n+1) X (n+1) board (n>0). (Cf. A000027 for rooks and queens (n>3), A008794 for kings or A030978 for knights.) - Martin Renner, Jan 26 2020
Integer k is even positive iff phi(2k) > phi(k), where phi is Euler's totient (A000010) [see reference De Koninck & Mercier]. - Bernard Schott, Dec 10 2020
Number of 3-permutations of n elements avoiding the patterns 132, 213, 312 and also number of 3-permutations avoiding the patterns 213, 231, 321. See Bonichon and Sun. - Michel Marcus, Aug 20 2022
a(n) gives the y-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The x-value is given by 2*n^2 + 1 (see Tattersall). - Stefano Spezia, Jul 24 2025

			G.f. = 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 16*x^8 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 28.
J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 529a pp. 71 and 257, Ellipses, 2004, Paris.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
Kevin Fagan, Drabble cartoon, Jun 15 1987: Intelligence Test
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012, J. Int. Seq. 15 (2012) # 12.6.2
Milan Janjic, Two Enumerative Functions
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Even Number
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros
Wikipedia, Alkane
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

a(n)=2*A001477(n). - Juri-Stepan Gerasimov, Dec 12 2009
Cf. A000027, A002061, A005408, A001358, A077553, A077554, A077555, A002024, A087112, A157888, A157889, A140811, A157872, A157909, A157910, A165900.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), this sequence (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A231200 (boustrophedon transform).

a005843 = (* 2)
a005843_list = [0, 2 ..]  -- Reinhard Zumkeller, Feb 11 2012

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

A005843 := n->2*n;
A005843:=2/(z-1)**2; # Simon Plouffe in his 1992 dissertation

Range[0,120,2] (* Harvey P. Dale, Aug 16 2011 *)

PARI
```
A005843(n) = 2*n
```

def a(n): return 2*n # Martin Gergov, Oct 20 2022

R
```
seq(0,200,2)
```

G.f.: 2*x/(1-x)^2.
E.g.f.: 2*x*exp(x). - Geoffrey Critzer, Aug 25 2012
G.f. with interpolated zeros: 2x^2/((1-x)^2 * (1+x)^2); e.g.f. with interpolated zeros: x*sinh(x). - Geoffrey Critzer, Aug 25 2012
Inverse binomial transform of A036289, n*2^n. - Joshua Zucker, Jan 13 2006
a(0) = 0, a(1) = 2, a(n) = 2a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
a(n) = Sum_{k=1..n} floor(6n/4^k + 1/2). - Vladimir Shevelev, Jun 04 2009
a(n) = A034856(n+1) - A000124(n) = A000217(n) + A005408(n) - A000124(n) = A005408(n) - 1. - Jaroslav Krizek, Sep 05 2009
a(n) = Sum_{k>=0} A030308(n,k)*A000079(k+1). - Philippe Deléham, Oct 17 2011
Digit sequence 22 read in base n-1. - Jason Kimberley, Oct 30 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Dec 23 2011
a(n) = 2*n = Product_{k=1..2*n-1} 2*sin(Pi*k/(2*n)), n >= 0 (undefined product := 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - Wolfdieter Lang, Oct 10 2013
From Ilya Gutkovskiy, Aug 19 2016: (Start)
Convolution of A007395 and A057427.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 = (1/2)*A002162 = (1/10)*A016655. (End)
From Bernard Schott, Dec 10 2020: (Start)
Sum_{n>=1} 1/a(n)^2 = Pi^2/24 = A222171.
Sum_{n>=1} (-1)^(n+1)/a(n)^2 = Pi^2/48 = A245058. (End)

A141417 (-1)^(n+1)A091137(n)a(0,n), where a(i,j) = Integral_{x=i..i+1} x(x-1)(x-2)...(x-j+1)/j! dx.

Original entry on oeis.org

-1, 1, 1, 1, 19, 27, 863, 1375, 33953, 57281, 3250433, 5675265, 13695779093, 24466579093, 132282840127, 240208245823, 111956703448001, 205804074290625, 151711881512390095, 281550972898020815, 86560056264289860203, 161867055619224199787, 20953816286242674495191, 39427936010479474495191

Offset: 0

Paul Curtz, Aug 05 2008

sign

This is row i=0 of an array defined as T(i,j) = (-1)^(i+j+1)*A091137(j)*a(i,j), columns j >= 0, which starts
  -1,   1,   1,    1,    19,    27,   863, ...
   1,  -3,   5,    1,    11,    11,   271, ...
  -1,   5, -23,    9,    19,    11,   191, ...
   1,  -7,  53,  -55,   251,    27,   271, ...
  -1,   9, -95,  161, -1901,   475,   863, ...
   1, -11, 149, -351,  6731, -4277, 19087, ...
  ...
The first two rows are related via T(0,j) = A027760(j)*T(0,j-1) - T(1,j).

P. Curtz, Integration .., note 12, C.C.S.A., Arcueil, 1969.

Cf. A141047, A140811, A140825.

A091137 := proc(n) local a, i, p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; fi; a := a*p^floor(n/(p-1)) ; od: a ; end proc:
A048994 := proc(n, k) combinat[stirling1](n, k) ; end proc:
a := proc(i,j) add(A048994(j,k)*x^k,k=0..j) ; int(%,x=i..i+1) ; %/j! ; end proc:
A141417 := proc(n) (-1)^(n+1)*A091137(n)*a(0,n) ; end proc:
seq(A141417(n),n=0..40) ; # R. J. Mathar, Nov 17 2010

(* a7 = A091137 *) a7[n_] := a7[n] = Times @@ Select[ Divisors[n]+1, PrimeQ]*a7[n-1]; a7[0]=1; a[n_] := (-1)^(n+1) * a7[n] * Integrate[ (-1)^n*Pochhammer[-x, n], {x, 0, 1}]/n!; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Aug 10 2012 *)

a(n):=if n=0 then -1 else num(n*(n+1)*sum(((-1)^(n-k)*stirling2(n+k,k)*binomial(2*n-1,n-k))/((n+k)*(n+k-1)),k,1,n)); /* Vladimir Kruchinin, Dec 12 2016 */

a(i,j) = a(i-1,j) + a(i-1,j-1), see reference page 33.
(q+1-j)*Sum_{j=0..q} a(i,j)*(-1)^(q-j) = binomial(i,q), see reference page 35.
a(n) = numerator(n*(n+1)*Sum_{k=1..n} ((-1)^(n-k)*Stirling2(n+k,k)*binomial(2*n-1,n-k))/((n+k)*(n+k-1))), n>0, a(0)=-1. - Vladimir Kruchinin, Dec 12 2016

Erroneous formula linking A091137 and A002196 removed, and more terms and program added by R. J. Mathar, Nov 17 2010

A140825 Numerators of upper right triangle of a(i,j) = Integral_{x=i..i+1} Sum_{k=0..j} A048994(j,k)*x^k.

Original entry on oeis.org

1, 1, 3, -1, 5, 23, 1, -1, 9, 55, -19, 11, -19, 251, 1901, 27, -11, 11, -27, 475, 4277, -863, 271, -191, 271, -863, 19087, 198721, 1375, -351, 191, -191, 351, -1375, 36799, 434241, -33953, 7297, -3233, 2497, -3233, 7297, -33953, 1070017, 14097247, 57281, -10625, 3969

Offset: 0

Paul Curtz, Jul 17 2008

Denominators of the j-th column are A002790(j). Note that the fractions defined by division are not fully reduced to coprime numerator and denominator.

			The array a(i,j) starts with rows i>=0 and columns j>=0 as:
1 1/2 -1/6 1/4 -19/30 9/4 -863/84 1375/24 ...
1 3/2 5/6 -1/4 11/30 -11/12 271/84 -117/8 ...
1 5/2 23/6 9/4 -19/30 11/12 -191/84 191/24 ...
1 7/2 53/6 55/4 251/30 -9/4 271/84 -191/24 ...
1 9/2 95/6 161/4 1901/30 475/12 -863/84 117/8 ...
1 11/2 149/6 351/4 6731/30 4277/12 19087/84 -1375/24 ...
The sequence lists the numerators of the j-th column from row 0 down to row j.
The fractions of the j=5 column, 9/4, -11/12, 11/12, -9/4, 475/12, 4277/12, are listed with a common denominator A002790(5)=12 as 27, -11, 11, -27, 475, 4277.

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

Cf. A002790, A048994, A140811.

a[i_, j_] := Sum[((1+i)^(k+1)-i^(k+1))*StirlingS1[j, k]/(k+1), {k, 0, j}]; col[j_] := Total[Table[a[i, j], {i, 0, j} ]*x^Range[0, j]] // Together // Numerator // CoefficientList[#, x]&; Table[col[j], {j, 0, 9}] // Flatten (* Jean-François Alcover, Jan 10 2016 *)

Edited by R. J. Mathar, Aug 06 2008

A141310 The odd numbers interlaced with the constant-2 sequence.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 9, 2, 11, 2, 13, 2, 15, 2, 17, 2, 19, 2, 21, 2, 23, 2, 25, 2, 27, 2, 29, 2, 31, 2, 33, 2, 35, 2, 37, 2, 39, 2, 41, 2, 43, 2, 45, 2, 47, 2, 49, 2, 51, 2, 53, 2, 55, 2, 57, 2, 59, 2, 61, 2, 63, 2, 65, 2, 67, 2, 69, 2, 71, 2, 73, 2, 75, 2, 77, 2, 79, 2, 81, 2, 83, 2, 85, 2, 87, 2, 89, 2, 91, 2, 93, 2, 95, 2, 97

Offset: 0

Paul Curtz, Aug 02 2008

Similarly, the principle of interlacing a sequence and its first differences leads from A000012 and its differences A000004 to A059841, or from A140811 and its first differences A017593 to a sequence -1, 6, 5, 18, ...
If n is even then a(n) = n + 1 ; otherwise a(n) = 2. - Wesley Ivan Hurt, Jun 05 2013
Denominators of floor((n+1)/2) / (n+1), n > 0. - Wesley Ivan Hurt, Jun 14 2013
a(n) is also the number of minimum total dominating sets in the (n+1)-gear graph for n>1. - Eric W. Weisstein, Apr 11 2018
a(n) is also the number of minimum total dominating sets in the (n+1)-sun graph for n>1. - Eric W. Weisstein, Sep 09 2021
Denominators of Cesàro means sequence of A114112, corresponding numerators are in A354008. -  Bernard Schott, May 14 2022
Also, denominators of Cesàro means sequence of A237420, corresponding numerators are in A354280. -  Bernard Schott, May 22 2022

Antti Karttunen, Table of n, a(n) for n = 0..16384
Eric Weisstein's World of Mathematics, Gear Graph.
Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
Eric Weisstein's World of Mathematics, Sun Graph.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000004, A000012, A000142, A005408, A008586, A017593, A059841, A060747, A109613, A140811, A211374, A319702 (rgs-transform).
Cf. A114112, A354008.
Cf. A237420, A354280.

a:= n-> n+1-(n-1)*(n mod 2): seq(a(n), n=0..96); # Wesley Ivan Hurt, Jun 05 2013

Flatten[Table[{2 n - 1, 2}, {n, 40}]] (* Alonso del Arte, Jun 15 2013 *)
Riffle[Range[1, 79, 2], 2] (* Alonso del Arte, Jun 14 2013 *)
Table[((-1)^n (n - 1) + n + 3)/2, {n, 0, 20}] (* Eric W. Weisstein, Apr 11 2018 *)
Table[Floor[(n + 1)/2]/(n + 1), {n, 0, 20}] // Denominator (* Eric W. Weisstein, Apr 11 2018 *)
LinearRecurrence[{0, 2, 0, -1}, {2, 3, 2, 5}, {0, 20}] (* Eric W. Weisstein, Apr 11 2018 *)
CoefficientList[Series[(1 + 2 x + x^2 - 2 x^3)/(-1 + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Apr 11 2018 *)

A141310(n) = if(n%2,2,1+n); \\ (for offset=0 version) - Antti Karttunen, Oct 02 2018

A141310off1(n) = if(n%2,n,2); \\ (for offset=1 version) - Antti Karttunen, Oct 02 2018

def A141310(n): return 2 if n % 2 else n + 1 # Chai Wah Wu, May 24 2022

a(2n) = A005408(n). a(2n+1) = 2.
First differences: a(n+1) - a(n) = (-1)^(n+1)*A109613(n-1), n > 0.
b(2n) = -A008586(n), and b(2n+1) = A060747(n), where b(n) = a(n+1) - 2*a(n).
a(n) = 2*a(n-2) - a(n-4). - R. J. Mathar, Feb 23 2009
G.f.: (1+2*x+x^2-2*x^3)/((x-1)^2*(1+x)^2). - R. J. Mathar, Feb 23 2009
From Wesley Ivan Hurt, Jun 05 2013: (Start)
a(n) = n + 1 - (n - 1)*(n mod 2).
a(n) = (n + 1) * (n - floor((n+1)/2))! / floor((n+1)/2)!.
a(n) = A000142(n+1) / A211374(n+1). (End)

Edited by R. J. Mathar, Feb 23 2009

Term a(45) corrected, and more terms added by Antti Karttunen, Oct 02 2018

A141047 Numerators of A091137(n)T(n,n)/n! where T(i,j)=Integral (x= i.. i+1) x(x-1)(x-2) .. *(x-j+1) dx.

Original entry on oeis.org

1, 3, 23, 55, 1901, 4277, 198721, 434241, 14097247, 30277247, 2132509567, 4527766399, 13064406523627, 27511554976875, 173233498598849, 362555126427073, 192996103681340479, 401972381695456831, 333374427829017307697, 691668239157222107697, 236387355420350878139797

Offset: 0

Paul Curtz, Jul 31 2008

nonn

Numerators of the main diagonal of the array A091137(j)*T(i,j)/j! where T(i,j)=Integral (x= i.. i+1) x*(x-1)*(x-2)* .. *(x-j+1) dx.
The reduced fractions of the array T(i,j) are shown in A140825, which also describes how the integrand is a generating function of Stirling numbers.
The sequence A027760 plays a role i) in relating to A091137 as described there and
ii) in a(n+1)-A027760(n+1)*a(n)= A002657(n+1), numerators of the diagonal T(n,n+1).

P. Curtz, Integration numerique des systemes differentiels a conditions initiales. Note 12, Centre de Calcul Scientifique de l' Armement, Arcueil (1969), p. 36.

Cf. A140811, A140825, A141045.

T := proc(i,j) local var,k ; var := x ; for k from 1 to j-1 do var := var*(x-k) ; od: int(var,x=i..i+1) ; simplify(A091137(j)*%/j!) ; numer(%) ; end:
A141047 := proc(n) T(n,n) ; end: for n from 0 to 20 do printf("%a,",A141047(n) ) ; od: # R. J. Mathar, Feb 23 2009

b[n_] := b[n] = (* A091137 *) If[n==0, 1, Product[d, {d, Select[Divisors[n] + 1, PrimeQ]}]*b[n-1]]; T[i_, j_] := Integrate[Product[x-k, {k, 0, j-1}], {x, i, i+1}]; a[n_] := b[n]*T[n, n]/n!; Table[a[n] // Numerator, {n, 0, 20}] (* Jean-François Alcover, Jan 10 2016 *)

a(n) = numerator( A091137(n)*T(n,n)/n!) where T(n,n) = sum_{k=0..n} A048994(n,k)*((n+1)^(k+1)-n^(k+1))/(k+1).

Edited and extended by R. J. Mathar, Feb 23 2009

A157371 a(n) = (n+1)(9-9n+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

Paul Curtz, Feb 28 2009

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.

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

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

[(n+1)*(9-9*n+5*n^2-n^3): n in [0..40] ]; // Vincenzo Librandi, Jul 14 2011

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 *)

a(n)=(n+1)*(9-9*n+5*n^2-n^3) \\ Charles R Greathouse IV, Oct 16 2015

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

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

A157411 a(n) = 30n^4 - 120n^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

Paul Curtz, Feb 28 2009

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).

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

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

[30*n^4 - 120*n^3 + 120*n^2 - 19: n in [0..50]]; // Vincenzo Librandi, Aug 07 2011

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 *)

a(n)=30*n^4-120*n^3+120*n^2-19 \\ Charles R Greathouse IV, Oct 16 2015

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.

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

Paul Curtz, Sep 14 2009

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)

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

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

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

(* 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 *)

A157872 a(n) = 9*n^2 - 3.

Original entry on oeis.org

6, 33, 78, 141, 222, 321, 438, 573, 726, 897, 1086, 1293, 1518, 1761, 2022, 2301, 2598, 2913, 3246, 3597, 3966, 4353, 4758, 5181, 5622, 6081, 6558, 7053, 7566, 8097, 8646, 9213, 9798, 10401, 11022, 11661, 12318, 12993, 13686, 14397, 15126, 15873, 16638, 17421, 18222

Offset: 1

Vincenzo Librandi, Mar 08 2009

The identity (6*n^2 - 1)^2 - (9*n^2 - 3)*(2*n)^2 = 1 can be written as A140811(n-1)^2 - a(n)*A005843(n+1)^2 = 1. - Vincenzo Librandi, Feb 05 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1. [broken link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A080663, A140811.

I:=[6, 33, 78]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 05 2012

LinearRecurrence[{3, -3, 1}, {6, 33, 78}, 50] (* Vincenzo Librandi, Feb 05 2012 *)
9*Range[50]^2-3 (* Harvey P. Dale, Aug 04 2024 *)

for(n=1, 40, print1(9*n^2 - 3", ")); \\ Vincenzo Librandi, Feb 05 2012

From Vincenzo Librandi, Feb 05 2012: (Start)
G.f.: -3*x*(2 + 5*x - x^2)/(x - 1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = a(n-1) + 18*n - 9. (End)
From Amiram Eldar, May 28 2022: (Start)
a(n) = 3*A080663(n).
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(3)*cot(Pi/sqrt(3))))/6.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(3))*csc(Pi/sqrt(3)) - 1)/6. (End)
E.g.f.: 3*(exp(x)*(3*x^2 + 3*x - 1) + 1). - Elmo R. Oliveira, Jan 25 2025

A158544 a(n) = 24*n^2 - 1.

Original entry on oeis.org

23, 95, 215, 383, 599, 863, 1175, 1535, 1943, 2399, 2903, 3455, 4055, 4703, 5399, 6143, 6935, 7775, 8663, 9599, 10583, 11615, 12695, 13823, 14999, 16223, 17495, 18815, 20183, 21599, 23063, 24575, 26135, 27743, 29399, 31103, 32855, 34655, 36503, 38399, 40343, 42335

Offset: 1

Vincenzo Librandi, Mar 21 2009

The identity (24*n^2 - 1)^2 - (144*n^2 - 12)*(2*n)^2 = 1 can be written as a(n)^2 - A158543(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A140811, A158543.

I:=[23, 95, 215]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 14 2012

LinearRecurrence[{3, -3, 1}, {23, 95, 215}, 50] (* Vincenzo Librandi, Feb 14 2012 *)

for(n=1, 40, print1(24*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 14 2012

From Vincenzo Librandi, Feb 14 2012: (Start)
G.f.: -x*(23 + 26*x - x^2)/(x - 1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 06 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(6)))*Pi/(2*sqrt(6)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(6)))*Pi/(2*sqrt(6)) - 1)/2. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(24*x^2 + 24*x - 1) + 1.
a(n) = A140811(2*n). (End)

Comment rewritten by R. J. Mathar, Oct 16 2009

cp's OEIS Frontend

A005843 The nonnegative even numbers: a(n) = 2n.

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A141417 (-1)^(n+1)*A091137(n)*a(0,n), where a(i,j) = Integral_{x=i..i+1} x*(x-1)*(x-2)*...*(x-j+1)/j! dx.

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A140825 Numerators of upper right triangle of a(i,j) = Integral_{x=i..i+1} Sum_{k=0..j} A048994(j,k)*x^k.

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A141310 The odd numbers interlaced with the constant-2 sequence.

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A141047 Numerators of A091137(n)*T(n,n)/n! where T(i,j)=Integral (x= i.. i+1) x*(x-1)*(x-2)* .. *(x-j+1) dx.

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

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A141417 (-1)^(n+1)A091137(n)a(0,n), where a(i,j) = Integral_{x=i..i+1} x(x-1)(x-2)...(x-j+1)/j! dx.

A141047 Numerators of A091137(n)T(n,n)/n! where T(i,j)=Integral (x= i.. i+1) x(x-1)(x-2) .. *(x-j+1) dx.

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

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