A114285 -id:A114285 - cp's OEIS frontend

A114112 a(1)=1, a(2)=2; thereafter a(n) = n+1 if n odd, n-1 if n even.

1, 2, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, 18, 17, 20, 19, 22, 21, 24, 23, 26, 25, 28, 27, 30, 29, 32, 31, 34, 33, 36, 35, 38, 37, 40, 39, 42, 41, 44, 43, 46, 45, 48, 47, 50, 49, 52, 51, 54, 53, 56, 55, 58, 57, 60, 59, 62, 61, 64, 63, 66, 65, 68, 67, 70, 69, 72, 71

Offset: 1

Leroy Quet, Nov 13 2005

a(1)=1; for n>1, a(n) is the smallest positive integer not occurring earlier in the sequence such that a(n) does not divide Sum_{k=1..n-1} a(k). - Leroy Quet, Nov 13 2005 (This was the original definition. A simple induction argument shows that this is the same as the present definition. - N. J. A. Sloane, Mar 12 2018)
Define b(1)=2; for n>1, b(n) is the smallest number not yet in the sequence which shares a prime factor with the sum of all preceding terms. Then a simple induction argument shows that the b(n) sequence is the same as the present sequence with the first term omitted. - David James Sycamore, Feb 26 2018
Here are the details of the two induction arguments (Start)
For a(n), let A(n) = a(1)+...+a(n). The claim is that for n>2 a(n)=n+1 if n odd, n-1 if n even.
The induction hypotheses are: for iFor b(n), the argument is very similar, except that the missing numbers when looking for b(n) are slightly different, and  (setting B(n) = b(1)+...b(n)), we have B(2i)=(i+1)(2i+1), B(2i+1)=(i+2)(2i+1). - N. J. A. Sloane, Mar 14 2018
When sequence a(n) is increasing, then the Cesàro means sequence c(n) = (a(1)+...+a(n))/n is also increasing, but the converse is false. This sequence is a such an example where c(n) is increasing, while a(n) is not increasing (Arnaudiès et al.). See proof in A354008. - Bernard Schott, May 11 2022

J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, Exercice 10, pp. 14-16.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000.
ProofWiki, Cesàro mean.
Wikipedia, Ernesto Cesàro.
Wikipédia, Lemme de Cesàro (in French).
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

All of A014681, A103889, A113981, A114112, A114285 are essentially the same sequence. - N. J. A. Sloane, Mar 12 2018
Cf. A114113 (partial sums).
See A084265 for the partial sums of the b(n) sequence.
About Cesàro mean theorem: A033999, A141310, A237420, A354008.
Cf. A244009.

a[1] = 1; a[n_] := a[n] = Block[{k = 1, s, t = Table[ a[i], {i, n - 1}]}, s = Plus @@ t; While[ Position[t, k] != {} || Mod[s, k] == 0, k++ ]; k]; Array[a, 72] (* Robert G. Wilson v, Nov 18 2005 *)

a(n) = if (n<=2, n, if (n%2, n+1, n-1)); \\ Michel Marcus, May 16 2022

def A114112(n): return n + (0 if n <= 2 else -1+2*(n%2)) # Chai Wah Wu, May 24 2022

G.f.: x*(x^4-2*x^3+x^2+x+1)/((1-x)*(1-x^2)). - N. J. A. Sloane, Mar 12 2018
The g.f. for the b(n) sequence is x*(x^3-3*x^2+2*x+2)/((1-x)*(1-x^2)). - Conjectured (correctly) by Colin Barker, Mar 04 2018
E.g.f.: 1 - x + x^2/2 + (x - 1)*cosh(x) + (x + 1)*sinh(x). - Stefano Spezia, Sep 02 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 1 - log(2) (A244009). - Amiram Eldar, Jun 29 2025

More terms from Robert G. Wilson v, Nov 18 2005

Entry edited (with simpler definition) by N. J. A. Sloane, Mar 12 2018

A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3x - sqrt(1+6x+x^2)) / (4*x) (A001003).

Original entry on oeis.org

0, 1, -5, 49, -725, 14401, -360005, 10863889, -384415925, 15612336481, -715930020005, 36592369889329, -2062911091119125, 127170577711282561, -8510569547826528005, 614491222512504748369, -47615614242877583230325, 3941408640018910366196641

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..353
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, triassociative operad "Trias".

Cf. A002050, A006531, A084099, A101851, A114285, A225883, A383985, A383986, A383988, A383989, A383991.

Composition of A001003 with exp(x)-1.

nn = 17; f[x_] := (1 + 3*x - Sqrt[1 + 6*x + x^2])/(4*x); Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

Original entry on oeis.org

0, 1, -11, 61, -467, 4381, -49091, 643021, -9615827, 161844541, -3026079971, 62243374381, -1396619164787, 33949401567901, -888725861445251, 24926889744928141, -745755560487363347, 23705772035082494461, -797875590555470224931, 28346366547928396344301

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..407
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, operad "FF6".

Cf. A002050, A006531, A084099, A101851, A114285, A225883, A383985, A383986, A383987, A383988, A383995.

nn = 19; f[x_] := x*(1 - 3*x - x^2 + x^3)/(1 + 3*x + x^2 - x^3);
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A383992 Series expansion of the exponential generating function exp(arbustive(x)) - 1 where arbustive(x) = (log(1+x) - x^2) / (1+x).

Original entry on oeis.org

0, 1, -4, 3, 40, -330, 1626, -3150, -54592, 1060920, -13022280, 127171440, -889086648, -283184616, 179750627616, -4895777544840, 99124001788800, -1721513264431680, 25736021675994816, -292896125040673728, 639149345262276480, 106178474282318726400

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..448
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, operad "Arbustive".

Cf. A003725, A097388, A111884, A112242, A114285, A177885, A318215, A383990, A383991, A383993, A383994, A383995.

nn = 21; f[x_] := Exp[x] - 1;
Range[0, nn]! * CoefficientList[Series[f[(Log[1 + x] - x^2)/(1 + x)], {x, 0, nn}], x]

A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

Original entry on oeis.org

0, 1, -1, 4, -23, 181, -1812, 22037, -315569, 5201602, -97009833, 2019669961, -46432870222, 1168383075471, -31939474693297, 942565598033196, -29866348653695203, 1011335905644178273, -36446897413531401020, 1392821757824071815641, -56259101478392975833333

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..200
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, permutative commutative operad "Perm/Com2".

Cf. A002050, A006531, A084099, A101851, A114285, A177885, A225883, A383986, A383987, A383988, A383989.

Composition of A000169 with signs and 1-exp(x).

nn = 20; f[x_] := -Sum[k^(k - 1)*(1 - Exp[x])^k/k!, {k, nn}];
Range[0, nn]! * CoefficientList[Series[f[x], {x, 0, nn}], x]

A383986 Expansion of the exponential generating function sqrt(4exp(x) - exp(2x) - 2) - 1.

Original entry on oeis.org

0, 1, -1, 1, -13, 61, -601, 5881, -73333, 1021861, -16334401, 290146561, -5707536253, 122821558861, -2873553719401, 72586328036041, -1969306486088773, 57106504958139061, -1762735601974347601, 57705363524117482321, -1996916624448159410893

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..409
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, operad "NAC_2".

Cf. A002050, A006531, A084099, A101851, A114285, A182037, A225883, A383985, A383987, A383988, A383989.

nn = 20; f[x_] := -1 + Sqrt[1 + 2 x - x^2];
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A383988 Series expansion of the exponential generating function -postLie(1-exp(x)) where postLie(x) = -log((1 + sqrt(1-4*x)) / 2) (given by A006963).

Original entry on oeis.org

0, 1, -2, 12, -110, 1380, -22022, 426972, -9747950, 256176660, -7617417302, 252851339532, -9268406209790, 371843710214340, -16206868062692582, 762569209601624892, -38525315595630383630, 2079964082064837282420, -119513562475103977951862

Offset: 0

Michael De Vlieger, May 16 2025

The series -postLie(-x) is the inverse for the substitution of the series comTrias(x), given by the suspension of the Koszul dual of comTrias. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..374
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, commutative triassociative operad "ComTrias".

Cf. A002050, A006531, A084099, A097388, A101851, A114285, A225883, A383985, A383986, A383987, A383989. Composition of -A006963(-x) and exp(x)-1.

nn = 18; f[x_] := Log[(1 + Sqrt[1 + 4*x])/2];
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A114284 Riordan array ((1-3*x)/(1-x), x).

Original entry on oeis.org

1, -2, 1, -2, -2, 1, -2, -2, -2, 1, -2, -2, -2, -2, 1, -2, -2, -2, -2, -2, 1, -2, -2, -2, -2, -2, -2, 1, -2, -2, -2, -2, -2, -2, -2, 1, -2, -2, -2, -2, -2, -2, -2, -2, 1, -2, -2, -2, -2, -2, -2, -2, -2, -2, 1, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, 1

Offset: 0

Paul Barry, Nov 20 2005

Inverse of A114283. Row sums are 1-2n. Diagonal sums are A114285. Sequence array of 3*0^n-2.

			Triangle begins:
  1;
  -2,1;
  -2,-2,1;
  -2,-2,-2,1;
  -2,-2,-2,-2,1;

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Ângela Mestre, José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.

Table[-2 - 3 Floor[1/2 (-1 + Sqrt[1 + 8 x])] + 3 Floor[1/2 (-1 + Sqrt[9 + 8 x])], {x, 0, 65}] (* Jackson Xier, Oct 07 2011 *)

T(n, k) = if(k<=n, 3*0^(n-k)-2, 0).

a(n) = -3*floor((1/2)*sqrt(8*n+1)-1/2)+3*floor((1/2)*sqrt(8*n+9)-1/2)-2. - Jackson Xier, Oct 07 2011

A243860 a(n) = 2^(n+1) - (n-1)^2.

Original entry on oeis.org

1, 4, 7, 12, 23, 48, 103, 220, 463, 960, 1967, 3996, 8071, 16240, 32599, 65340, 130847, 261888, 523999, 1048252, 2096791, 4193904, 8388167, 16776732, 33553903, 67108288, 134217103, 268434780, 536870183, 1073741040, 2147482807, 4294966396, 8589933631, 17179868160, 34359737279, 68719475580

Offset: 0

Juri-Stepan Gerasimov, Jun 12 2014

Sequences of the form (k-1)^m - m^(k+1):
k\m | 0 |  1 |     2 |       3 |         4 |          5 |           6 |
-----------------------------------------------------------------------
 | 1 | -2 |    -1 |      -4 |        -3 |         -6 |          -5 |
 | 1 | -1 |    -4 |      -9 |       -16 |        -25 |         -36 |
 | 1 |  0 |    -7 |     -26 |       -63 |       -124 |        -215 |
 | 1 |  1 |   -12 |     -73 |      -240 |       -593 |       -1232 |
 | 1 |  2 |   -23 |    -216 |      -943 |      -2882 |       -7047 |
 | 1 |  3 |   -43 |    -665 |     -3840 |     -14601 |      -42560 |
 | 1 |  4 |  -103 |   -2062 |    -15759 |     -75000 |     -264311 |
 | 1 |  5 |  -220 |   -6345 |    -64240 |    -382849 |    -1632960 |
 | 1 |  6 |  -463 |  -19340 |   -259743 |   -1936318 |    -9960047 |
 | 1 |  7 |  -960 |  -58537 |  -1044480 |   -9732857 |   -60204032 |
| 1 |  8 | -1967 | -176418 |  -4187743 |  -48769076 |  -362265615 |
| 1 |  9 | -3996 | -530441 | -16767216 | -244040625 | -2175782336 |

			1 = 2^(0+1) - (0-1)^2, 4 = 2^(1+1) - (1-1)^2, 7 = 2^(2+1) - (2-1)^2.

Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Sequences of the form (k-1)^m - m^(k+1): A000012 (m = 0), A023444 (m = 1), (-1)*(this sequence) for m = 2, A114285 (k = 0),(A000007-A000290) for k = 1, A024001 (k = 2), A024014 (k = 3), A024028 (k = 4), A024042 (k = 5), A024056 (k = 6), A024070 (k = 7), A024084 (k = 8), A024098 (k = 9), A024112 (k = 10), A024126 (k = 11).

Magma
```
[2^(n+1) - (n-1)^2: n in [0..35]];
```

A243860:=n->2^(n + 1) - (n - 1)^2; seq(A243860(n), n=0..30); # Wesley Ivan Hurt, Jun 12 2014

Table[2^(n + 1) - (n - 1)^2, {n, 0, 30}] (* Wesley Ivan Hurt, Jun 12 2014 *)
LinearRecurrence[{5,-9,7,-2},{1,4,7,12},40] (* Harvey P. Dale, Nov 29 2015 *)

Vec((6*x^3-4*x^2-x+1)/((x-1)^3*(2*x-1)) + O(x^100)) \\ Colin Barker, Jun 12 2014

a(n) = 5*a(n-1)-9*a(n-2)+7*a(n-3)-2*a(n-4). - Colin Barker, Jun 12 2014

G.f.: (6*x^3-4*x^2-x+1) / ((x-1)^3*(2*x-1)). - Colin Barker, Jun 12 2014

A289870 a(n) = n(n + 1) for n odd, otherwise a(n) = (n - 1)(n + 1).

Original entry on oeis.org

-1, 2, 3, 12, 15, 30, 35, 56, 63, 90, 99, 132, 143, 182, 195, 240, 255, 306, 323, 380, 399, 462, 483, 552, 575, 650, 675, 756, 783, 870, 899, 992, 1023, 1122, 1155, 1260, 1295, 1406, 1443, 1560, 1599, 1722, 1763, 1892, 1935, 2070, 2115, 2256, 2303, 2450, 2499

Offset: 0

Jean-François Alcover and Paul Curtz, Jul 14 2017

a(n) is a fifth-order linear recurrence whose main interest is that it is related to (at least) eight other sequences (see the formula section).

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

After -1, subsequence of A035106, A198442 and A214297.

Cf. A000466, A002378, A093178, A109613, A114285, A124356, A193356, A289296.

a[n_] := (n + 1)(n - 1 + Mod[n, 2]); Table[a[n], {n, 0, 50}]

a(n)=if(n%2, n, n-1)*(n+1) \\ Charles R Greathouse IV, Jul 14 2017

a(n) = (n + 1)*(n - 1 + (n mod 2)).
a(n) = n * A109613(n-1) for n>0.
a(n) = -A114285(n) * A109613(n).
a(n) = A002378(n) - A193356(n).
a(n) = A289296(-n).
a(n) = n^2 - (-1)^n * A093178(n).
a(2*k) = A000466(k).
G.f.: (1-3*x-3*x^2-3*x^3)/((-1+x)^3*(1+x)^2).

Showing 1-10 of 10 results.

cp's OEIS Frontend

A114112 a(1)=1, a(2)=2; thereafter a(n) = n+1 if n odd, n-1 if n even.

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A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3*x - sqrt(1+6*x+x^2)) / (4*x) (A001003).

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A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3*x-x^2+x^3) / (1+3*x+x^2-x^3).

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A383992 Series expansion of the exponential generating function exp(arbustive(x)) - 1 where arbustive(x) = (log(1+x) - x^2) / (1+x).

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A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

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A383986 Expansion of the exponential generating function sqrt(4*exp(x) - exp(2*x) - 2) - 1.

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A383988 Series expansion of the exponential generating function -postLie(1-exp(x)) where postLie(x) = -log((1 + sqrt(1-4*x)) / 2) (given by A006963).

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A114284 Riordan array ((1-3*x)/(1-x), x).

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A243860 a(n) = 2^(n+1) - (n-1)^2.

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A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3x - sqrt(1+6x+x^2)) / (4*x) (A001003).

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

A383986 Expansion of the exponential generating function sqrt(4exp(x) - exp(2x) - 2) - 1.

A289870 a(n) = n(n + 1) for n odd, otherwise a(n) = (n - 1)(n + 1).