A114113 -id:A114113 - 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

A210796 Triangle of coefficients of polynomials v(n,x) jointly generated with A210795; see the Formula section.

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 3, 7, 6, 5, 5, 10, 16, 12, 8, 5, 16, 26, 34, 23, 13, 7, 21, 47, 64, 70, 43, 21, 7, 29, 68, 123, 147, 140, 79, 34, 9, 36, 104, 200, 304, 324, 274, 143, 55, 9, 46, 140, 324, 538, 714, 690, 527, 256, 89, 11, 55, 195, 480, 932, 1366, 1616, 1431

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with A109613(n) and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Column 2: A114113
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1...2
3...3....3
3...7....6....5
5...10...16...12...8
First three polynomials v(n,x): 1, 1 + 2x, 3 + 3x + 3x^2

Cf. A210795, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 1; h = 2; p = -1; f = 0;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210795 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210796 *)

u(n,x)=u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.

A210798 Triangle of coefficients of polynomials v(n,x) jointly generated with A210797; see the Formula section.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 2, 5, 7, 5, 1, 6, 12, 13, 8, 2, 8, 20, 29, 25, 13, 1, 9, 27, 51, 62, 46, 21, 2, 11, 39, 84, 125, 129, 84, 34, 1, 12, 48, 126, 224, 284, 258, 151, 55, 2, 14, 64, 182, 374, 562, 622, 505, 269, 89, 1, 15, 75, 250, 580, 1008, 1328, 1315, 969, 475

Offset: 1

Clark Kimberling, Mar 26 2012

Row n starts with A109613(n) and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Column 2: A114113
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2...2
1...3...3
2...5...7....5
1...6...12...13...8
First three polynomials v(n,x): 1, 2 + 2x, 1 + 3x + 3x^2

Cf. A210797, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 0; c = 0; h = 2; p = -1; f = 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210797 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210798 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A099232 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A006130 *)
Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A008346 *)
Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A039834 *)

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A376901 a(n) = (n*(n-1)+(-1)^n+5)/2.

Original entry on oeis.org

3, 2, 4, 5, 9, 12, 18, 23, 31, 38, 48, 57, 69, 80, 94, 107, 123, 138, 156, 173, 193, 212, 234, 255, 279, 302, 328, 353, 381, 408, 438, 467, 499, 530, 564, 597, 633, 668, 706, 743, 783, 822, 864, 905, 949, 992, 1038, 1083, 1131, 1178, 1228, 1277, 1329, 1380, 1434

Offset: 0

Eric W. Weisstein, Oct 08 2024

For n >= 3, also the disorder number of the pan graph.

Eric Weisstein's World of Mathematics, Disorder Number.
Eric Weisstein's World of Mathematics, Pan Graph.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081353, A114113, A236453.

Table[(n (n - 1) + 5 + (-1)^n)/2, {n, 20}]
LinearRecurrence[{2, 0, -2, 1}, {2, 4, 5, 9}, {0, 20}]

a(n) = 2*a(n-1)-2*a(n-3)+1*a(n-4).
G.f.: x*(-2+3*x^2-3*x^3)/((-1+x)^3*(1+x)).
After initial terms same as {A114113}+2, {A236453}+1, ({A081353}+1)/2 + 2. Hugo Pfoertner, Oct 10 2024.

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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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A210796 Triangle of coefficients of polynomials v(n,x) jointly generated with A210795; see the Formula section.

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A210798 Triangle of coefficients of polynomials v(n,x) jointly generated with A210797; see the Formula section.

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Examples

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Formula

A376901 a(n) = (n*(n-1)+(-1)^n+5)/2.

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