A111284 -id:A111284 - cp's OEIS frontend

A113127 Expansion of (1 + x + x^2 + x^3)/(1-x)^2.

1, 3, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 0

Paul Barry, Oct 14 2005

Row sums of number triangle A113126.
Equals binomial transform of [1, 2, 1, 0, -1, 2, -3, 4, -5, ...]. - Gary W. Adamson, Feb 14 2009
From 6 on the same as A016825. - R. J. Mathar, Jul 21 2013
The size of a maximal 4-degenerate graph of order n-2 (this class includes 4-trees). - Allan Bickle, Nov 14 2021
Maximum size of an apex graph of order n-2 (an apex graph can be made planar by deleting a single vertex). - Allan Bickle, Nov 14 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016825, A073760, A111284, A131034.

a(n) - a(n-1) = A158411(n+1). - Jaume Oliver Lafont, Mar 27 2009

[4*n-2+2*Binomial(0, n)+Binomial(1, n): n in [0..80]]; // Vincenzo Librandi, Nov 03 2018

CoefficientList[Series[(1 + x + x^2 + x^3) / (1 - x)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Nov 03 2018 *)
LinearRecurrence[{2,-1},{1,3,6,10},60] (* Harvey P. Dale, Jul 08 2019 *)

x='x+O('x^66); Vec((1+x+x^2+x^3)/(1-x)^2) \\ Joerg Arndt, May 06 2013

a(n) = 4*n - 2 + 2*binomial(0, n) + binomial(1, n);
a(n) = binomial(n+1, n) + binomial(n, n-1) + binomial(n-1, n-2) + binomial(n-2, n-3).
Row sums of triangle A131034. - Gary W. Adamson, Jun 10 2007
G.f.: (x^2-1)/Q(0), where Q(k)= 4*x - 1 + x*k - x*(x-1)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 05 2013
a(n) = A111284(n+1) for n >= 2. - Georg Fischer, Nov 02 2018
a(n) = 4*(n+2) - 10 for n >= 2. - Allan Bickle, Nov 14 2021

A130824 a(n) = 2*A004273(n).

Original entry on oeis.org

0, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 0

Paul Curtz, Jul 17 2007

Equals A111284 from the 2nd term on. - R. J. Mathar, Jun 13 2008
Besides the first term, this sequence gives the denominators of the alternating series Pi/8 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + .... - Mohammad K. Azarian, Oct 14 2011 [edited by Jon E. Schoenfield, Mar 07 2015]
Numbers that cannot be a side of a primitive Pythagorean triangle. - Torlach Rush, Nov 07 2019
Simple continued fraction expansion of tanh(1/2) = (e - 1)/(e + 1) = 1/(2 + 1/(6 + 1/(10 + 1/(14 + ...)))). - Peter Bala, Oct 01 2023

Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016825, A111284, A160327.

Concatenation([0], List([1..60], n-> 4*n-2 )); # G. C. Greubel, Dec 30 2019

[4*n-2*Floor((n+2) mod (n+1)):n in [0..60]]; // Vincenzo Librandi, Sep 22 2011

A130827 := proc(n) if n =0 then 0 ; else 4*n-2 ; fi ; end: seq(A130827(n),n=0..120) ; # R. J. Mathar, Oct 28 2007

2 Join[{0}, Range[1, 200, 2]] (* Michael De Vlieger, Mar 07 2015 *)

vector(61, n, if(n==1, 0, 4*(n-1) -2) ) \\ G. C. Greubel, Dec 30 2019

[0]+[4*n-2 for n in (1..60)] # G. C. Greubel, Dec 30 2019

From Stefano Spezia, Dec 09 2019: (Start)
G.f.: 2*x*(1+x)/(1-x)^2.
a(n) = 2*a(n-1) - a(n-2) for n > 0.
a(n) = 4*n - 1 - (-1)^(2^n). (End)
E.g.f: 2*(1 - (1-2*x)*exp(x)). - G. C. Greubel, Dec 30 2019

More terms from R. J. Mathar, Oct 28 2007

A133653 A007318^(-1) * A003261.

Original entry on oeis.org

1, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154

Offset: 1

Gary W. Adamson, Sep 19 2007

It appears this sequence gives the positive integers m such that the sum of the first m Fibonacci numbers divides their product. For example, if n=2 and m=a(2)=6, we have the sum 1+1+2+3+5+8=20 which clearly divides the corresponding product 480. See A175553 for the analogous sequence when using the triangular numbers. Sum_{k=1..n} Fibonacci(k) divides Product_{k=1..n} Fibonacci(k). - John W. Layman, Jul 10 2010

			a(4) = 14 = (1, 3, 3, 1) dot (1, 5, -1, 1) = (1, 15, -3, 1).

Cf. A003261, A007318, A175553.

Essentially the same as A130824, A113127, A111284, A073760, A016825.

Inverse binomial transform of A003261: (1, 7, 23, 63, 159, 383, ...).
Binomial transform of [1, 5, -1, 1, -1, 1, ...].
"1" followed by 2 * [3, 5, 7, 9, 11, ...].
O.g.f.: x*(1+4x-x^2)/(1-x)^2. a(n) = 4n-2, n > 1. - R. J. Mathar, Jun 08 2008
1/(1+1/(6+1/(10+1/(14+1/(...(continued fraction)))))) = (e-1)/2 with e = 2.718281...- Philippe Deléham, Mar 09 2013

More terms from R. J. Mathar, Jun 08 2008

A167666 Triangle read by rows given by [1,1,-4,2,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 0, 4, 5, 1, 0, 0, 6, 7, 1, 0, 0, 0, 8, 9, 1, 0, 0, 0, 0, 10, 11, 1, 0, 0, 0, 0, 0, 12, 13, 1, 0, 0, 0, 0, 0, 0, 14, 15, 1, 0, 0, 0, 0, 0, 0, 0, 16, 17, 1, 0, 0, 0, 0, 0, 0, 0, 0, 18, 19, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 23, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Nov 08 2009

Row sums = A111284(n+1), Diagonal sums = A109613(n).

			Triangle begins :
1 ;
1, 1 ;
2, 3, 1 ;
0, 4, 5, 1 ;
0, 0, 6, 7, 1 ;
0, 0, 0, 8, 9, 1 ;
0, 0, 0, 0, 10, 11, 1 ; ...

Cf. A000012, A005408, A005843

T(n,k) = 2*T(n-1,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(3,0) = 0, T(3,1) = 4. - Philippe Deléham, Feb 18 2012
G.f.: (1+(1-y)*x+(2+y)*x^2)/(1-y*x)^2. - Philippe Deléham, Feb 18 2012
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A130779(n), A111284(n+1), A167667(n), A167682(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Feb 18 2012

A306264 a(n) = 1 + d*a(n/d); a(1)=0. If n has only one prime divisor, then d=n, otherwise d is the greatest proper unitary divisor of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 5, 1, 8, 6, 1, 1, 10, 1, 6, 8, 12, 1, 9, 1, 14, 1, 8, 1, 16, 1, 1, 12, 18, 8, 10, 1, 20, 14, 9, 1, 22, 1, 12, 10, 24, 1, 17, 1, 26, 18, 14, 1, 28, 12, 9, 20, 30, 1, 21, 1, 32, 10, 1, 14, 34, 1, 18, 24, 36, 1, 10, 1, 38, 26, 20, 12, 40, 1, 17

Offset: 1

David James Sycamore, Feb 01 2019

nonn

Name related to recursive formula of A006022.
a(n) = 1 if and only if n is a prime power; p^t; t >= 1.
The sequence of indices k on which a(k) is a record (1,2,6,10,14,18,22,26,30,...), appears to be A111284.

			a(8) = a(25) = 1 because 8 and 25 are prime powers.
a(30) = 16 because 15 is the greatest proper unitary divisor of 30, so a(30) = 1 + 15*a(2) = 1 + 15 = 16.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A006022, A077610, A111284, A324388.

f[n_] := If[PrimePowerQ[n], n,
  SelectFirst[Transpose@
    {Reverse@ #[[-Ceiling[Length[#]/2] ;; -2]],
      #[[2 ;; Ceiling[Length[#]/2]]]} &@ Divisors[n],
    CoprimeQ @@ # &][[1]] ]; f[1] = 1;
a[n_] := 1 + #*a[n/#] &[f[n]]; a[1] = 0;
Array[a, 120] (* Michael De Vlieger, Jun 24 2025 *)

d(n) = if (omega(n) == 1, n, my(v=select(x->(gcd(x, n/x)==1), divisors(n))); v[#v-1]);
lista(nn) = {va = vector(nn); va[1] = 0; for (n=2, nn, dn = d(n); va[n] = 1 + dn*va[n/dn];); va;} \\ Michel Marcus, Feb 10 2019

A324388(n) = if(1>=omega(n),n,fordiv(n,d,if((d>1)&&(1==gcd(d,n/d)),return(n/d))));
A306264(n) = if(1==n,0,my(d=A324388(n)); 1+(d*A306264(n/d))); \\ Antti Karttunen, Feb 28 2019

a(1) = 0; for n > 1, a(n) = 1 + (A324388(n) * a(n/A324388(n))). - Antti Karttunen, Feb 28 2019

A161718 Expansion of (1+3*x^2)/(1+x)^2.

Original entry on oeis.org

1, -2, 6, -10, 14, -18, 22, -26, 30, -34, 38, -42, 46, -50, 54, -58, 62, -66, 70, -74, 78, -82, 86, -90, 94, -98, 102, -106, 110, -114, 118, -122, 126, -130, 134, -138, 142, -146, 150, -154, 158, -162, 166, -170, 174, -178, 182, -186, 190, -194, 198, -202, 206, -210, 214, -218, 222, -226, 230, -234

Offset: 0

Pr Mosbah Amlouk (Mosbah.Amlouk(AT)fsb.rnu.tn), Jun 17 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A016825, A111284 (unsigned version), A135929, A137276.

CoefficientList[Series[(1+3*x^2)/(1+x)^2,{x,0,80}],x] (* or *) LinearRecurrence[ {-2,-1},{1,-2,6},80] (* Harvey P. Dale, Mar 19 2016 *)

x='x+O('x^99); Vec((1+3*x^2)/(1+x)^2) \\ Altug Alkan, Apr 17 2018

From R. J. Mathar, Aug 27 2009: (Start)
a(n) = -2*a(n-1)-a(n-2), n>3.
G.f.: (1+3*x^2)/(1+x)^2.
a(n) = 4*(-1)^n*n+2*(-1)^(n+1) = (-1)^n*A016825(n-1), n>0. (End)
E.g.f.: 3 - 2*exp(-x)*(1 + 2*x). - Stefano Spezia, Feb 02 2023

Spurious commas in sequence deleted by N. J. A. Sloane, Aug 02 2009
Offset corrected, extended by R. J. Mathar, Aug 27 2009
Edited by Joerg Arndt, Sep 04 2011

A247555 A permutation of the nonnegative numbers: a(4n) = 8n, a(4n+1) = 2n + 1, a(4n+2) = 4n + 2, a(4n+3) = 8n + 4.

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 6, 12, 16, 5, 10, 20, 24, 7, 14, 28, 32, 9, 18, 36, 40, 11, 22, 44, 48, 13, 26, 52, 56, 15, 30, 60, 64, 17, 34, 68, 72, 19, 38, 76, 80, 21, 42, 84, 88, 23, 46, 92, 96, 25, 50, 100, 104, 27, 54, 108, 112, 29, 58, 116, 120

Offset: 0

Paul Curtz, Sep 19 2014

A permutation of the nonnegative integers.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A005408, A008590, A016825, A017113, A225055, A001477, A111284.

&cat[[4*(i-1),i,2*i,4*i]: i in [1..50 by 2]]; // Bruno Berselli, Sep 19 2014

a[n_]:=Switch[Mod[n,4],0,2 n,1,(n+1)/2,2,n,3,2 n-2]; Table[a[n],{n,0,60}] (* Jean-François Alcover, Oct 09 2014 *)
LinearRecurrence[{0,0,0,2,0,0,0,-1}, {0,1,2,4,8,3,6,12}, 50] (* G. C. Greubel, May 01 2018 *)

Vec(x*(4*x^6+2*x^5+x^4+8*x^3+4*x^2+2*x+1)/((x-1)^2*(x+1)^2*(x^2+1)^2) + O(x^100)) \\ Colin Barker, Sep 19 2014

a(n) = a(n-4) + a(n-8) - a(n-12).
a(n) = 2*a(n-4) - a(n-8). - Colin Barker, Sep 19 2014
G.f.: x*(4*x^6 + 2*x^5 + x^4 + 8*x^3 + 4*x^2 + 2*x + 1) / ((x-1)^2*(x+1)^2*(x^2+1)^2). - Colin Barker, Sep 19 2014
a(n) = (11*n-3+(n+3)*(-1)^n+(4*n-1+(-1)^n)*cos(n*Pi/2)+2*(9-3*n+4(-1)^n)* sin(n*Pi/2))/8. - Wesley Ivan Hurt, May 07 2021

A302986 Number of partitions of n into two distinct parts (p,q) such that p, q and |q-p| are all squarefree.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 2, 2, 2, 0, 1, 2, 2, 0, 3, 3, 4, 0, 2, 3, 3, 0, 4, 4, 5, 0, 4, 3, 4, 0, 4, 5, 5, 0, 4, 7, 5, 0, 6, 6, 7, 0, 8, 7, 9, 0, 6, 7, 8, 0, 5, 7, 7, 0, 6, 6, 8, 0, 8, 7, 9, 0, 11, 7, 9, 0, 8, 10, 8, 0, 10, 13, 12, 0, 10, 11, 11, 0, 11, 11, 15, 0, 9

Offset: 1

Wesley Ivan Hurt, Apr 16 2018

Index entries for sequences related to partitions

Cf. A008683, A111284, A294247, A303051.

[0,0] cat [&+[(MoebiusMu(k)^2*MoebiusMu(n-k)^2)*MoebiusMu(n-2*k)^2: k in [1..Floor((n-1)/2)]]: n in [3..100]]; // Vincenzo Librandi, Apr 17 2018

Table[Sum[MoebiusMu[i]^2*MoebiusMu[n - i]^2*MoebiusMu[n - 2 i]^2, {i, Floor[(n - 1)/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[{#[[1]],#[[2]],#[[1]] - #[[2]]},SquareFreeQ]&)],{n,90}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jan 21 2021 *)

a(n) = sum(i=1, (n-1)\2, moebius(i)^2*moebius(n-i)^2*moebius(n-2*i)^2); \\ Michel Marcus, Apr 17 2018

a(n) = Sum_{i=1..floor((n-1)/2)} mu(i)^2 * mu(n-i)^2 * mu(n-2*i)^2, where mu is the Möbius function (A008683).

a(n) = 0 for n in A111284. - Michel Marcus, Apr 17 2018

cp's OEIS Frontend

A113127 Expansion of (1 + x + x^2 + x^3)/(1-x)^2.

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A130824 a(n) = 2*A004273(n).

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A133653 A007318^(-1) * A003261.

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A167666 Triangle read by rows given by [1,1,-4,2,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A306264 a(n) = 1 + d*a(n/d); a(1)=0. If n has only one prime divisor, then d=n, otherwise d is the greatest proper unitary divisor of n.

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A161718 Expansion of (1+3*x^2)/(1+x)^2.

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A247555 A permutation of the nonnegative numbers: a(4n) = 8n, a(4n+1) = 2n + 1, a(4n+2) = 4n + 2, a(4n+3) = 8n + 4.

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