A152271 -id:A152271 - cp's OEIS frontend

A194590 a(n) = (-1)^n(A056040(n+1)A152271(n)-2^n)/2.

0, 0, 1, -2, 7, -14, 38, -76, 187, -374, 874, -1748, 3958, -7916, 17548, -35096, 76627, -153254, 330818, -661636, 1415650, -2831300, 6015316, -12030632, 25413342, -50826684, 106853668, -213707336, 447472972, -894945944, 1867450648, -3734901296, 7770342787

Offset: 0

Peter Luschny, Aug 30 2011

sign

The binomial transform of a(n) are the complementary Riordan numbers A194589 (see link).

Peter Luschny, The lost Catalan numbers

Cf. A107373 (has offset 1).

A056040 := n -> n!/iquo(n,2)!^2:
A152271 := n -> `if`(n mod 2 = 0, 1, (n+1)/2):
A194590 := n -> (-1)^n*(A056040(n+1)*A152271(n)-2^n)/2:

sf[n_] := n!/Quotient[n, 2]!^2;
a[n_] := (-1)^n (sf[n + 1] * If[EvenQ[n], 1, (n + 1)/2] - 2^n)/2;
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 26 2019 *)

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*cr(k), where cr(k) are the complementary Riordan numbers A194589.

A057979 a(n) = 1 for even n and (n-1)/2 for odd n.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 1, 16, 1, 17, 1, 18, 1, 19, 1, 20, 1, 21, 1, 22, 1, 23, 1, 24, 1, 25, 1, 26, 1, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 1, 39, 1, 40, 1, 41, 1, 42, 1, 43, 1

Offset: 0

Labos Elemer, Nov 13 2000

a(n) = b(n)/c(n) where b(n) = A001405(n+1) - A001405(n), c(n) = gcd(A001405(n+1), A001405(n)).
Also the minimal number of disjoint edge-paths into which the complete graph on n edges can be partitioned - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jan 19 2001
For n >= 2, number of partitions of n-2 into parts that are distinct mod 2. - Giovanni Resta, Feb 06 2006
Sequence starting with a(3) obeys the rule "smallest positive value such that the ordered pair (a(n-1),a(n)) has not occurred previously", or the rule "smallest positive value such that the ratio a(n-1)/a(n) has not occurred previously". The same subsequence has its ordinal transform equal to itself, shifted left. (The ordinal transform has as its n-th term the number of values in a(1),...,a(n) that are equal to a(n).) - Franklin T. Adams-Watters, Dec 13 2006
Numerators of floor(n/2)/n, n > 0. - Wesley Ivan Hurt, Jun 14 2013
Number of nonisomorphic outer planar graphs of order n >= 3, maximum degree 3, and largest possible size. The size is (3n-2)/2 when n is even and (3n-3)/2 when n is odd. - Christian Barrientos and Sarah Minion, Feb 27 2018

			For n=12, C(12,6) - C(11,5) = 924 - 462 = 462, gcd(C(12,6), C(11,5)) = 462, and the quotient is 1.
For n=13, C(13,6) - C(12,6) = 792, gcd(C(13,6), C(12,6)) = 132, and the quotient is 6.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A001405, A007879, A059222, A000035, A027656, A037952, A067992, A152271.

import Data.List (transpose)
a057979 n = 1 - rest * (1 - n') where (n', rest) = divMod n 2
a057979_list = concat $ transpose [repeat 1, [0..]]
-- Reinhard Zumkeller, Aug 11 2014

[Floor(n/2)^(n mod 2): n in [0..100]]; // Vincenzo Librandi, Mar 17 2015

A057979:=n->(n+1)/4+(3-n)*(-1)^n/4; seq(A057979(k), k=0..100); # Wesley Ivan Hurt, Oct 14 2013

With[{no=45},Riffle[Table[1,{no}],Range[0,no-1]]] (* Harvey P. Dale, Feb 18 2011 *)

a(n)=if(n%2,n-1,2)/2 \\ Charles R Greathouse IV, Sep 02 2015

def A057979(n): return n>>1 if n&1 else 1 # Chai Wah Wu, Jan 04 2024

a(n) = (n+1)/4+(3-n)*(-1)^n/4. - Paul Barry, Mar 21 2003, corrected by Hieronymus Fischer, Sep 25 2007
a(n) = (a(n-2) + a(n-3)) / a(n-1).
From Paul Barry, Oct 21 2004: (Start)
G.f.: (1-x^2+x^3)/((1+x)^2(1-x)^2);
a(n) = 2*a(n-2) - a(n-4);
a(n) = 0^n + Sum_{k=0..floor((n-2)/2)} C(n-k-2,k) * C(1,n-2k-2). (End)
a(n) = gcd(n-1, floor((n-1)/2)). - Paul Barry, May 02 2005
a(n) = binomial((2n-3)/4-(-1)^n/4, (1-(-1)^n)/2). - Paul Barry, Jun 29 2006
G.f.: (x^3-x^2+1)/(1-x^2)^2 = 1 + x^2*G(0) where G(k) = 1 + x*(k+1)/(1 - x/(x + (k+1)/G(k+1) )); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 29 2012
a(n) = binomial(floor(n/2), n mod 2). - Wesley Ivan Hurt, Oct 14 2013
a(n) = 1 - n mod 2 * (1 - floor(n/2)). - Reinhard Zumkeller, Aug 11 2014
a(n) = floor(n/2)^(n mod 2). - Wesley Ivan Hurt, Mar 16 2015
E.g.f.: ((2 + x)*cosh(x) - sinh(x))/2. - Stefano Spezia, Mar 26 2022

A117142 Number of partitions of n in which any two parts differ by at most 2.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 14, 15, 20, 21, 27, 28, 35, 36, 44, 45, 54, 55, 65, 66, 77, 78, 90, 91, 104, 105, 119, 120, 135, 136, 152, 153, 170, 171, 189, 190, 209, 210, 230, 231, 252, 253, 275, 276, 299, 300, 324, 325, 350, 351, 377, 378, 405, 406, 434, 435, 464, 465

Offset: 1

Emeric Deutsch, Feb 27 2006

Equals row sums of triangle A177991. - Gary W. Adamson, May 16 2010

Positive numbers that are either triangular (A000217) or triangular minus 1 (A000096). - Jon E. Schoenfield, Jun 12 2010

			a(6) = 9 because we have
  1: [6],
  2: [4, 2],
  3: [3, 3],
  4: [3, 2, 1],
  5: [3, 1, 1, 1],
  6: [2, 2, 2],
  7: [2, 2, 1, 1],
  8: [2, 1, 1, 1, 1],
  9: [1, 1, 1, 1, 1, 1]
([5,1] and [4,1,1] do not qualify).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Jonathan Bloom and Nathan McNew, Counting pattern-avoiding integer partitions, arXiv:1908.03953 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000096, A000217, A117143, A152271, A177991, A152919.

Column k=2 of A194621. - Alois P. Heinz, Oct 17 2012

List([1..60],n->(2*n^2+10*n+3+(-1)^n*(2*n-3))/16); # Muniru A Asiru, Dec 21 2018

[(2*n*(n+5) +3 +(-1)^n*(2*n-3))/16: n in [1..60]]; // G. C. Greubel, Jul 18 2023

g:=sum(x^k/(1-x^k)/(1-x^(k+1))/(1-x^(k+2)),k=1..75): gser:=series(g,x=0,70): seq(coeff(gser,x^n),n=1..65); with(combinat): for n from 1 to 7 do P:=partition(n): A:={}: for j from 1 to nops(P) do if P[j][nops(P[j])]-P[j][1]<3 then A:=A union {P[j]} else A:=A fi od: print(A); od: # this program yields the partitions

Table[Count[IntegerPartitions[n], ?(Max[#] - Min[#] <= 2 &)], {n, 30}] (* _Birkas Gyorgy, Feb 20 2011 *)
Table[(2*n^2 +10*n +3 +(-1)^n*(2*n-3))/16, {n,30}] (* Birkas Gyorgy, Feb 20 2011 *)
Table[Sum[If[EvenQ[k], 1, (k+1)/2], {k,0,n}], {n,0,60}] (* Jon Maiga, Dec 21 2018 *)

Vec(x*(x^2-x-1)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Mar 05 2015

[(2*n*(n+5) +3 +(-1)^n*(2*n-3))/16 for n in range(1,61)] # G. C. Greubel, Jul 18 2023

G.f.: Sum_{k>=1} x^k/((1 - x^k)*(1 - x^(k + 1))*(1 - x^(k + 2))). More generally, the g.f. of the number of partitions in which any two parts differ by at most b is Sum_{k>=1} (x^k/(Product_{j=k..k+b} 1 - x^j)).
a(n) = (2*n^2 + 10*n + 3 + (-1)^n * (2*n - 3))/16. - Birkas Gyorgy, Feb 20 2011
G.f.: (1 + x)/(1 - x)/(Q(0) - x^2 - x^3), where Q(k) = 1 + x^2 + x^3 + k*x*(1 + x^2) - x^2*(1 + x*(k + 2))*(1 + k*x)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jan 05 2014
G.f.: x*(1 + x - x^2)/((1 - x)^3*(1 + x)^2). - Colin Barker, Mar 05 2015
a(n) = Sum_{k=0..n-1} A152271(k). - Jon Maiga, Dec 21 2018
E.g.f.: (1/16)*( (3 + 2*x)*exp(-x) + (3 + 12*x + 2*x^2)*exp(x) ). - G. C. Greubel, Jul 18 2023
a(n) = A152919(n+1)/2. - Ridouane Oudra, Oct 29 2024

A124625 Even numbers sandwiched between 1's.

Original entry on oeis.org

1, 0, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 14, 1, 16, 1, 18, 1, 20, 1, 22, 1, 24, 1, 26, 1, 28, 1, 30, 1, 32, 1, 34, 1, 36, 1, 38, 1, 40, 1, 42, 1, 44, 1, 46, 1, 48, 1, 50, 1, 52, 1, 54, 1, 56, 1, 58, 1, 60, 1, 62, 1, 64, 1, 66, 1, 68, 1, 70, 1, 72, 1, 74, 1, 76, 1, 78, 1, 80, 1, 82, 1, 84

Offset: 0

N. J. A. Sloane, Jun 13 2007

Interleaving of A000012 and A005843.
Created to simplify the definition of A129952.
a(n) = abs(A009531(n-1)).
Starting (1, 2, 1, 4,...): square (1 + x - x^2 - x^3 + x^4 + x^5 - ...) = (1 + 2x - x^2 - 4x^3 + x^4 + 6x^5 - ...).
With a(3) taken as 0, a(n+2) = n^k+1 mod 2*n, n>=1, for any k>=2, also for k=n. - Wolfdieter Lang, Dec 21 2011
Also !(n+2) mod n for n>0 where !n is a subfactorial number (A000166). - Michel Lagneau, Sep 05 2012
Greatest common divisor of n-1 and (n-1) mod 2. - Bruno Berselli, Mar 07 2017

Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000012 (all 1's), A005843 (even numbers), A009531, A093178, A152271.

Magma
```
&cat[[1, 2*k]: k in [0..42]];
```

A124625:=n->(n-(n-2)*(-1)^n)/2; seq(A124625(k), k=0..100); # Wesley Ivan Hurt, Oct 19 2013

Join[{1},Riffle[2Range[0,50],1]] (* Harvey P. Dale, Nov 02 2011 *)

{for(n=0, 85, print1(if(n%2>0, n-1, 1), ","))}

print([(n-1)**(n%2) for n in range(0, 86)]) # Karl V. Keller, Jr., Jul 26 2020

a(n) = 1 for even n, a(n) = n-1 for odd n.
a(2*k) = 1, a(2*k+1) = 2*k.
G.f.: (1 - x^2 + 2*x^3)/((1 - x)^2*(1 + x)^2).
a(n) = (n - (n - 2)*(-1)^n)/2. - Bruno Berselli, May 06 2011
E.g.f.: 1 + x^2*U(0)/2 where U(k) = 1 + 2*x*(k+1)/(2*k + 3 - x*(2*k+3)/(x + 4*(k+2)*(k+1)/U(k+1))) (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Oct 20 2012
a(n) = 2*floor(n/2) - (n-1)*((n-1) mod 2). - Wesley Ivan Hurt, Oct 19 2013
a(n) = (n-1)^((1-(-1)^n)/2). - Wesley Ivan Hurt, Mar 21 2015
a(n) = (n-1) - a(a(n-1))*a(n-1), a(0) = 0. - Eli Jaffe, Jun 07 2016
E.g.f.: (x + 1)*cosh(x) - sinh(x). - Ilya Gutkovskiy, Jun 07 2016
a(n) = (-1)^n mod n for n > 0. - Franz Vrabec, Mar 06 2020
a(n) = (n-1)^(n mod 2). - Karl V. Keller, Jr., Aug 01 2020

More terms from Klaus Brockhaus, Jun 16 2007

Edited by N. J. A. Sloane, May 21 2008 at the suggestion of R. J. Mathar

A158416 Expansion of g.f. (1+x-x^3)/(1-x^2)^2.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 1, 16, 1, 17, 1, 18, 1, 19, 1, 20, 1, 21, 1, 22, 1, 23, 1, 24, 1, 25, 1, 26, 1, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 1, 39, 1, 40, 1, 41, 1, 42, 1, 43, 1, 44, 1

Offset: 0

Paul Barry, Mar 18 2009

Binomial transform is A111297. Binomial transform of [1,1,1,2,1,3,1,...] is A109975.
Essentially the same as A152271 and A133622. - R. J. Mathar, Mar 20 2009
Also defined by: a(0)=1; thereafter, a(n) = number of copies of a(n-1) in the list [a(0), a(1), ..., a(n-1)]. - N. J. A. Sloane, Feb 12 2016

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A109975, A111297, A133622, A152271.

Related to A268696.

CoefficientList[Series[(1+x-x^3)/(1-x^2)^2,{x,0,100}],x] (* or *) LinearRecurrence[{0,2,0,-1},{1,1,2,1},100] (* Harvey P. Dale, Aug 17 2016 *)

a(n)=1+!(n%2)*n/2 \\ Jaume Oliver Lafont, Mar 21 2009

a(n) = Sum_{k=0..floor(n/2)} binomial(k+1,n-k).
G.f.: Q(0)/x - 1/x, where Q(k)= 1 + (k+1)*x/(1 - x/(x + (k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 23 2013
E.g.f.: cosh(x) + (2 + x)*sinh(x)/2. - Stefano Spezia, Sep 06 2023

A145677 Triangle T(n, k) read by rows: T(n, 0) = 1, T(n, n) = n, n>0, T(n,k) = 0, 0 < k < n-1.

Original entry on oeis.org

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

Offset: 0

Gary W. Adamson and Roger L. Bagula, Mar 28 2009

The first entry in each row is 1, the last entry in each of the rows consist of the positive integers (starting 1,1,2,3,...), and all other entries in the triangle are 0's (see example).

The vector of (1, 1, 2, 5, 16, 65, 326,...), which is 1 followed by A000522, is an eigenvector of the matrix: 1 + Sum_{k=1..n} T(n,k)*A000522(k-1) = A000522(n).

			First few rows of the triangle:
  1;
  1, 1;
  1, 0, 2;
  1, 0, 0, 3;
  1, 0, 0, 0, 4;
  1, 0, 0, 0, 0, 5;
  1, 0, 0, 0, 0, 0, 6;
  1, 0, 0, 0, 0, 0, 0, 7;
  1, 0, 0, 0, 0, 0, 0, 0, 8;
  1, 0, 0, 0, 0, 0, 0, 0, 0, 9;
  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A128229, A002061, A000522, A152271, A158416.

T[n_, k_]:= If[k==0, 1, If[k==n, n, 0]];
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 23 2021 *)

def A145677(n,k):
    if (k==0): return 1
    elif (k==n): return n
    else: return 0
flatten([[A145677(n,k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Dec 23 2021

T(n, k) = A158821(n,n-k).
1 + Sum_{k= 1..n} T(n,k) *(k-1) = A002061(n).
From G. C. Greubel, Dec 23 2021: (Start)
Sum_{k=0..n} T(n, k) = A000027(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A158416(n) = A152271(n+1). (End)

Edited by R. J. Mathar, Oct 02 2009

A296168 Decimal expansion of BesselJ(1,2)/BesselJ(0,2).

Original entry on oeis.org

2, 5, 7, 5, 9, 2, 0, 3, 2, 1, 3, 6, 8, 2, 2, 1, 9, 5, 6, 8, 5, 7, 4, 9, 6, 7, 8, 2, 3, 1, 5, 0, 4, 4, 4, 9, 0, 6, 1, 2, 9, 8, 1, 9, 5, 3, 2, 6, 0, 0, 1, 5, 1, 4, 6, 2, 7, 8, 2, 7, 2, 4, 1, 9, 9, 3, 2, 0, 0, 2, 4, 9, 9, 1, 3, 9, 2, 2, 7, 4, 2, 3, 2, 1, 3, 5, 1, 5, 6, 4, 0, 1, 0, 9, 3, 0, 1, 4, 5, 3

Offset: 1

Ilya Gutkovskiy, Dec 06 2017

			2.575920321368221956857496782315044490612981953260015...

Eric Weisstein's World of Mathematics, Continued Fraction Constants
Index entries for sequences related to Bessel functions or polynomials

Cf. A052119, A091681, A152271.

RealDigits[BesselJ[1, 2]/BesselJ[0, 2], 10, 100] [[1]]
RealDigits[Sum[(-1)^k/((k + 1) (k!)^2), {k, 0, Infinity}]/Sum[(-1)^k/(k!)^2, {k, 0, Infinity}], 10, 100][[1]]

besselj(1,2)/besselj(0,2) \\ Charles R Greathouse IV, Oct 23 2023

Equals 2 + 1/(1 + 1/(1 + 1/(2 + 1/(1 + 1/(3 + 1/(1 + 1/(4 + 1/(1 + 1/(5 + 1/(1 + 1/(6 + ...))))))))))).

A048619 a(n) = LCM(binomial(n,0), ..., binomial(n,n)) / binomial(n,floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 3, 4, 2, 10, 5, 30, 15, 7, 7, 56, 28, 252, 126, 60, 30, 330, 165, 396, 198, 286, 143, 2002, 1001, 15015, 15015, 7280, 3640, 1768, 884, 15912, 7956, 3876, 1938, 38760, 19380, 406980, 203490, 99484, 49742, 1144066, 572033, 1961256, 980628

Offset: 0

Labos Elemer

			If n=10 then A002944(10)=2520, A001405(10)=252, the quotient a(10)=10.

Index entries for sequences related to lcm's

Cf. A001405, A002944.

[Lcm([1..n+1]) div (Floor((n+3)/2)*Binomial(n+1,Floor((n+3)/2))): n in [0..50]]; // Vincenzo Librandi, Jul 10 2019

Table[Apply[LCM, Binomial[n, Range[0, n]]]/Binomial[n, Floor[n/2]], {n, 0, 48}] (* Michael De Vlieger, Jun 29 2017 *)

{A048619(n) = lcm(vector(n+1, i, i)) / binomial(n+1, (n+1)\2) / ((n+2)\2);}

a(n) = A002944(n)/A001405(n).
a(n) = lcm(1..n+1)/(floor((n+3)/2)*binomial(n+1,floor((n+3)/2))). - Paul Barry, Jul 03 2006
a(n) = lcm(1,2,...,n+1) / (ceiling((n+1)/2)*binomial(n+1,floor((n+1)/2))) = A003418(n+1) / A100071(n+1). - Max Alekseyev, Oct 23 2015
a(n) = A263673(n+1) / A110654(n+1) = A180000(n+1) / A152271(n). - Max Alekseyev, Oct 23 2015
a(2*n-1) = A068553(n) = A068550(n)/n.

Definition corrected and a(0)=1 prepended by Max Alekseyev, Oct 23 2015

A128218 First differences of A128217.

Original entry on oeis.org

1, 3, 1, 3, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

John W. Layman, Feb 19 2007

nonn

a(A130883(n-1)) = 2*n-1 and a(m) != 2*n-1 for m < A130883(n-1). - Reinhard Zumkeller, Jun 20 2015

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A128127.

Cf. A130883, A152271 (run lengths after initial term).

a128218 n = a128218_list !! (n-1)
a128218_list = zipWith (-) (tail a128217_list) a128217_list
-- Reinhard Zumkeller, Jun 20 2015

nsrQ[n_]:=Module[{sr=Sqrt[n]},Abs[First[sr-Nearest[{Floor[sr], Ceiling[ sr]}, sr]]]<1/4];Differences[Select[Range[0,250],nsrQ]] (* Harvey P. Dale, May 02 2012 *)

default(realprecision, 10000);
is_A128217(n) = ((abs(sqrt(n)-sqrtint(n))<(1/4)) || (abs(sqrt(n)-(1+sqrtint(n)))<(1/4)));
k=0; n=0; prevm=0; while(k<20000, n++; if(is_A128217(n), k++; write("b128218.txt", k, " ", (n-prevm)); prevm = n)); \\ Antti Karttunen, Jan 16 2025

Let A(1)={1}. Then, for k=2,3,4,..., form A(k) by appending to A(k-1) the term k-1 followed by k-1 1's, if k is even, or by appending to A(k-1) the term k followed by k-1 1's, if k is odd. {a(n)} appears to be the limit of {A(k)} as k->infinity.

Offset changed by Reinhard Zumkeller, Jun 20 2015

A175686 a(n) = binomial(n-j-1,j) + binomial(n-j,j-1) with j= floor((n-1)/2).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 7, 7, 14, 11, 25, 16, 41, 22, 63, 29, 92, 37, 129, 46, 175, 56, 231, 67, 298, 79, 377, 92, 469, 106, 575, 121, 696, 137, 833, 154, 987, 172, 1159, 191, 1350, 211, 1561, 232, 1793, 254, 2047, 277, 2324, 301, 2625, 326, 2951, 352, 3303, 379

Offset: 0

Roger L. Bagula, Dec 04 2010

The column m=1 in the array A175685, where the sum over the binomials reduces to only two terms.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A175685, A057979, A152271.

Table[Sum[Binomial[n - j - 1, j], {j, Floor[(n - 1)/2] - 1, Floor[(
    n - 1)/2]}], {n, 0, 30}]
CoefficientList[Series[-(x^3-x^2-x)(x^4-x^2+1)/(x^2-1)^4, {x, 0, 30}],x] (* Benedict W. J. Irwin, Oct 31 2016 *)
Table[(42+20n+6n^2+n^3+(-1)^n(-42+20n-6n^2+n^3))/96, {n, 0, 30}] (* Benedict W. J. Irwin, Oct 31 2016 *)
LinearRecurrence[{0,4,0,-6,0,4,0,-1},{0,1,1,2,3,4,7,7},60] (* Harvey P. Dale, Jul 29 2018 *)

concat(0, Vec(x*(1+x-x^2)*(1-x^2+x^4)/((1-x)^4*(1+x)^4) + O(x^100))) \\ Colin Barker, Oct 31 2016

a(n) = A057979(n+1) + binomial(n-j,j-1) with j = A004526(n-1), n>0.
From Benedict W. J. Irwin, Oct 31 2016: (Start)
G.f.: -(x^3 - x^2 - x)*(x^4 - x^2 + 1)/(x^2 - 1)^4.
E.g.f.: ((6*x + 3*x^2)*cosh(x) + (42 + 21*x + 6*x^2 + x^3)*sinh(x))/48.
a(n) = (42 + 20*n + 6*n^2 + n^3 + (-1)^n*(-42 + 20*n - 6*n^2 + n^3))/96. (End)
a(n) = 4*a(n-2)-6*a(n-4)+4*a(n-6)-a(n-8) for n>7. - Colin Barker, Oct 31 2016

More terms from Colin Barker, Oct 31 2016

cp's OEIS Frontend

A194590 a(n) = (-1)^n*(A056040(n+1)*A152271(n)-2^n)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A057979 a(n) = 1 for even n and (n-1)/2 for odd n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

A117142 Number of partitions of n in which any two parts differ by at most 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

SageMath

Formula

A124625 Even numbers sandwiched between 1's.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A158416 Expansion of g.f. (1+x-x^3)/(1-x^2)^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A145677 Triangle T(n, k) read by rows: T(n, 0) = 1, T(n, n) = n, n>0, T(n,k) = 0, 0 < k < n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A194590 a(n) = (-1)^n(A056040(n+1)A152271(n)-2^n)/2.