A109613 -id:A109613 - cp's OEIS frontend

A158821 Triangle read by rows: row n (n>=0) ends with 1, and for n>=1 begins with n; other entries are zero.

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

Offset: 0

Gary W. Adamson, Mar 30 2008

			Triangle begins:
  1;
  1, 1;
  2, 0, 1;
  3, 0, 0, 1;
  4, 0, 0, 0, 1;
  5, 0, 0, 0, 0, 1;
  6, 0, 0, 0, 0, 0, 1;
  7, 0, 0, 0, 0, 0, 0, 1;

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

Cf. A000027, A109613, A145677.

A158821:= proc(n,k)
    if n = k then 1;
    elif k = 0 then n;
    else 0;
    end if;
end proc: # R. J. Mathar, Jan 08 2015

T[n_, k_]:= If[k==0, Boole[n==0] +n, If[k==n, 1, 0]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 22 2021 *)
Join[{1},Table[Join[{n},PadLeft[{1},n,0]],{n,15}]]//Flatten (* Harvey P. Dale, Apr 05 2023 *)

def A158821(n,k):
    if (k==0): return n + bool(n==0)
    elif (k==n): return 1
    else: return 0
flatten([[A158821(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 22 2021

T(n, k) = A145677(n, n-k-1). - R. J. Mathar, Apr 01 2009
From G. C. Greubel, Dec 22 2021: (Start)
Sum_{k=0..n} T(n, k) = A000027(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A109613(n). (End)

A182579 Triangle read by rows: T(0,0) = 1, for n>0: T(n,n) = 2 and for k<=floor(n/2): T(n,2k) = n/(n-k) binomial(n-k,k), T(n,2k+1) = (n-1)/(n-1-k) binomial(n-1-k,k).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 2, 1, 1, 5, 4, 5, 2, 1, 1, 6, 5, 9, 5, 2, 1, 1, 7, 6, 14, 9, 7, 2, 1, 1, 8, 7, 20, 14, 16, 7, 2, 1, 1, 9, 8, 27, 20, 30, 16, 9, 2, 1, 1, 10, 9, 35, 27, 50, 30, 25, 9, 2, 1, 1, 11, 10, 44, 35, 77, 50, 55, 25, 11, 2

Offset: 0

Reinhard Zumkeller, May 06 2012

A000204(n+1) = sum of n-th row, Lucas numbers;
A000204(n+3) = alternating row sum of n-th row;
A182584(n) = T(2*n,n), central terms;
A000012(n) = T(n,0), left edge;
A040000(n) = T(n,n), right edge;
A054977(n-1) = T(n,1) for n > 0;
A109613(n-1) = T(n,n-1) for n > 0;
A008794(n) = T(n,n-2) for n > 1.

			Starting with 2nd row = [1 2] the rows of the triangle are defined recursively without computing explicitely binomial coefficients; demonstrated for row 8, (see also Haskell program):
   (0) 1  1  7  6 14  9  7  2      [A]  row 7 prepended by 0
    1  1  7  6 14  9  7  2 (0)     [B]  row 7, 0 appended
    1  0  1  0  1  0  1  0  1      [C]  1 and 0 alternating
    1  0  7  0 14  0  7  0  0      [D]  = [B] multiplied by [C]
    1  1  8  7 20 14 16  7  2      [E]  = [D] added to [A] = row 8.
The triangle begins:                 | A000204
              1                      |       1
             1  2                    |       3
            1  1  2                  |       4
           1  1  3  2                |       7
          1  1  4  3  2              |      11
         1  1  5  4  5  2            |      18
        1  1  6  5  9  5  2          |      29
       1  1  7  6 14  9  7  2        |      47
      1  1  8  7 20 14 16  7  2      |      76
     1  1  9  8 27 20 30 16  9  2    |     123
    1  1 10  9 35 27 50 30 25  9  2  |     199 .

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271, with corrections.

Cf. A065941, A059841.

a182579 n k = a182579_tabl !! n !! k
a182579_row n = a182579_tabl !! n
a182579_tabl = [1] : iterate (\row ->
  zipWith (+) ([0] ++ row) (zipWith (*) (row ++ [0]) a059841_list)) [1,2]

T[_, 0] = 1;
T[n_, n_] /; n > 0 = 2;
T[_, 1] = 1;
T[n_, k_] := T[n, k] = Which[
     OddQ[k],  T[n - 1, k - 1],
     EvenQ[k], T[n - 1, k - 1] + T[n - 1, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)

T(n+1,2*k+1) = T(n,2*k), T(n+1,2*k) = T(n,2*k-1) + T(n,2*k).

A186421 Even numbers interleaved with repeated odd numbers.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 6, 3, 8, 5, 10, 5, 12, 7, 14, 7, 16, 9, 18, 9, 20, 11, 22, 11, 24, 13, 26, 13, 28, 15, 30, 15, 32, 17, 34, 17, 36, 19, 38, 19, 40, 21, 42, 21, 44, 23, 46, 23, 48, 25, 50, 25, 52, 27, 54, 27, 56, 29, 58, 29, 60, 31, 62, 31, 64, 33, 66, 33, 68, 35, 70, 35, 72, 37, 74, 37, 76, 39, 78, 39, 80, 41, 82, 41, 84, 43, 86, 43

Offset: 0

Reinhard Zumkeller, Feb 21 2011

A005843 interleaved with A109613.

Row sum of superimposed binary filled triangle. - Craig Knecht, Aug 07 2015

			A005843: 0   2   4   6   8   10   12   14   16   18   20    22
A109613:   1   1   3   3   5    5    7    7    9    9    11    11
this   : 0 1 2 1 4 3 6 3 8 5 10 5 12 7 14 7 16 9 18 9 20 11 22 ... .

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Craig Knecht, Row sums of superimposed binary triangles.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A005843, A109043, A109613.

Cf. A186422 (first differences), A186423 (partial sums), A240828 (row sum of superimposed binary triangle).

a186421 n = a186421_list !! n
a186421_list = interleave [0,2..] $ rep [1,3..] where
   interleave (x:xs) ys = x : interleave ys xs
   rep (x:xs) = x : x : rep xs

[IsEven(n) select n else 2*Floor(n/4)+1: n in [0..100]]; // Vincenzo Librandi, Jul 13 2015

A186421:=n->n-(1-(-1)^n)*(n+(-1)^(n*(n+1)/2))/4: seq(A186421(n), n=0..100); # Wesley Ivan Hurt, Aug 07 2015

Table[n - (1 - (-1)^n)*(n + I^(n (n + 1)))/4, {n, 0, 87}] (* or *)
CoefficientList[Series[x (1 + 2 x + 2 x^3 + x^4)/((1 + x^2) (x - 1)^2 (1 + x)^2), {x, 0, 87}], x] (* or *)
n = 88; Riffle[Range[0, n, 2], Flatten@ Transpose[{Range[1, n, 2], Range[1, n, 2]}]] (* Michael De Vlieger, Jul 14 2015 *)

makelist(n-(1-(-1)^n)*(n+%i^(n*(n+1)))/4, n, 0, 90); /* Bruno Berselli, Mar 04 2013 */

def A186421(n): return (n>>1)|1 if n&1 else n # Chai Wah Wu, Jan 31 2023

a(2*k) = 2*k, a(4*k+1) = a(4*k+3) = 2*k+1.
a(n) = n if n is even, else 2*floor(n/4)+1.
a(2*n-(2*k+1)) + a(2*n+2*k+1) = 2*n, 0 <= k < n.
a(n+2) = A109043(n) - a(n).
G.f.: x*(1+2*x+2*x^3+x^4) / ( (1+x^2)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, Feb 23 2011
a(n) = n-(1-(-1)^n)*(n+i^(n(n+1)))/4, where i=sqrt(-1). - Bruno Berselli, Feb 23 2011
a(n) = a(n-2)+a(n-4)-a(n-6) for n>5. - Wesley Ivan Hurt, Aug 07 2015
E.g.f.: (x*cosh(x) + sin(x) + 2*x*sinh(x))/2. - Stefano Spezia, May 09 2021

Edited by Bruno Berselli, Mar 04 2013

A186422 First differences of A186421.

Original entry on oeis.org

1, 1, -1, 3, -1, 3, -3, 5, -3, 5, -5, 7, -5, 7, -7, 9, -7, 9, -9, 11, -9, 11, -11, 13, -11, 13, -13, 15, -13, 15, -15, 17, -15, 17, -17, 19, -17, 19, -19, 21, -19, 21, -21, 23, -21, 23, -23, 25, -23, 25, -25, 27, -25, 27, -27, 29, -27, 29, -29, 31, -29, 31, -31, 33, -31, 33, -33, 35, -33, 35, -35, 37, -35, 37, -37, 39, -37, 39, -39, 41, -39, 41, -41, 43

Offset: 0

Reinhard Zumkeller, Feb 21 2011

a(n) = A186421(n+1) - A186421(n);
a(2*n) = - A109613(n-1) for n>0; a(2*n+1) = A109613(n);
a(3*k) = A047270(floor((k+1)/2)) * (-1)^(k+1);
a(3*k+1) = A007310(floor((k+2)/2)) * (-1)^k;
a(3*k+2) = A047241(floor((k+3)/2)) * (-1)^(k+1).

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

Cf. A007310, A047241, A047270, A109613, A186421.

a186422 n = a186422_list !! n
a186422_list = zipWith (-) (tail a186421_list) a186421_list

/* By definition: */
A186421:=func;
[A186421(n+1)-A186421(n): n in [0..90]]; // Bruno Berselli, Mar 04 2013

Differences@ CoefficientList[Series[x (1 + 2 x + 2 x^3 + x^4)/((1 + x^2) (x - 1)^2 (1 + x)^2), {x, 0, 84}], x] (* Michael De Vlieger, Oct 02 2017 *)

makelist(-((2*n+1)*(-1)^n-2*%i^(n*(n+1))-3)/4,n,0,83); /* Bruno Berselli, Mar 04 2013 */

G.f.: -(x^4+2*x^3+2*x+1) / ((x-1)*(x+1)^2*(x^2+1)). - Colin Barker, Mar 04 2013
a(n) = -((2*n+1)*(-1)^n-2*i^(n*(n+1))-3)/4, where i=sqrt(-1). [Bruno Berselli, Mar 04 2013]
a(n) = cos((n-1)*Pi)*(2*n+1-2*cos(n*Pi/2)-3*cos(n*Pi)-2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 02 2017
E.g.f.: (cos(x) + (1 + x)*cosh(x) - sin(x) - (x - 2)*sinh(x))/2. - Stefano Spezia, May 09 2021

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

A182432 Recurrence a(n)a(n-2) = a(n-1)(a(n-1) + 3) with a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 28, 217, 1705, 13420, 105652, 831793, 6548689, 51557716, 405913036, 3195746569, 25160059513, 198084729532, 1559517776740, 12278057484385, 96664942098337, 761041479302308, 5991666892320124, 47172293659258681, 371386682381749321

Offset: 0

Peter Bala, Apr 30 2012

The non-linear recurrence equation a(n)*a(n-2) = a(n-1)*(a(n-1) + r) with initial conditions a(0) = 1, a(1) = 1 + r has the solution a(n) = 1/2 + (1/2)*Sum_{k = 0..n} (2*r)^k*binomial(n+k,2*k) = 1/2 + b(n,2*r)/2, where b(n,x) are the Morgan-Voyce polynomials of A085478. The recurrence produces sequences A101265 (r = 1), A011900 (r = 2) and A054318 (r = 4), as well as signed versions of A133872 (r = -1), A109613 (r = -2), A146983 (r = -3) and A084159(r = -4).
Also the indices of centered pentagonal numbers (A005891) which are also centered triangular numbers (A005448). - Colin Barker, Jan 01 2015
Also positive integers y in the solutions to 3*x^2 - 5*y^2 - 3*x + 5*y = 0. - Colin Barker, Jan 01 2015

Seiichi Manyama, Table of n, a(n) for n = 0..1116 (first 201 terms from Vincenzo Librandi)
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Cf. A011900, A054318, A070997, A101265, A109613, A133872, A146983, A211955.

I:=[1, 4, 28]; [n le 3 select I[n] else 9*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..25]]; // Vincenzo Librandi, May 18 2012

RecurrenceTable[{a[0]==1,a[1]==4,a[n]==(a[-1+n] (3+a[-1+n]))/a [-2+n]}, a[n],{n,30}] (* or *) LinearRecurrence[{9,-9,1},{1,4,28},30] (* Harvey P. Dale, May 14 2012 *)

Vec((1-5*x+x^2)/((1-x)*(1-8*x+x^2)) + O(x^100)) \\ Colin Barker, Jan 01 2015

a(n) = 1/2 + (1/2)*Sum_{k = 0..n} 6^k*binomial(n+k,2*k).
a(n) = R(n,3) where R(n,x) denotes the row polynomials of A211955.
a(n) = (1/u)*T(n,u)*T(n+1,u) with u = sqrt(5/2) and T(n,x) the n-th Chebyshev polynomial of the first kind.
Recurrence equation: a(n) = 8*a(n-1) - a(n-2) - 3 with a(0) = 1 and a(1) = 4.
O.g.f.: (1 - 5*x + x^2)/((1 - x)*(1 - 8*x + x^2)) = 1 + 4*x + 28*x^2 + ....
Sum_{n >= 0} 1/a(n) = sqrt(5/3); 5 - 3*(Sum_{k = 0..2*n} 1/a(k))^2 = 2/A070997(n)^2.
a(0) = 1, a(1) = 4, a(2) = 28, a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3). - Harvey P. Dale, May 14 2012

A098557 Expansion of e.g.f. (1/2)(1+x)log((1+x)/(1-x)).

Original entry on oeis.org

0, 1, 2, 2, 8, 24, 144, 720, 5760, 40320, 403200, 3628800, 43545600, 479001600, 6706022400, 87178291200, 1394852659200, 20922789888000, 376610217984000, 6402373705728000, 128047474114560000, 2432902008176640000, 53523844179886080000, 1124000727777607680000

Offset: 0

Paul Barry, Sep 14 2004

G. C. Greubel, Table of n, a(n) for n = 0..450

From Johannes W. Meijer, Nov 12 2009: (Start)
Cf. A109613 (odd numbers repeated).
Equals the first left hand column of A167552.
Equals the first right hand column of A167556.
A098557(n)*A064455(n) equals the second right hand column of A167556(n).
(End)
Cf. A052558, A073742.

[0,1] cat [Factorial(n-1) + Factorial(n-2)*(1+(-1)^n)/2: n in [2..30]]; // G. C. Greubel, Jan 17 2018

Join[{0,1}, Table[(n-1)! + (n-2)!*(1+(-1)^n)/2, {n,2,30}]] (* or *) With[{nmax = 50}, CoefficientList[Series[(1/2)*(1 + x)*Log[(1 + x)/(1 - x)], {x,0,nmax}], x]*Range[0,nmax]!] (* G. C. Greubel, Jan 17 2018 *)

for(n=0, 30, print1(if(n==0,0, if(n==1, 1, (n-1)! + (n-2)!*(1 + (-1)^n)/2)), ", ")) \\ G. C. Greubel, Jan 17 2018

a(n+1) = n! + (n-1)! * (1-(-1)^n)/2.
a(n+2) = 2*A052558(n).
conjecture: -a(n) +a(n-1) +(n-1)*(n-3)*a(n-2)=0. - R. J. Mathar, Nov 14 2011
G.f.: 1-G(0), where G(k)= 1 + x*(2*k-1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 11 2013
Sum_{n>=1} 1/a(n) = sinh(1) + 1 = A073742 + 1. - Amiram Eldar, Jan 22 2023

A212120 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the sum of the divisors d of n with min(d, n/d) = k.

Original entry on oeis.org

1, 3, 5, 7, 1, 9, 1, 11, 3, 13, 3, 15, 5, 17, 5, 1, 19, 7, 1, 21, 7, 1, 23, 9, 3, 25, 9, 3, 27, 11, 3, 29, 11, 5, 31, 13, 5, 1, 33, 13, 5, 1, 35, 15, 7, 1, 37, 15, 7, 1, 39, 17, 7, 3, 41, 17, 9, 3, 43, 19, 9, 3, 45, 19, 9, 3, 47, 21, 11, 5, 49, 21, 11, 5, 1

Offset: 1

Omar E. Pol, Jul 02 2012

Column k lists the odd numbers repeated k times starting in row k^2.
1 together with the first differences of the row sums give the divisor function A000005.
T(n,k) is also the total number of divisors of all positive integers <= n on the edges of k-th triangle in the diagram of divisors (see link section). See also A212119.

			Written as an irregular triangle the sequence begins:
1;
3;
5;
7,   1;
9,   1;
11,  3;
13,  3;
15,  5;
17,  5,  1;
19,  7,  1;
21,  7,  1;
23,  9,  3;
25,  9,  3;
27, 11,  3;
29, 11,  5;
31, 13,  5,  1;
33, 13,  5,  1;
35, 15,  7,  1;
37, 15,  7,  1;
39, 17,  7,  3;
41, 17,  9,  3;
43, 19,  9,  3;
45, 19,  9,  3;
47, 21, 11,  5;
49, 21, 11,  5,  1;

Omar E. Pol, Diagram of divisors, figure 1, figure 2.

Row sums give A006218, n >= 1.
Columns (1-5): A005408, A109613, A130823, A129756, A130497.
Cf. A000005, A027750, A010766, A147861, A163100, A212119.

T(n,k) = Sum_{j=1..n} A212119(j,k).

Definition changed by Franklin T. Adams-Watters, Jul 12 2012

A063196 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 7 ).

Original entry on oeis.org

0, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 39, 41, 41, 43, 43, 45, 45, 47, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 63, 65, 65, 67, 67, 69, 69, 71, 71, 73, 73, 75, 75, 77, 77, 79, 79, 81, 81, 83

Offset: 1

N. J. A. Sloane, Jul 10 2001

Also, for n>1, number of involutions (i.e. elements of order 2) in the dihedral group D_(n-1). - Lekraj Beedassy, Oct 22 2004

Also, the chromatic number of the n-th triangular graph; i.e., the chromatic index (edge chromatic number) of the n-th complete graph. - Danny Rorabaugh, Nov 26 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
J. Sondow and E. W. Weisstein, MathWorld: Wallis Formula.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Eric Weisstein's World of Mathematics, Chromatic Number, Edge Chromatic Number, and Triangular Graph.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A109613.

CoefficientList[Series[-x (x^2 - 2 x - 1) / ((x - 1)^2 (x + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 27 2018 *)
LinearRecurrence[{1,1,-1},{0,1,3,3},90] (* Harvey P. Dale, Sep 11 2024 *)

concat([0], Vec(-x^2*(x^2-2*x-1)/((x-1)^2*(x+1)) + O(x^100))) \\ Colin Barker, Sep 08 2013

For n > 1, a(n-1) = (2n + 1 + (-1)^n)/2 (odd numbers appearing twice). - Lekraj Beedassy, Oct 22 2004
For n > 1, a(n) = 2*n - a(n-1), (with a(1)=1). - Vincenzo Librandi, Dec 06 2010
From Colin Barker, Sep 08 2013: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n > 4.
G.f.: -x^2*(x^2-2*x-1) / ((x-1)^2*(x+1)). (End)

A179207 a(n) = n - 1 + ceiling((-3 + n^2)/2) if n > 1 with a(1)=1, complement of A182835.

Original entry on oeis.org

1, 2, 5, 10, 15, 22, 29, 38, 47, 58, 69, 82, 95, 110, 125, 142, 159, 178, 197, 218, 239, 262, 285, 310, 335, 362, 389, 418, 447, 478, 509, 542, 575, 610, 645, 682, 719, 758, 797, 838, 879, 922, 965, 1010, 1055, 1102, 1149, 1198, 1247, 1298, 1349, 1402, 1455

Offset: 1

Clark Kimberling, Jan 07 2011

Guenther Schrack, Table of n, a(n) for n = 1..10010
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A182835, A047838.

First differences: A109613(n) for n > 2. - Guenther Schrack, Jun 06 2018

a:=[2,5,10,15];; for n in [5..60] do a[n]:=2*a[n-1]-2*a[n-3]+a[n-4]; od; a:=Concatenation([1],a); # Muniru A Asiru, Aug 05 2018

a:=n->n-1+ceil((-3+n^2)/2): 1,seq(a(n),n=2..60); # Muniru A Asiru, Aug 05 2018

Table[n-1+Ceiling[(n*n-3)/2], {n,60}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011 *)
Join[{1},LinearRecurrence[{2,0,-2,1},{2,5,10,15},52]] (* Ray Chandler, Jul 15 2015 *)

a(n) = n - 1 + ceiling((-3 + n^2)/2) if n > 1.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Joerg Arndt, Apr 02 2011
From Guenther Schrack, Jun 06 2018: (Start)
a(n) = (2*n^2 + 4*n - 9 + (-1)^n)/4 for n > 1.
a(n) = a(n-2) + 2*n for n > 3.
a(-n) = a(n-2) for n > 1.
a(n) = n - 1 + A047838(n) for n > 1. (End)
G.f.: x * (1 + x^2 + 2*x^3 - 2*x^4) / (1 - 2*x + 2*x^3 - x^4). - Michael Somos, Oct 28 2018
Sum_{n>=1} 1/a(n) = 8/3 + tan(sqrt(5)*Pi/2)*Pi/(2*sqrt(5)) - cot(sqrt(3/2)*Pi)*Pi/(2*sqrt(6)). - Amiram Eldar, Sep 16 2022

cp's OEIS Frontend

A158821 Triangle read by rows: row n (n>=0) ends with 1, and for n>=1 begins with n; other entries are zero.

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A182579 Triangle read by rows: T(0,0) = 1, for n>0: T(n,n) = 2 and for k<=floor(n/2): T(n,2*k) = n/(n-k) * binomial(n-k,k), T(n,2*k+1) = (n-1)/(n-1-k) * binomial(n-1-k,k).

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A186421 Even numbers interleaved with repeated odd numbers.

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Maxima

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A186422 First differences of A186421.

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

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A182432 Recurrence a(n)*a(n-2) = a(n-1)*(a(n-1) + 3) with a(0) = 1, a(1) = 4.

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A182579 Triangle read by rows: T(0,0) = 1, for n>0: T(n,n) = 2 and for k<=floor(n/2): T(n,2k) = n/(n-k) binomial(n-k,k), T(n,2k+1) = (n-1)/(n-1-k) binomial(n-1-k,k).

A182432 Recurrence a(n)a(n-2) = a(n-1)(a(n-1) + 3) with a(0) = 1, a(1) = 4.

A098557 Expansion of e.g.f. (1/2)(1+x)log((1+x)/(1-x)).