A119910 -id:A119910 - cp's OEIS frontend

A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2(-1)^((n-k)/2)(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

1, 0, 1, 2, 0, 1, 0, 1, 0, 1, -2, 0, 0, 0, 1, 0, -3, 0, -1, 0, 1, 2, 0, -3, 0, -2, 0, 1, 0, 5, 0, -2, 0, -3, 0, 1, -2, 0, 8, 0, 0, 0, -4, 0, 1, 0, -7, 0, 10, 0, 3, 0, -5, 0, 1, 2, 0, -15, 0, 10, 0, 7, 0, -6, 0, 1, 0, 9, 0, -25, 0, 7, 0, 12, 0, -7, 0, 1, -2, 0, 24, 0, -35, 0, 0, 0, 18, 0, -8, 0, 1, 0, -11, 0, 49, 0, -42, 0, -12, 0

Offset: 0

Roger L. Bagula and Gary W. Adamson, Mar 13 2008

Polynomial coefficients of P(n,x) in increasing powers, read by rows, where P(0,x)=1, P(1,x)=x, P(2,x)=2+x^2, P(3,x)=x+x^3, P(4,x)=-2+x^4, and  P(n,x) = x*P(n-1,x) - P(n-2,x) for n>=5.
The row-reversed version of A135929.
Row sums are repeating 1, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1..., see A138034 and A119910.

			{1}, = 1
{0, 1}, = x
{2, 0, 1}, = 2+x^2
{0, 1, 0, 1}, = x+x^3
{-2, 0, 0, 0, 1}, = -2+x^4
{0, -3, 0, -1, 0, 1}, = -3x-x^3+x^5
{2, 0, -3, 0, -2, 0, 1},
{0, 5, 0, -2, 0, -3, 0, 1},
{-2, 0, 8, 0, 0, 0, -4, 0, 1},
{0, -7, 0, 10, 0, 3, 0, -5, 0, 1},
{2, 0, -15, 0, 10, 0, 7, 0, -6, 0, 1},
{0, 9, 0, -25, 0, 7, 0, 12, 0, -7, 0, 1}

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31, MR 1439165

Cf. A123956, A137289.

A137276 := proc(n,k) local nmk,npk; if n = 0 then 1; elif (n-k) mod 2 <> 0 then 0; else nmk := (n-k)/2 ; npk := (n+k)/2 ; (-1)^nmk*(2*k-n)/npk*binomial(npk,nmk) ; fi; end:
seq( seq(A137276(n,k),k=0..n),n=0..13) ;

T(n,k)= 0 if n-k odd. T(n,k)= 2*(-1)^((n-k)/2)*(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.
P(n,x) = x*P(n-1,x)-P(n-2,x), n>=5.
P(n,2*x) = -2*T(n,x)+4*x*U(n-1,x), where T(n,x) is A053120 and U(n,x) is A053117.

Fourth row inserted by the Associate Editors of the OEIS, Aug 27 2009

A138034 Expansion of (1+3*x^2)/(1-x+x^2).

Original entry on oeis.org

1, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2, 1, 3

Offset: 0

Karem Boubaker (mmbb11112000(AT)yahoo.fr), Mar 01 2008; corrected Mar 03 2008

Essentially a duplicate of A119910: 1, followed by A119910. - Joerg Arndt, Nov 14 2014

Karem Boubaker and Lin Zhang, Fermat-linked relations for the Boubaker polynomial sequences via Riordan matrices analysis, arXiv preprint arXiv:1203.2082, 2012. - From _N. J. A. Sloane_, Sep 15 2012
Index entries for linear recurrences with constant coefficients, signature (1,-1).

Cf. A135929, A135936, A137276.

CoefficientList[Series[(1 + 3*x^2)/(1 - x + x^2), {x, 0, 100}], x] (* Wesley Ivan Hurt, Jan 15 2017 *)
LinearRecurrence[{1,-1},{1,1,3},120] (* Harvey P. Dale, Jun 14 2024 *)

a(n) = A119910(n), n>=1.

G.f.: (1+3*x^2)/(1-x+x^2). a(n)=a(n-1)-a(n-2), n>2.

A130624 Binomial transform of A101000.

Original entry on oeis.org

0, 1, 5, 12, 23, 43, 84, 169, 341, 684, 1367, 2731, 5460, 10921, 21845, 43692, 87383, 174763, 349524, 699049, 1398101, 2796204, 5592407, 11184811, 22369620, 44739241, 89478485, 178956972, 357913943, 715827883, 1431655764, 2863311529, 5726623061, 11453246124

Offset: 0

Paul Curtz, Jun 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, -3, 2).

Cf. A101000, A119910, A130625 (first differences), A130626 (second differences).

m:=32; S:=[[0, 1, 3][(n-1) mod 3 +1]: n in [1..m]]; [&+[Binomial(i-1, k-1)*S[k]: k in [1..i]]: i in [1..m]]; /* Klaus Brockhaus, Jun 21 2007 */

I:=[0,1,5]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+2*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Nov 15 2018

LinearRecurrence[{3,-3,2},{0,1,5},40] (* Harvey P. Dale, Mar 05 2013 *)
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(2^n) + a[n-1] - a[n-2]}, a, {n, 50}] (* Vincenzo Librandi, Nov 15 2018 *)

{m=32; v=concat([0, 1, 5], vector(m-3)); for(n=4, m, v[n]=3*v[n-1]-3*v[n-2]+2*v[n-3]); v} /* Klaus Brockhaus, Jun 21 2007 */

a(0)=0, a(1)=1, a(2)=5; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
a(0)=0; a(n+1) = 2*a(n) + A119910(n).
G.f.: x*(1 + 2*x)/((1 - 2*x)*(1 - x + x^2)).
a(n) = 2^n + a(n-1) - a(n-2). - Jon Maiga, Nov 14 2018

Edited and extended by Klaus Brockhaus, Jun 21 2007

A124038 Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

Original entry on oeis.org

1, -2, 1, -1, -2, 1, 2, -2, -2, 1, 1, 4, -3, -2, 1, -2, 3, 6, -4, -2, 1, -1, -6, 6, 8, -5, -2, 1, 2, -4, -12, 10, 10, -6, -2, 1, 1, 8, -10, -20, 15, 12, -7, -2, 1, -2, 5, 20, -20, -30, 21, 14, -8, -2, 1, -1, -10, 15, 40, -35, -42, 28, 16, -9, -2, 1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 03 2006

			Triangular sequence begins as:
   1;
  -2,   1;
  -1,  -2,   1;
   2,  -2,  -2,   1;
   1,   4,  -3,  -2,   1;
  -2,   3,   6,  -4,  -2,   1;
  -1,  -6,   6,   8,  -5,  -2,  1;
   2,  -4, -12,  10,  10,  -6, -2,  1;
   1,   8, -10, -20,  15,  12, -7, -2,  1;
  -2,   5,  20, -20, -30,  21, 14, -8, -2,  1;
  -1, -10,  15,  40, -35, -42, 28, 16, -9, -2, 1;

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

Cf. A005809, A045721, A066170.
Row reversal of: A374439.
Columns are related to: A000034 (k=0), A029578 (k=1), A131259 (k=2).
Diagonals are related to: A113679 (k=n-1), A022958 (k=n-2), A005843 (k=n-3), A000217 (k=n-4), -A002378 (k=n-5).
Sums include: (-1)^floor((n+1)/2)*A016116 (signed diagonal), A057079 (row), A119910 (signed row), (-1)^n*A130706 (diagonal).

function T(n,k) // T = A124038
  if k lt 0 or k gt n then return 0;
  elif k ge n-2 then return k-n + (-1)^(n+k);
  else return T(n-1,k-1) - T(n-2,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 22 2025

(* First program *)
t[n_, m_, d_]:= If[n==m && n>1 && m>1, x, If[n==m-1 || n==m+1, -1, If[n==m== 1, x-2, 0]]];
M[d_]:= Table[t[n,m,d], {n,d}, {m,d}];
Join[{{1}}, Table[CoefficientList[Table[Det[M[d]], {d,10}][[d]], x], {d,10}]]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k] = If[k<0 || k>n, 0, If[k>n-2, k-n+(-1)^(n-k), T[n-1, k- 1] -T[n-2,k]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 22 2025 *)

@CachedFunction
def A124038(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A124038(n-1,k) if n==1 else 0
    return A124038(n-1,k-1) - A124038(n-2,k) - h
for n in (0..9): [A124038(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

from sage.combinat.q_analogues import q_stirling_number2
def A124038(n,k): return (1 + ((n-k)%2))*q_stirling_number2(n+1, n-k+1, -1)
print(flatten([[A124038(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 22 2025

From G. C. Greubel, Jan 22 2025: (Start)
T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.
T(n, k) = (-1)^floor((n-k+1)/2)*(1 + (n-k mod 2))*qStirling2(n+1, n-k+1,-1).
T(2*n, n) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)* A045721((n-1)/2) ). (End)

Edited by G. C. Greubel, Jan 22 2025

A134977 Period 6: repeat [1, 4, 2, 3, 0, 2].

Original entry on oeis.org

1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2

Offset: 0

Paul Curtz, Feb 04 2008

Northwest diagonal sums of A134658, omitting row 0.

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

Cf. A010892, A119910, A130784, A131756, A134658.

&cat[[1, 4, 2, 3, 0, 2]^^20]; // Wesley Ivan Hurt, Jun 18 2016

A134977:=n->[1, 4, 2, 3, 0, 2][(n mod 6)+1]: seq(A134977(n), n=0..100); # Wesley Ivan Hurt, Jun 18 2016

Flatten[Table[{1, 4, 2, 3, 0, 2}, {20}]] (* Wesley Ivan Hurt, Jun 18 2016 *)
PadRight[{}, 100, {1, 4, 2, 3, 0, 2}] (* Vincenzo Librandi, Jun 19 2016 *)

O.g.f.: -1/(x+1)-2/(x-1)+x/(x^2-x+1). a(n) = 2-(-1)^n+A010892(n-1). - R. J. Mathar, Feb 08 2008
From Wesley Ivan Hurt, Jun 18 2016: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
a(n) = (6-3*cos(n*Pi)+2*sqrt(3)*sin(n*Pi/3))/3. (End)

A135694 Period 6: repeat [1, -1, -1, -1, 0, 2].

Original entry on oeis.org

1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2, 1, -1, -1, -1, 0, 2

Offset: 0

Paul Curtz, Feb 24 2008

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

Cf. A000079, A049347, A119910, A135575.

&cat [[1, -1, -1, -1, 0, 2]^^20]; // Wesley Ivan Hurt, Jun 22 2016

A135694 := proc(n) op((n mod 6)+1,[1,-1,-1,-1,0,2]) ; end: seq(A135694(n), n=0..150) ; # R. J. Mathar, Feb 07 2009

PadRight[{}, 100, {1, -1, -1, -1, 0, 2}] (* Wesley Ivan Hurt, Jun 22 2016 *)

From R. J. Mathar, Mar 31 2008: (Start)
a(n) = a(n-6) for n>5. a(n) = -a(n-2) - a(n-4) for n>3.
a(n) = (A119910(n+3) - A049347(n+1))/2 for n>0.
G.f.: (1-x-2*x^3)/((x^2-x+1)*(x^2+x+1)). (End)
a(n) = (3*cos(n*Pi/3) - 8*sqrt(3)*cos(n*Pi/6)^3*sin(n*Pi/6))/3. - Wesley Ivan Hurt, Jun 22 2016

More periods from R. J. Mathar, Feb 07 2009

A155751 A variation on 10^n mod 17.

Original entry on oeis.org

1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5, 1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5, 1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5, 1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5, 1, -7, -2, -3, 4, 6, -8, 5, -1, 7, 2, 3, -4, -6, 8, -5

Offset: 0

Ferruccio Guidi (fguidi(AT)cs.unibo.it), Jan 26 2009, Feb 08 2009

This is 10^n mod 17, using values -8,-7,...,7,8 (instead of 0..16). - Don Reble, Sep 02 2017.
This sequence can be employed in a test for divisibility by 17 and works like A033940 works for 7.
The use of negative coefficients ensures the termination of the test because the modulus of the intermediate sum at each step of the test decreases strictly.
The test is successful if the final sum is 0.
The negative coefficients have the form (10^n mod 17) - 17 when 10^n mod 17 > 8.
Example: 9996 is divisible by 17 since |6*1 + 9*(-7) + 9*(-2) + 9*(-3)| = 102 and 2*1 + 0*(-7) + 1*(-2) = 0.

Cf. A033940, A119910, A117378.

a(n)= -a(n-8). G.f.:(1-7x-2x^2-3x^3+4x^4+6x^5-8x^6+5x^7)/(1+x^8). [From R. J. Mathar, Feb 13 2009]

A188125 Number of strictly increasing arrangements of 6 nonzero numbers in -(n+4)..(n+4) with sum zero.

Original entry on oeis.org

4, 16, 52, 137, 308, 624, 1154, 1999, 3278, 5144, 7772, 11387, 16230, 22602, 30830, 41303, 54440, 70734, 90706, 114963, 144146, 178984, 220244, 268797, 325548, 391514, 467756, 555449, 655816, 770208, 900020, 1046787, 1212094, 1397668

Offset: 0

R. H. Hardin, Mar 21 2011

nonn

Row 6 of A188122.

			4 + 16*x + 52*x^2 + 137*x^3 + 308*x^4 + 624*x^5 + 1154*x^6 + 1999*x^7 + 3278*x^8 + ...
Some solutions for n=6
-10...-8...-7...-8...-8...-9...-9...-9...-9...-7..-10...-9...-7..-10...-9...-9
.-8...-6...-5...-5...-6...-3...-7...-3...-2...-5...-6...-5...-5...-6...-4...-5
.-1....1...-1...-1...-1...-2...-2....1...-1...-2...-2...-1...-1...-2...-2...-4
..4....3....1....1....2....3....3....2....1....1....2....1....3....3....1....3
..7....4....2....3....5....4....5....4....2....4....6....6....4....5....5....6
..8....6...10...10....8....7...10....5....9....9...10....8....6...10....9....9

R. H. Hardin, Table of n, a(n) for n = 0..200 (a(0) inserted by _Georg Fischer_, Jan 24 2019)

Cf. A188122

{a(n) = local(v, c, m); m = n+4; forvec( v = vector( 6, i, [-m, m]), if( 0==prod( k=1, 6, v[k]), next); if( 0==sum( k=1, 6, v[k]), c++), 2); c} /* Michael Somos, Apr 11 2011 */

Empirical: a(n)=2*a(n-1)-a(n-3)-a(n-5)+2*a(n-8)-a(n-11)-a(n-13)+2*a(n-15)-a(n-16)
= 168587/43200 +187*n/32 +3593*n^3/2160 +619*n^2/144 +457*n^4/1440 +11*n^5/450 -(-1)^n/64-3*n*(-1)^n/32 +4*(-1)^n*A119910(n+1)/27 -2*A117444(n+2)/25 +A057077(n)/8.
Empirical: G.f. -x*(-16 -20*x -33*x^2 -50*x^3 -60*x^4 -59*x^5 -51*x^6 -41*x^7 -18*x^8 -3*x^9 -x^10 +x^11 +4*x^12 -2*x^13 -7*x^14 +4*x^15) / ( (1+x+x^2) *(x^4+x^3+x^2+x+1) *(x^2+1) *(1+x)^2 *(x-1)^6 ). - R. J. Mathar, Mar 21 2011

A155754 A variation on 10^n mod 19.

Original entry on oeis.org

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

Offset: 0

Ferruccio Guidi (fguidi(AT)cs.unibo.it), Jan 26 2009

This sequence can be employed in a test for divisibility by 19 and works like A033940 works for 7.
The use of negative coefficients ensures the termination of the test because the modulus of the intermediate sum at each step of the test decreases strictly.
The test is successful if the final sum is 0.
The negative coefficients have the form (10^n mod 19) - 19 when 10^n mod 19 > 9.
Example: 8284 is divisible by 19 since |4*1 + 8*(-9) + 2*5 + 8*(-7)| = 114 and 4*1 + 1*(-9) + 1*5 = 0.

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

Cf. A033940, A119910, A117378.

a(n) = -a(n-9). G.f.: (-2*x^8-4*x^7-8*x^6+3*x^5+6*x^4-7*x^3+5*x^2-9*x+1) / (x^9+1). [Colin Barker, Feb 14 2013]

cp's OEIS Frontend

A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2*(-1)^((n-k)/2)*(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.

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A138034 Expansion of (1+3*x^2)/(1-x+x^2).

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A130624 Binomial transform of A101000.

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A124038 Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

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A134977 Period 6: repeat [1, 4, 2, 3, 0, 2].

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Magma

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A135694 Period 6: repeat [1, -1, -1, -1, 0, 2].

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Magma

Maple

Mathematica

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A155751 A variation on 10^n mod 17.

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A137276 Triangle T(n,k), read by rows: T(n,k)= 0 if n-k odd. T(n,k)= 2(-1)^((n-k)/2)(2k-n)/(n+k)*binomial((n+k)/2,(n-k)/2) if n-k even.