A079935 -id:A079935 - cp's OEIS frontend

A123971 Triangle T(n,k), read by rows, defined by T(n,k)=3*T(n-1,k)-T(n-1,k-1)-T(n-2,k), T(0,0)=1, T(1,0)=2, T(1,1)=-1, T(n,k)=0 if k<0 or if k>n.

1, 2, -1, 5, -5, 1, 13, -19, 8, -1, 34, -65, 42, -11, 1, 89, -210, 183, -74, 14, -1, 233, -654, 717, -394, 115, -17, 1, 610, -1985, 2622, -1825, 725, -165, 20, -1, 1597, -5911, 9134, -7703, 3885, -1203, 224, -23, 1, 4181, -17345, 30691, -30418, 18633, -7329

Offset: 0

Gary W. Adamson and Roger L. Bagula, Oct 30 2006

This entry is the result of merging two sequences, this one and a later submission by Philippe Deléham, Nov 29 2013 (with edits from Ralf Stephan, Dec 12 2013). Most of the present version is the work of Philippe Deléham, the only things remaining from the original entry are the sequence data and the Mathematica program. - N. J. A. Sloane, May 31 2014
Subtriangle of the triangle given by (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Apart from signs, equals A126124.
Row sums = 1.
Sum_{k=0..n} T(n,k)*(-x)^k = A001519(n+1), A079935(n+1), A004253(n+1), A001653(n+1), A049685(n), A070997(n), A070998(n), A072256(n+1), A078922(n+1), A077417(n), A085260(n+1), A001570(n+1) for x=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.

			Triangle begins:
  1
  2, -1
  5, -5, 1
  13, -19, 8, -1
  34, -65, 42, -11, 1
  89, -210, 183, -74, 14, -1
  233, -654, 717, -394, 115, -17, 1
Triangle (0, 2, 1/2, 1/2, 0, 0, ...) DELTA (1, -2, 0, 0, ...) begins:
  1
  0, 1
  0, 2, -1
  0, 5, -5, 1
  0, 13, -19, 8, -1
  0, 34, -65, 42, -11, 1
  0, 89, -210, 183, -74, 14, -1
  0, 233, -654, 717, -394, 115, -17, 1

Cf. A094954, A098495, A123971, A126124, A152063, A001519, A079935, A004253, A001653, A049685, A070997, A070998, A072256, A078922, A077417, A085260, A001570, A001870, A126124.

Mathematica ( general k th center) Clear[M, T, d, a, x, k] k = 3 T[n_, m_, d_] := If[ n == m && n < d && m < d, k, If[n == m - 1 || n == m + 1, -1, If[n == m == d, k - 1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[ Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a] Table[NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x], {d, 1, 10}] Table[x /. NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x][[d]], {d, 1, 10}]

T(n,k)=polcoeff(polcoeff(Ser((1-x)/(1+(y-3)*x+x^2)),n,x),n-k,y) \\ Ralf Stephan, Dec 12 2013

@CachedFunction
def A123971(n,k): # With T(0,0) = 1!
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A123971(n-1,k) if n==1 else 3*A123971(n-1,k)
    return A123971(n-1,k-1) - A123971(n-2,k) - h
for n in (0..9): [A123971(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

T(n,k) = (-1)^n*A126124(n+1,k+1).
T(n,k) = (-1)^k*Sum_{m=k..n} binomial(m,k)*binomial(m+n,2*m). - Wadim Zudilin, Jan 11 2012
G.f.: (1-x)/(1+(y-3)*x+x^2).
T(n,0) = A001519(n+1) = A000045(2*n+1).
T(n+1,1) = -A001870(n).

Edited by N. J. A. Sloane, May 31 2014

A052677 Expansion of e.g.f. (1-x)/(1-4*x+x^2).

Original entry on oeis.org

1, 3, 22, 246, 3672, 68520, 1534320, 40083120, 1196737920, 40196580480, 1500156806400, 61585275628800, 2758072531737600, 133812468652262400, 6991529043750451200, 391391124208051968000, 23371064978815217664000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..365
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 625

Cf. A000142, A079935.

[n le 2 select 3^(n-1) else 4*(n-1)*Self(n-1) - (n-1)*(n-2)*Self(n-2): n in [1..31]]; // G. C. Greubel, Jun 11 2022

spec := [S,{S=Sequence(Union(Z,Prod(Sequence(Z),Union(Z,Z))))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

With[{nn=20},CoefficientList[Series[(1-x)/(1-4x+x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 28 2016 *)

[factorial(n)*sum( binomial(2*n-k, k)*2^(n-k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 11 2022

E.g.f.: (1-x)/(1-4*x+x^2).
Recurrence: a(0)=1, a(1)=3, a(n+2) = 4*(n+2)*a(n+1) - (n+2)*(n+1)*a(n).
a(n) = (n!/6)*Sum_{alpha=RootOf(1 -4*Z +Z^2)} (1 + alpha)*alpha^(-1-n).
a(n) = n!*A079935(n). - R. J. Mathar, Nov 27 2011
a(n) = (-1)^n * n! * ChebyshevU(2*n, i/sqrt(2)). - G. C. Greubel, Jun 11 2022

A167479 Convolution of the Catalan numbers A000108(n) and (-2)^n.

Original entry on oeis.org

1, -1, 4, -3, 20, 2, 128, 173, 1084, 2694, 11408, 35970, 136072, 470756, 1732928, 6228989, 22899692, 83845406, 309947888, 1147367414, 4269385592, 15927495836, 59627571968, 223804469714, 842295207896, 3177355985660, 12012641100832

Offset: 0

Paul Barry, Nov 04 2009

Hankel transform is A079935.

G. C. Greubel, Table of n, a(n) for n = 0..500
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

Cf. A000108, A079935.

CoefficientList[Series[(1 - Sqrt[1 - 4*t])/(2*t*(1 + 2*t)), {t, 0, 50}], t] (* G. C. Greubel, Jun 13 2016 *)

G.f.: c(x)/(1+2x), c(x) the g.f. of A000108.
a(n) = Sum_{k=0..n} (-2)^(n-k)*A000108(k).
(n+1)*a(n) + 2*(2-n)*a(n-1) + 4*(1-2*n)*a(n-2)=0. - R. J. Mathar, Nov 16 2011 [Proof in Ekhad/Yang, Theorem 26]
a(n) ~ 2^(2*n + 1) / (3 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2018

A375098 Diagonals of a Euclidian solid such that there exists a Pythagorean quadruple d^2=a^2+b^2+c^2 that is more cube-like than any prior value of d.

Original entry on oeis.org

3, 9, 11, 41, 123, 153, 571, 1713, 2131, 7953, 23859, 29681, 110771, 332313, 413403, 1542841, 4628523, 5757961, 21489003, 64467009, 80198051, 299303201, 897909603, 1117014753, 4168755811, 12506267433, 15558008491, 58063278153, 174189834459, 216695104121

Offset: 1

Christian N. K. Anderson, Mark K. Transtrum, and David D. Allred, Jul 29 2024

nonn

To determine how "cube-like" a Pythagorean quad is, we use the quotient C/(C-a*b*c) where C is the volume of an ideal cube for a given diagonal d, C=(d/sqrt(3))^3. A ratio of 100 indicates that the best {a,b,c} combination creates a solid 1 part per 100 smaller than the ideal cube volume.
Contains all the terms in A001835 and A079935 except the leading 1s. For such a term b(m) contained in a(n) from those sequences, 2*b(m) will tie the current record, while 3*b(m) will tie but will also break the current record exactly half the time and thus appear as a(n+1). This means round((2+sqrt(3))*b(m)) will also be in the sequence as either a(n+1) or a(n+2) if a better quad for 3*b(n) was found.
The constant (2+sqrt(3)) follows from the recurrence relationship b(n)=4*b(n-1)-b(n-2). The constant can be represented as the continued fraction 3,1,2,1,2,1,2,1,2,.... The equivalents for the 2D case (A001653) are 3+2*sqrt(2) and {5,1,4,1,4,1,4,1,4,...}.
We conjecture this method provides complete solutions. This was confirmed directly by brute-force testing up to 1e7 (optimized using the sum-of-two-squares theorem for a given d^2-a^2 = b^2 + c^2; see link below).

			3 is in the sequence because 3^2=1^2+2^2+2^2 is the smallest Pythagorean quad, with an error of one part in 4.344.
6 is NOT in the sequence because {6,2,4,4} is the most cube-like Pythagorean quad, but only ties the previous record without breaking it.
7 is NOT in the sequence because the most cube-like quad {7,2,3,6} has an error of one part in 2.2, worse than that for d=3.
9 is in the sequence NOT because of {9,3,6,6} which ties the previous record, but because {9,4,4,7} improves on the previous record with an error of one part in 4.958.

Christian N. K. Anderson, Table of n, a(n) for n = 1..1000
Christian N. K. Anderson, Table of most cube-like Pythagorean quadruples and the corresponding and the error quotient for d=1..10000
Christian N. K. Anderson, Table of d, a, b, c and error quotient for n = 1..50.
Christian N. K. Anderson, Mathematica code for generating this sequence with different amounts of rigorous error checking.

Cf. A096907, A096908, A096909, A096910 for lists of a, b, c, and d of the 10,000 first Pythgorean quads, sorted by ascending d.
Contains all terms in A001835 and A079935 except the leading 1s.
Cf. A001653: The 2D equivalent of this sequence (i.e., right triangle whose legs are closest to equal)

(* An efficient program is provided in the links section. *)

For n == 0 (mod 3), a(n) = 4*a(n-2)-a(n-3) OR a(n) = floor(a(n-1)*(2+sqrt(3))/3),
For n == 1 (mod 3), a(n) = 4*a(n-2)-a(n-1) OR a(n) = floor(a(n-1)*(2+sqrt(3))),
For n == 2 (mod 3), a(n) = 3*a(n-1).

cp's OEIS Frontend

A123971 Triangle T(n,k), read by rows, defined by T(n,k)=3*T(n-1,k)-T(n-1,k-1)-T(n-2,k), T(0,0)=1, T(1,0)=2, T(1,1)=-1, T(n,k)=0 if k<0 or if k>n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

Extensions

A052677 Expansion of e.g.f. (1-x)/(1-4*x+x^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

A167479 Convolution of the Catalan numbers A000108(n) and (-2)^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A375098 Diagonals of a Euclidian solid such that there exists a Pythagorean quadruple d^2=a^2+b^2+c^2 that is more cube-like than any prior value of d.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula