cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Dec 10 2011

Keywords

Comments

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
{} {1} {1,3} . . .
{2} {1,4}
{3} {2,4}
{4} {1,2,4}
{1,2} {1,3,4}
{2,3}
{3,4}
{1,2,3}
{2,3,4}
{1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

Examples

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Crossrefs

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

Formula

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if n
T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

A025170 Expansion of g.f.: 1/(1 + 2*x + 9*x^2).

Original entry on oeis.org

1, -2, -5, 28, -11, -230, 559, 952, -6935, 5302, 51811, -151340, -163619, 1689298, -1906025, -11391632, 39937489, 22649710, -404736821, 605626252, 2431378885, -10313394038, -1255621889, 95331790120, -179362983239, -499260144602, 2612787138355, -732232975292
Offset: 0

Views

Author

Wouter Meeussen

Keywords

Comments

Reciprocal Chebyshev polynomial of second kind evaluated at 3 multiplied by (-1)^n.
From Sharon Sela (sharonsela(AT)hotmail.com), Jan 19 2002: (Start)
a(n) is (-1)^n times the determinant of the following tridiagonal n X n matrix:
[2 3 0 . . . . . . .]
[3 2 3 0 . . . . . .]
[0 3 2 3 0 . . . . .]
[. 0 3 2 3 0 . . . .]
[. . . . . . . . . .]
[. . . . . . . . . .]
[. . . . 0 3 2 3 0 .]
[. . . . . 0 3 2 3 0]
[. . . . . . 0 3 2 3]
[. . . . . . . 0 3 2]
(End)

Links

Crossrefs

Variant is A127357.
Cf. A087455, A088137.

Programs

  • Magma
    [(-3)^n*Evaluate(ChebyshevU(n+1),1/3): n in [0..50]]; // G. C. Greubel, Jan 02 2024
  • Mathematica
    Table[3^n ChebyshevU[n, -1/3], {n, 0, 24}]
  • PARI
    a(n)=if(n<0,0,polcoeff(1/(1+2*x+9*x^2)+x*O(x^n),n))
  • PARI
    a(n)=if(n<0, 0, 3^n*subst(poltchebi(n+1)+3*poltchebi(n),'x,-1/3)*3/8) /* Michael Somos, Sep 15 2005 */
  • PARI
    a(n)=if(n<0, 0, (-1)^n*matdet(matrix(n,n,i,j, if(abs(i-j)<2, 2+abs(i-j))))) /* Michael Somos, Sep 15 2005 */
  • SageMath
    [3^n*chebyshev_U(n,-1/3) for n in range(41)] # G. C. Greubel, Jan 02 2024

Formula

a(n) = 3^n * ChebyshevU(n, -1/3).
a(n) = ( A088137(n+1) )^2 + ( A087455(n+1)/2 )^2 - ( A087455(n+2)/2 )^2. - Creighton Dement, Aug 20 2004
a(n) = -(2*a(n-1) + 9*a(n-2)) for n>1, with a(0)=1, a(1)=-2. - Philippe Deléham, Sep 19 2009
a(n) = (-2)^n*Product_{k=1..n}(1 + 3*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
From G. C. Greubel, Jan 02 2024: (Start)
a(n) = (-1)^n * A127357(n).
E.g.f.: (1/4)*exp(-x)*(4*cos(2*sqrt(2)*x) - sqrt(2)*sin(2*sqrt(2)*x)). (End)

A266046 Real part of Q^n, where Q is the quaternion 2 + j + k.

Original entry on oeis.org

1, 2, 2, -4, -28, -88, -184, -208, 272, 2336, 7712, 16832, 21056, -16768, -193408, -673024, -1531648, -2088448, 836096, 15875072, 58483712, 138684416, 203835392, -16764928, -1290072064, -5059698688, -12498362368, -19635257344, -3550855168, 103608123392
Offset: 0

Views

Author

Stanislav Sykora, Dec 20 2015

Keywords

Comments

In general, given a quaternion Q = r+u*i+v*j+w*k with integer coefficients [r,u,v,w], its powers Q^n = R(n)+U(n)*i+V(n)*j+W(n)*k define four integer sequences R(n),U(n),V(n),W(n). The process can be also transcribed as a four-term, first order recurrence for the elements of the four sequences. Since |Q^n| = |Q|^n, we have, for any n, R(n)^2+U(n)^2+V(n)^2+W(n)^2 = (L^2)^n, where L^2 = r^2+u^2+v^2+w^2 is a constant. The normalized sequence Q^n/L^n describes a unitary quaternion undergoing stepwise rotations by the angle phi = arctan(sqrt(u^2+v^2+w^2)/r). Consequently, the four sequences exhibit sign changes with the mean period of P = 2*Pi/phi steps.
When Q has a symmetry with respect to permutations and/or inversions of the imaginary axes, the four sequences become even more interdependent.
In this particular case Q = 2+j+k, and Q^n = a(n)+b(n)*(j+k), where b(n) is the sequence A190965. The first-order recurrence reduces to two-terms, namely a(n+1)=2*a(n)-2*b(n), b(n+1)=2*b(n)+a(n). This implies further a single-term, second order recurrence a(n+2)=4*a(n+1)-6*a(n), shared by both a(n) and b(n), but with different starting terms. The mean period of sign changes is P = 10.208598624... steps.
The following OEIS sequences can be also cast as quaternion powers:
Q = 1+i+j+k: Q^n = A128018(n)+A088138(n)*(i+j+k), P = 6.000,
Q = 1+j+k : Q^n = A087455(n)+A088137(n)*(j+k), P = 6.577071086...,
Q = 2+i+j+k: Q^n = A213421(n)+A168175(n)*(i+j+k), P = 8.803377735...

Links

Crossrefs

Cf. A087455 (Inv. Bin. Transf.), A088137, A088138, A128018, A168175, A190965, A213421.

Programs

  • Magma
    [n le 2 select n else  4*Self(n-1)-6*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Dec 22 2015
  • Mathematica
    LinearRecurrence[{4, -6}, {1, 2}, 30] (* Bruno Berselli, Dec 22 2015 *)
  • PARI
    \\ A simple function to generate quaternion powers:
QuaternionToN(r, u, v, w, nmax) = {local (M); M = matrix(nmax+1, 4); M[1, 1]=1; for(n=2, nmax+1, M[n, 1]=r*M[n-1, 1]-u*M[n-1, 2]-v*M[n-1, 3]-w*M[n-1, 4]; M[n, 2]=u*M[n-1, 1]+r*M[n-1, 2]+w*M[n-1, 3]-v*M[n-1, 4]; M[n, 3]=v*M[n-1, 1]-w*M[n-1, 2]+r*M[n-1, 3]+u*M[n-1, 4]; M[n, 4]=w*M[n-1, 1]+v*M[n-1, 2]-u*M[n-1, 3]+r*M[n-1, 4]; ); return (M); }
a=QuaternionToN(2, 0, 1, 1, 1000)[,1]; \\ Select the real parts
  • PARI
    Vec((1-2*x)/(1-4*x+6*x^2) + O(x^40)) \\ Colin Barker, Dec 21 2015

Formula

a(n)^2 + 2*A190965(n)^2 = 6^n.
From Colin Barker, Dec 21 2015: (Start)
a(n) = ((2-i*sqrt(2))^n+(2+i*sqrt(2))^n)/2, where i=sqrt(-1).
a(n) = 4*a(n-1) - 6*a(n-2) for n>1.
G.f.: (1-2*x) / (1-4*x+6*x^2). (End)

A106632 Expansion of g.f. -(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2)).

Original entry on oeis.org

-1, 1, -25, 49, -1, 529, -1849, 289, -9025, 58081, -38809, 108241, -1560001, 2283121, -525625, 35796289, -95863681, 2666689, -681575449, 3261894769, -1289169025, 9906021841, -94109673529, 99199171681, -84332740801, 2327696411041, -4753075824025, 46970592529, -48635546218561
Offset: 0

Views

Author

Creighton Dement, May 11 2005

Keywords

Comments

Floretion Algebra Multiplication Program, FAMP Code: 1tesseq[A*B] with A = + .5'i - .5'k + .5i' - .5k' - 3'jj' - .5'ij' - .5'ji' - .5'jk' - .5'kj' and B = + .5'i + .5'j + .5i' + .5j' + .5'kk' + .5'ij' + .5'ji' + .5e

References

  • S. Severini, A note on two integer sequences arising from the 3-dimensional hypercube, Technical Report, Department of Computer Science, University of Bristol, Bristol, UK (October 2003).

Links

Crossrefs

Cf. A087455, A025172.

Programs

  • GAP
    a:=[-1,1,-25];; for n in [4..40] do a[n]:=-a[n-1]-3*a[n-2] - 27*a[n-3]; od; a; # G. C. Greubel, Feb 19 2019
  • Magma
    m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( -(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2)) )); // G. C. Greubel, Feb 19 2019
  • Mathematica
    CoefficientList[Series[-(1+27x^2)/((1+3x)(1-2x+9x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{-1,-3,-27},{-1,1,-25},40] (* Harvey P. Dale, Oct 03 2014 *)
  • PARI
    my(x='x+O('x^40)); Vec(-(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2))) \\ G. C. Greubel, Feb 19 2019
  • SageMath
    (-(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 19 2019

Formula

a(n) = (3^(n+1)/2)*(cos((n+1)*arccos(1/3)) + (-1)^(n+1) ).
a(n) = - a(n-1) - 3*a(n-2) - 27*a(n-3), a(0) = -1, a(1) = 1, a(2) = -25.
a(n) = 1/4( p^(n+1) + q^(n+1) ) + (-3)^(n+1)/2 with p = 1 + 2*sqrt(2)i and q = 1 - 2*sqrt(2)i ( i^2 = -1 ).
a(n) = ((-1)^(n+1))*(A087455(n+1))^2; 2*a(n) = A025172(n) + (-3)^(n+1).

Extensions

Edited by Ralf Stephan, Apr 09 2009
Definition corrected by Harvey P. Dale, Oct 03 2014

A221131 Table, T, read by antidiagonals where T(-j,k) = ((1+sqrt(j))^k + (1-sqrt(j))^k)/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -2, 1, 1, 1, -2, -5, -4, 1, 1, 1, -3, -8, -7, -4, 1, 1, 1, -4, -11, -8, 1, 0, 1, 1, 1, -5, -14, -7, 16, 23, 8, 1, 1, 1, -6, -17, -4, 41, 64, 43, 16, 1, 1, 1, -7, -20, 1, 76, 117, 64, 17, 16, 1, 1, 1, -8, -23, 8, 121, 176, 29, -128, -95, 0, 1
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com) and Robert G. Wilson v, Jan 02 2013

Keywords

Comments

.j\k.........0..1...2....3...4....5....6......7.......8......9......10
.0: A000012..1..1...1....1...1....1....1......1.......1......1.......1
-1: A146559..1..1...0...-2..-4...-4....0......8......16.....16.......0
-2: A087455..1..1..-1...-5..-7....1...23.....43......17....-95....-241
-3: A138230..1..1..-2...-8..-8...16...64.....64....-128...-512....-512
-4: A006495..1..1..-3..-11..-7...41..117.....29....-527..-1199.....237
-5: A138229..1..1..-4..-14..-4...76..176...-104...-1264..-1904....3776
-6: A090592..1..1..-5..-17...1..121..235...-377...-2399..-2159...12475
-7: A090590..1..1..-6..-20...8..176..288...-832...-3968..-1280...29184
-8: A025172..1..1..-7..-23..17..241..329..-1511...-5983...1633...57113
-9: A120743..1..1..-8..-26..28..316..352..-2456...-8432...7696...99712
-10: ........1..1..-9..-29..41..401..351..-3709..-11279..18241..160551

Crossrefs

Cf. A084097,
Rows: A000012, A146559, A087455, A138230, A006495, A138229, A090592, A090590, A025172, A120743.
Columns: A000012, A000012, A024000, 1-3n, A028884, A134593.

Programs

  • Mathematica
    T[j_, k_] := Expand[((1 + Sqrt[j])^k + (1 - Sqrt[j])^k)/2]; Table[ T[ -j + k, k], {j, 0, 11}, {k, 0, j}] // Flatten
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