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.

Showing 1-5 of 5 results.

A060924 Bisection of Lucas triangle A060922: odd-indexed members of column sequences of A060922 (not counting leading zeros).

Original entry on oeis.org

3, 7, 6, 18, 38, 9, 47, 158, 120, 12, 123, 566, 753, 280, 15, 322, 1880, 3612, 2568, 545, 18, 843, 5964, 15040, 16220, 7043, 942, 21, 2207, 18342, 57366, 83780, 57560, 16536, 1498, 24, 5778, 55162, 206115
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Row sums give A060927. Column sequences (without leading zeros) are, for m=0..5: A005248(n+1), 2*A061171, A061172, 4*A061173, A061174, 2*A061175.
Companion triangle A060923 (even part).

Examples

			{3}; {7,6}; {18,38,9}; {47,158,120,12}; ...; pLo(2,x)= 2*(3+x-2*x^2).

Crossrefs

Cf. A005248.

Formula

a(n, m) = A060922(2*n+1-m, m).
a(n, m) = ((2*n-m+1)*A060923(n, m-1) + 2*(2*(2*n+1)-3*m)*a(n-1, m-1) + 4*(2*n-m)*A060923(n-1, m-1))/(5*m), m >= n >= 1; a(n, 0) = A005248(n); otherwise 0.
G.f. for column m >= 0: x^m*pLo(m+1, x)/(1-3*x+x^2)^(m+1), where pLo(n, x) := Sum_{m=0..n+floor((n-1)/2)} A061187(n-1, m)*x^m are the row polynomials of the (signed) staircase A061187.
T(n,k) = 3*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) + 2*T(n-2,k-1) - T(n-2,k-2) + 4*T(n-3,k-2), T(0,0) = 3, T(1,0) = 7, T(1,1) = 6, T(2,0) = 18, T(2,1) = 38, T(2,2) = 9, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Jan 21 2014

A061172 Third column of Lucas bisection triangle (odd part).

Original entry on oeis.org

9, 120, 753, 3612, 15040, 57366, 206115, 709152, 2360943, 7659870, 24340184, 76031100, 234116493, 712166952, 2143779645, 6394719216, 18923041360, 55601888562, 162350117703, 471371537040, 1361642740059
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator of g.f. is row polynomial Sum_{m=0..4} A061187(2,m)*x^m.

Links

Crossrefs

Cf. A002548, 2*A061171.

Programs

  • Magma
    I:=[9, 120, 753, 3612, 15040, 57366]; [n le 6 select I[n] else 9*Self(n-1)-30*Self(n-2)+45*Self(n-3)-30*Self(n-4)+9*Self(n-5) - Self(n-6): n in [1..30]]; // G. C. Greubel, Dec 21 2017
  • Mathematica
    CoefficientList[Series[(3-2*x)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^3, {x,0,50}], x] (* or *) LinearRecurrence[{9,-30,45,-30,9,-1}, {9, 120, 753, 3612, 15040, 57366}, 30] (* G. C. Greubel, Dec 21 2017 *)
  • PARI
    x='x+O('x^30); Vec((3-2*x)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^3) \\ G. C. Greubel, Dec 21 2017

Formula

a(n) = A060924(n+2, 2).
G.f.: (3-2*x)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^3.

A061173 One-fourth of fourth column of Lucas bisection triangle (odd part).

Original entry on oeis.org

3, 70, 642, 4055, 20945, 95372, 398290, 1561683, 5836190, 21001410, 73300478, 249476600, 831342517, 2720979140, 8768966810, 27881856697, 87610794135, 272424413508, 839229287580, 2563768831145, 7773145679478
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator of g.f. is row polynomial sum(A061187(3,m)*x^m,m=0..5)/4.

Crossrefs

Cf. A005248, 2*A061171, A061172.

Formula

a(n)=A060924(n+3, 3)/4.
G.f.: (x+1)*(3-2*x)*(1+11*x-11*x^2+4*x^3)/(1-3*x+x^2)^4.

A061174 Fifth column of Lucas bisection triangle (odd part).

Original entry on oeis.org

15, 545, 7043, 57560, 365045, 1970905, 9520315, 42385132, 177293730, 705980760, 2701362950, 10001654350, 36020160943, 126701700755, 436709397085, 1478813477920, 4930328078835, 16212542696607
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator of g.f. is row polynomial sum(A061187(4,m)*x^m,m=0..7).

Crossrefs

Cf. A002548, 2*A061171, A061172, 4*A061173.

Formula

a(n)=A060924(n+4, 4).
G.f.: (3-2*x)*(16*x^6-56*x^5+181*x^4-306*x^3+171*x^2+110*x+5)/(1-3*x+x^2)^5.

A061175 One half of sixth column of Lucas bisection triangle (odd part).

Original entry on oeis.org

9, 471, 8268, 85962, 662773, 4215123, 23440212, 118073914, 551281476, 2423731704, 10148667670, 40812739230, 158644493079, 599051383561, 2206150654944, 7949311477362, 28098758599203, 97645872621753
Offset: 0

Views

Author

Wolfdieter Lang, Apr 20 2001

Keywords

Comments

Numerator of G.f. is one half of row polynomial sum(A061187(5,m)*x^m,m=0..8).

Crossrefs

Cf. A002548, 2*A061171, A061172, 4*A061173, A061174.

Formula

a(n)= A060924(n+5, 5)/2.
G.f.: (1+x)*(3-2*x)*(12*x^3-35*x^2+29*x+1)*(4*x^3-9*x^2+15*x+3)/(1-3*x+x^2)^6.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.