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 1-4 of 4 results.

A162516 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x+d)^n + (x-d)^n)/2, where d=sqrt(x+4).

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 1, 3, 12, 0, 1, 6, 25, 8, 16, 1, 10, 45, 40, 80, 0, 1, 15, 75, 121, 252, 48, 64, 1, 21, 119, 287, 644, 336, 448, 0, 1, 28, 182, 588, 1457, 1360, 1888, 256, 256, 1, 36, 270, 1092, 3033, 4176, 6240, 2304, 2304, 0, 1, 45, 390, 1890, 5925, 10801, 17780, 11680, 12160, 1280, 1024
Offset: 0

Views

Author

Clark Kimberling, Jul 05 2009

Keywords

Examples

			First six rows:
  1;
  1,  0;
  1,  1,  4;
  1,  3, 12,  0;
  1,  6, 25,  8, 16;
  1, 10, 48, 40, 80, 0;

Links

Crossrefs

Cf. A162514, A162515, A162517.
Cf. A000217, A026150, A084057, A125818, A199572.
For fixed k, the sequences P(n,k), for n=1,2,3,4,5, are A084057, A084059, A146963, A081342, A081343, respectively.

Programs

  • Magma
    m:=12;
p:= func< n,x | ((x+Sqrt(x+4))^n + (x-Sqrt(x+4))^n)/2 >;
R:=PowerSeriesRing(Rationals(), m+1);
T:= func< n,k | Coefficient(R!( p(n,x) ), n-k) >;
[T(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Jul 09 2023
  • Mathematica
    P[n_, x_]:= P[n, x]= ((x+Sqrt[x+4])^n + (x-Sqrt[x+4])^n)/2;
T[n_, k_]:= Coefficient[Series[P[n, x], {x,0,n-k+1}], x, n-k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 08 2020; Jul 09 2023 *)
  • SageMath
    def p(n,x): return ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2
def T(n,k):
    P. = PowerSeriesRing(QQ)
    return P( p(n,x) ).list()[n-k]
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 09 2023

Formula

P(n,x) = 2*x*P(n-1,x) - (x^2 -x -4)*P(n-2,x).
From G. C. Greubel, Jul 09 2023: (Start)
T(n, k) = [x^(n-k)] ( ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2 ).
T(n, 1) = A000217(n-1), n >= 1.
T(n, n) = A199572(n).
Sum_{k=0..n} T(n, k) = A084057(n).
Sum_{k=0..n} 2^k*T(n, k) = A125818(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A026150(n).
Sum_{k=0..n} (-2)^k*T(n, k) = A133343(n). (End)

A191348 Array read by antidiagonals: ((ceiling(sqrt(n)) + sqrt(n))^k + (ceiling(sqrt(n)) - sqrt(n))^k)/2 for columns k >= 0 and rows n >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 6, 2, 1, 0, 8, 20, 7, 2, 1, 0, 16, 68, 26, 8, 3, 1, 0, 32, 232, 97, 32, 14, 3, 1, 0, 64, 792, 362, 128, 72, 15, 3, 1, 0, 128, 2704, 1351, 512, 376, 81, 16, 3, 1, 0
Offset: 0

Views

Author

Charles L. Hohn, May 31 2011

Keywords

Examples

			1, 0,  0,   0,    0,     0,      0,      0,       0,        0,         0, ...
1, 1,  2,   4,    8,    16,     32,     64,     128,      256,       512, ...
1, 2,  6,  20,   68,   232,    792,   2704,    9232,    31520,    107616, ...
1, 2,  7,  26,   97,   362,   1351,   5042,   18817,    70226,    262087, ...
1, 2,  8,  32,  128,   512,   2048,   8192,   32768,   131072,    524288, ...
1, 3, 14,  72,  376,  1968,  10304,  53952,  282496,  1479168,   7745024, ...
1, 3, 15,  81,  441,  2403,  13095,  71361,  388881,  2119203,  11548575, ...
1, 3, 16,  90,  508,  2868,  16192,  91416,  516112,  2913840,  16450816, ...
1, 3, 17,  99,  577,  3363,  19601, 114243,  665857,  3880899,  22619537, ...
1, 3, 18, 108,  648,  3888,  23328, 139968,  839808,  5038848,  30233088, ...
1, 4, 26, 184, 1316,  9424,  67496, 483424, 3462416, 24798784, 177615776, ...
1, 4, 27, 196, 1433, 10484,  76707, 561236, 4106353, 30044644, 219825387, ...
1, 4, 28, 208, 1552, 11584,  86464, 645376, 4817152, 35955712, 268377088, ...
1, 4, 29, 220, 1673, 12724,  96773, 736012, 5597777, 42574180, 323800109, ...
1, 4, 30, 232, 1796, 13904, 107640, 833312, 6451216, 49943104, 386642400, ...
...

Crossrefs

Row 1 is A000007, row 2 is A011782, row 3 is A006012, row 4 is A001075, row 5 is A081294, row 6 is A098648, row 7 is A084120, row 8 is A146963, row 9 is A001541, row 10 is A081341, row 11 is A084134, row 13 is A090965.
Row 3*2 is A056236, row 4*2 is A003500, row 5*2 is A155543, row 9*2 is A003499.
Cf. A191347 which uses floor() in place of ceiling().

Programs

  • PARI
    T(n, k) = if (k==0, 1, if (k==1, ceil(sqrt(n)), T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2));
matrix(9, 9, n, k, T(n-1, k-1)) \\ Charles L. Hohn, Aug 23 2019

Formula

For each row n >= 0 let T(n,0)=1 and T(n,1) = ceiling(sqrt(n)), then for each column k >= 2: T(n,k) = T(n,k-2)*(n-T(n,1)^2) + T(n,k-1)*T(n,1)*2. - Charles L. Hohn, Aug 23 2019

A146964 a(n) = ((4 + sqrt(7))^n + (4 - sqrt(7))^n)/2.

Original entry on oeis.org

1, 4, 23, 148, 977, 6484, 43079, 286276, 1902497, 12643492, 84025463, 558412276, 3711069041, 24662841844, 163903113383, 1089259330468, 7238946623297, 48108239012164, 319715392487639, 2124748988791636, 14120553377944337
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Nov 03 2008

Keywords

Comments

Binomial transform of A146963.
Inverse binomial transform of A146965.

Links

Crossrefs

Cf. A098158, A146963, A146965.

Programs

  • GAP
    a:=[1,4];; for n in [3..25] do a[n]:=8*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, Jan 08 2020
  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-7); S:=[ ((4+r7)^n+(4-r7)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 05 2008
  • Maple
    seq(coeff(series((1-4*x)/(1-8*x+9*x^2), x, n+1), x, n), n = 0..25); # G. C. Greubel, Jan 08 2020
  • Mathematica
    LinearRecurrence[{8,-9}, {1,4}, 25] (* G. C. Greubel, Jan 08 2020 *)
  • PARI
    my(x='x+O('x^25)); Vec((1-4*x)/(1-8*x+9*x^2)) \\ G. C. Greubel, Jan 08 2020
  • Sage
    def A146964_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-4*x)/(1-8*x+9*x^2) ).list()
A146964_list(25) # G. C. Greubel, Jan 08 2020

Formula

From Philippe Deléham and Klaus Brockhaus, Nov 05 2008: (Start)
a(n) = 8*a(n-1) - 9*a(n-2) with a(0)=1, a(1)=4.
G.f.: (1-4*x)/(1-8*x+9*x^2). (End)
a(n) = (Sum_{k=0..n} A098158(n,k)*4^(2*k)*7^(n-k))/4^n. - Philippe Deléham, Nov 06 2008
E.g.f.: exp(4*x)*cosh(sqrt(7)*x). - G. C. Greubel, Jan 08 2020

Extensions

Extended beyond a(7) by Klaus Brockhaus, Nov 05 2008
Edited by Klaus Brockhaus, Jul 16 2009

A356200 Number of edge covers in the n-gear graph.

Original entry on oeis.org

3, 25, 162, 969, 5613, 32062, 181989, 1030017, 5821902, 32886505, 185714829, 1048619646, 5920559661, 33426829321, 188721717102, 1065481514817, 6015458406741, 33961820796094, 191740095366885, 1082517159435249, 6111623364952302, 34504707439240921
Offset: 1

Views

Author

Eric W. Weisstein, Jul 29 2022

Keywords

Comments

Sequence extended to a(1) using the formula/recurrence.

Links

Programs

  • Mathematica
    Table[(3 - Sqrt[7])^n + (3 + Sqrt[7])^n - LucasL[2 n], {n, 30}] // Expand
CoefficientList[Series[(3 - 2 x)/((1 - 3 x + x^2) (1 - 6 x + 2 x^2)), {x, 0, 20}], x]
LinearRecurrence[{9, -21, 12, -2}, {3, 25, 162, 969}, 20]

Formula

a(n) = (3-sqrt(7))^n + (sqrt(7)+3)^n - Lucas(2*n).
a(n) = 9*a(n-1) - 21*a(n-2) + 12*a(n-3) - 2*a(n-4).
G.f.: x*(3-2*x)/((1-3*x+x^2)*(1-6*x+2*x^2)).
a(n) = 2*A146963(n) - A005248(n). - R. J. Mathar, Jan 25 2023
Showing 1-4 of 4 results.

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

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