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.

A350917 a(0) = 1, a(1) = 2, and a(n) = 23*a(n-1) - a(n-2) - 4 for n >= 2.

Original entry on oeis.org

1, 2, 41, 937, 21506, 493697, 11333521, 260177282, 5972743961, 137112933817, 3147624733826, 72258255944177, 1658792261982241, 38079963769647362, 874180374439907081, 20068068648348215497, 460691398537569049346, 10575834097715739919457, 242783492848924449098161, 5573444501427546589338242, 127946440039984647105681401, 2937194676418219336841333977
Offset: 0

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Author

Max Alekseyev, Jan 21 2022

Keywords

Comments

One of 10 linear second-order recurrence sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4 and together forming A350916.
Other properties for all n:
(a(n)+1)*(a(n+2)+1) = (a(n+1)+1)*(a(n+1)+26);
((105*a(n) - 20)^2 - 50^2) / 21 is an integer square.

Links

Crossrefs

Cf. A350916.
Other sequences satisfying (a(n)*a(n-1)-1) * (a(n)*a(n+1)-1) = (a(n)+1)^4: A103974, A350919, A350920, A350921, A350922, A350923, A350924, A350925, A350926.

Formula

a(n) = 17/42*A090731(n) - 15/2*A097778(n-1) + 4/21.
G.f.: ( -1+22*x-17*x^2 ) / ( (x-1)*(x^2-23*x+1) ). - R. J. Mathar, Feb 07 2022

A003690 Number of spanning trees in K_3 X P_n.

Original entry on oeis.org

3, 75, 1728, 39675, 910803, 20908800, 479991603, 11018898075, 252954664128, 5806938376875, 133306628004003, 3060245505715200, 70252340003445603, 1612743574573533675, 37022849875187828928, 849912803554746531675, 19510971631883982399603
Offset: 1

Views

Author

Frans J. Faase

Keywords

Comments

Column 3 of A173958. The sequence a(n)/3 is linear divisibility sequence of the fourth order; it is the case P1 = 25, P2 = 46, Q = 1 of the three parameter family of divisibility sequences found by Williams and Guy. - Peter Bala, Apr 27 2014

References

  • F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Links

Crossrefs

Cf. A100047, A173958 (column 3).

Programs

  • Magma
    I:=[3,75,1728]; [n le 3 select I[n] else 24*Self(n-1)-24*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Apr 28 2014
  • Mathematica
    CoefficientList[Series[3 (1 + x)/((1 - x) (1 - 23 x + x^2)), {x, 0, 20}], x] (* Vincenzo Librandi, Apr 28 2014 *)
  • PARI
    Vec(3*x*(1+x)/((1-x)*(1-23*x+x^2)) + O(x^25)) \\ Colin Barker, Mar 06 2016

Formula

a(n) = (A090731(n)-2)/7.
a(n) = 24*a(n-1) - 24*a(n-2) + a(n-3), n>3.
G.f.: 3*x*(1+x)/((1-x)*(1-23*x+x^2)). - R. J. Mathar, Dec 16 2008
a(n) = 3*(A004254(n))^2. - R. K. Guy, seqfan list, Mar 28 2009, - R. J. Mathar, Jun 03 2009
From Peter Bala, Apr 27 2014: (Start)
Product {n >= 2} (1 - 3/a(n)) = 1/2 + sqrt(21)/10.
a(n) = (2/7)*( T(n,23/2) - 1), where T(n,x) is the Chebyshev polynomial of the first kind.
a(n) = 3 * the bottom left entry of the 2 X 2 matrix T(n,M), where M is the 2 X 2 matrix [0, -23/2; 1, 25/2].
a(n) = 3*U(n-1,5/2)^2, where U(n,x) is the Chebyshev polynomial of the second kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
a(n) = (-2+(2/(23+5*sqrt(21)))^n+(1/2*(23+5*sqrt(21)))^n)/7. - Colin Barker, Mar 06 2016

A097778 Chebyshev polynomials S(n,23) with Diophantine property.

Original entry on oeis.org

1, 23, 528, 12121, 278255, 6387744, 146639857, 3366328967, 77278926384, 1774048977865, 40725847564511, 934920445005888, 21462444387570913, 492701300469125111, 11310667466402306640, 259652650426783927609
Offset: 0

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Author

Wolfdieter Lang, Aug 31 2004

Keywords

Comments

All positive integer solutions of Pell equation b(n)^2 - 525*a(n)^2 = +4 together with b(n)=A090731(n+1), n>=0. Note that D=525=21*5^2 is not squarefree.
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 23's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011
For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,22}. - Milan Janjic, Jan 25 2015

Examples

			(x,y) = (23;1), (527;23), (12098;528), ... give the positive integer solutions to x^2 - 21*(5*y)^2 =+4.

Links

Crossrefs

A004254, A049310.

Programs

  • Mathematica
    LinearRecurrence[{23,-1},{1,23},20] (* Harvey P. Dale, May 06 2016 *)
  • Sage
    [lucas_number1(n,23,1) for n in range(1,20)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = S(n, 23) = U(n, 23/2) = S(2*n+1, 5)/5 with S(n, x) = U(n, x/2) Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x)= 0 = U(-1, x). S(n, 5) = A004254(n+1).
a(n) = 23*a(n-1)-a(n-2), n >= 1; a(0)=1, a(1)=23; a(-1)=0.
a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap := (23+5*sqrt(21))/2 and am := (23-5*sqrt(21))/2.
G.f.: 1/(1-23*x+x^2).
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*22^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/21*(21 + 5*sqrt(21)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/46*(21 + 5*sqrt(21)). - Peter Bala, Dec 23 2012

A299741 Array read by antidiagonals upwards: a(i,0) = 2, i >= 0; a(i,1) = i+2, i >= 0; a(i,j) = (i+2) * a(i,j-1) - a(i,j-2), for i >= 0, j > 1.

Original entry on oeis.org

2, 2, 2, 2, 3, 2, 2, 4, 7, 2, 2, 5, 14, 18, 2, 2, 6, 23, 52, 47, 2, 2, 7, 34, 110, 194, 123, 2, 2, 8, 47, 198, 527, 724, 322, 2, 2, 9, 62, 322, 1154, 2525, 2702, 843, 2, 2, 10, 79, 488, 2207, 6726, 12098, 10084, 2207, 2, 2, 11, 98, 702, 3842, 15127, 39202, 57965, 37634, 5778, 2
Offset: 0

Views

Author

William W. Collier, Feb 18 2018

Keywords

Comments

Note the similarity in form of the recursive steps in the array definition above and the polynomial definition under FORMULA.

Examples

			i\j |0  1   2    3      4       5        6          7           8            9
----+-------------------------------------------------------------------------
   0|2  2   2    2      2       2        2          2           2            2
   1|2  3   7   18     47     123      322        843        2207         5778
   2|2  4  14   52    194     724     2702      10084       37634       140452
   3|2  5  23  110    527    2525    12098      57965      277727      1330670
   4|2  6  34  198   1154    6726    39202     228486     1331714      7761798
   5|2  7  47  322   2207   15127   103682     710647     4870847     33385282
   6|2  8  62  488   3842   30248   238142    1874888    14760962    116212808
   7|2  9  79  702   6239   55449   492802    4379769    38925119    345946302
   8|2 10  98  970   9602   95050   940898    9313930    92198402    912670090
   9|2 11 119 1298  14159  154451  1684802   18378371   200477279   2186871698
  10|2 12 142 1692  20162  240252  2862862   34114092   406506242   4843960812
  11|2 13 167 2158  27887  360373  4656962   60180133   777684767  10049721838
  12|2 14 194 2702  37634  524174  7300802  101687054  1416317954  19726764302
  13|2 15 223 3330  49727  742575 11088898  165590895  2472774527  36926027010
  14|2 16 254 4048  64514 1028176 16386302  261152656  4162056194  66331746448
  15|2 17 287 4862  82367 1395377 23639042  400468337  6784322687 114933017342
  16|2 18 322 5778 103682 1860498 33385282  599074578 10749957122 192900153618
  17|2 19 359 6802 128879 2441899 46267202  876634939 16609796639 314709501202
  18|2 20 398 7940 158402 3160100 63043598 1257711860 25091193602 500566160180
  19|2 21 439 9198 192719 4037901 84603202 1772629341 37140612959 778180242798

Links

Crossrefs

The array first appeared in A298675.
Rows 1 through 29 of the array appear in these OEIS entries: A005248, A003500, A003501, A003499, A056854, A086903, A056918, A087799, A057076, A087800, A078363, A067902, A078365, A090727, A078367, A087215, A078369, A090728, A090729, A090730, A090731, A090732, A090733, A090247, A090248, A090249, A090251. Also entries occur for rows 45, 121, and 320: A087265, A065705, A089775. Each of these entries asserts that a(i,j)=f(i+2,j) is true for that row.
A few of the columns appear in the OEIS: A008865 (for column 2), A058794 and A007754 (for column 3), and A230586 (for column 5).
Main diagonal gives A343261.

Programs

  • Maple
    A:= proc(i, j) option remember; `if`(min(i, j)=0, 2,
      `if`(j=1, i+2, (i+2)*A(i, j-1)-A(i, j-2)))
    end:
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Mar 05 2019
  • Mathematica
    a[, 0] = a[0, ] = 2; a[i_, 1] := i + 2;
a[i_, j_] := a[i, j] =(i + 2) a[i, j - 1] - a[i, j - 2];
Table[a[i - j, j], {i, 0, 10}, {j, 0, i}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

Formula

Let k be an integer, and let r1 and r2 be the roots of x + 1/x = k. Then f(k,n) = r1^n + r2^n is an integer, for integer n >= 0. Theorem: a(i,j) = f(i+2,j), for i,j >= 0. Proof: See the Collier link.
Define polynomials recursively by:
p[0](n) = 2, for n >= 0 ( [ and ] demark subscripts).
p[1](n) = n + 2, for n >= 0.
p[j](n) = p[j-1](n) * p[1](n) - p[j-2](n), for j > 1, n >= 0. The coefficients of these polynomials occur as the even numbered, upward diagonals in the OEIS Wiki link. Conjecture: a(i,j) = p[j](i), i,j >= 0.

Extensions

Edited by N. J. A. Sloane, Apr 04 2018
Showing 1-4 of 4 results.

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