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 1-3 of 3 results.

A259969 a(n) = n*A259968(n).

Original entry on oeis.org

0, 1, 2, 9, 24, 50, 102, 210, 424, 837, 1630, 3146, 6024, 11453, 21644, 40695, 76176, 142035, 263916, 488870, 903060, 1663998, 3059166, 5612483, 10277448, 18787150, 34287916, 62485371, 113715448, 206683725, 375211710, 680399005, 1232533696, 2230537914
Offset: 0

Views

Author

N. J. A. Sloane, Jul 11 2015

Keywords

References

  • R. K. Guy, Letter to N. J. A. Sloane, Feb 05 1986.

Links

Crossrefs

Cf. A259968, A259967.

Programs

  • Haskell
    a259967 n = a259967_list !! n
a259967_list = 3 : 2 : 2 : 5 : zipWith3 (((+) .) . (+))
   a259967_list (drop 2 a259967_list) (drop 3 a259967_list)
-- Reinhard Zumkeller, Jul 12 2015
  • PARI
    concat(0, Vec(x*(x^4-6*x^3+7*x^2-2*x+1)/(x^3-x^2+2*x-1)^2 + O(x^40))) \\ Colin Barker, Jul 12 2015

Formula

G.f.: x*(x^4-6*x^3+7*x^2-2*x+1) / (x^3-x^2+2*x-1)^2. - Colin Barker, Jul 12 2015

Extensions

a(28)-a(33) from Hiroaki Yamanouchi, Jul 12 2015

A259967 a(n) = a(n-1) + a(n-2) + a(n-4).

Original entry on oeis.org

3, 2, 2, 5, 10, 17, 29, 51, 90, 158, 277, 486, 853, 1497, 2627, 4610, 8090, 14197, 24914, 43721, 76725, 134643, 236282, 414646, 727653, 1276942, 2240877, 3932465, 6900995, 12110402, 21252274, 37295141, 65448410, 114853953, 201554637, 353703731, 620706778
Offset: 0

Views

Author

N. J. A. Sloane, Jul 11 2015

Keywords

Comments

Also the number of maximal independent vertex sets (and minimal vertex covers) in the n-gear graph. - Eric W. Weisstein, May 25 2017
Also the number of chordless cycles in the n-antiprism graph for n >= 4. - Eric W. Weisstein, Jan 02 2018

References

  • R. K. Guy, Letter to N. J. A. Sloane, Feb 05 1986.

Links

Crossrefs

Cf. A109377, A259968, A259969.

Programs

  • Haskell
    a259967 n = a259967_list !! n
a259967_list = 3 : 2 : 2 : 5 : zipWith3 (((+) .) . (+))
   a259967_list (drop 2 a259967_list) (drop 3 a259967_list)
-- Reinhard Zumkeller, Jul 12 2015
  • Magma
    I:=[3,2,2,5]; [n le 4 select I[n] else Self(n-1)+Self(n-2)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Sep 26 2017
  • Maple
    f:= gfun:-rectoproc({-a(n+3)+2*a(n+2)-a(n+1)+a(n), a(0) = 3, a(1) = 2, a(2) = 2},a(n),remember):
map(f, [$0..50]); # Robert Israel, Jul 18 2016
  • Mathematica
    Abs @ CoefficientList[Series[(x - 1) (x - 3)/(-1 + 2 x - x^2 + x^3), {x, 0, 36}], x] (* Michael De Vlieger, Jul 18 2016 *)
LinearRecurrence[{2, -1, 1}, {2, 2, 5}, 20] (* Eric W. Weisstein, May 25 2017 *)
Table[RootSum[-1 + # - 2 #^2 + #^3 &, #^n &], {n, 20}] (* Eric W. Weisstein, May 25 2017 *)
RootSum[-1 + # - 2 #^2 + #^3 &, #^Range[0, 20] &] (* Eric W. Weisstein, Jan 02 2018 *)
  • PARI
    x='x+O('x^50); Vec((x-1)*(x-3)/(1-2*x+x^2-x^3)) \\ G. C. Greubel, May 24 2017

Formula

G.f.: (x-1)*(x-3) / (1 -2*x +x^2 -x^3). - R. J. Mathar, Jul 15 2015
a(n) = -4*A005314(n) +3*A005314(n+1) +A005314(n-1). - R. J. Mathar, Jul 15 2015
a(n) = Sum_{i=1..3} r_i^n where r_i are the roots of x^3-2*x^2+x-1. - Robert Israel, Jul 18 2016
a(n) = A109377(n-2) for n > 1. - Georg Fischer, Oct 09 2018

A171843 Triangle read by rows = truncated columns of an array formed by variants of the natural number decrescendo triangle, A004736.

Original entry on oeis.org

1, 1, 3, 1, 3, 8, 1, 3, 6, 21, 1, 3, 6, 12, 55, 1, 3, 6, 10, 24, 144, 1, 3, 6, 10, 17, 48, 377, 1, 3, 6, 10, 15, 30, 96, 987, 1, 3, 6, 10, 15, 23, 53, 192, 2584, 1, 3, 6, 10, 15, 21, 37, 93, 384, 6765, 1, 3, 6, 10, 15, 21, 30, 61, 163, 768, 17711, 1, 3, 6, 10, 15, 21, 28, 45, 100, 286, 1536, 46368
Offset: 1

Views

Author

Gary W. Adamson, Dec 19 2009

Keywords

Comments

Rows tend to the triangular series, A000217.
Let T(n) be the variants of the natural number decrescendo triangle, A004736; such that T(n) = A004736, prepending n ones to the leftmost column. Then take Lim_{k=1..inf} ((T(n))^k, left-shifted vectors considered as sequences = rows of the array, deleting the first 1. The rows of this triangle sequence are the truncated columns of the array with one "1" per row.

Examples

			First few rows of the array are:
.
  1, 3, 8, 21, 55, 144, 377, 987, ...
  1, 1, 3,  6, 12,  24,  48,  96, ...
  1, 1, 1,  3,  6,  10,  17,  30, ...
  1, 1, 1,  1,  3,   6,  10,  15, ...
  1, 1, 1,  1,  1,   3,   6,  10, ...
  ...
First few rows of the triangle =
  1;
  1, 3;
  1, 3, 8;
  1, 3, 6, 21;
  1, 3, 6, 12, 55;
  1, 3, 6, 10, 24, 144;
  1, 3, 6, 10, 17, 48, 377;
  1, 3, 6, 10, 15, 30, 96, 987;
  1, 3, 6, 10, 15, 23, 53, 192, 2584;
  1, 3, 6, 10, 15, 21, 37, 93, 384, 6765;
  1, 3, 6, 10, 15, 21, 30, 61, 163, 768, 17711;
  1, 3, 6, 10, 15, 21, 28, 45, 100, 286, 1536, 46368;
  ...
Example: Row 2 of the array is generated from a variant of A004736, the leftmost column with two prepended 1's, = T(2):
  1;
  1;
  1;
  2, 1;
  3, 2, 1;
  ...
Take lim_{k->inf.} (P(2))^k, obtaining a left-shifted vector considered as a sequence; then delete the first 1, getting row 2 of the array.

Links

Crossrefs

Row sums are A171844.
Diagonals include A001906, A003945, A259968.
Cf. A004736.

Programs

  • PARI
    T(n)={[Vec(p) | p<-Vec(sum(k=1, n, x^k*y^(k-1)*(1 - x^k)/((1 - x)*(1 - 2*x + x^2 - x^k)) + O(x*x^n)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Apr 13 2021

Extensions

a(52) corrected and terms a(56) and beyond from Andrew Howroyd, Apr 13 2021
Showing 1-3 of 3 results.

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

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