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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A205493 Third row or column of table A205497.

Original entry on oeis.org

1, 14, 109, 623, 2951, 12331, 47191, 169416, 579889, 1914226, 6144668, 19298724, 59579803, 181448918, 546629054, 1632497850, 4841448042, 14277423006, 41912838982, 122587133760, 357476552161, 1039922075888, 3019280091491, 8752184436454, 25337900299765
Offset: 0

Views

Author

L. Edson Jeffery, Jan 28 2012

Keywords

Comments

See A205497 regarding association of this sequence with generating functions for the rows of the tabular form of A050446.

Links

Crossrefs

Cf. A205497, A050446, A050447.

Programs

Formula

Conjecture 1. a(n) = M_{n,3} = M_{3,n}, where M = A205497.
Conjecture 2. Let w=2*cos(Pi/9). Then lim_{n->oo} a(n+1)/a(n) = w^3-2*w = spectral radius of the 4 X 4 unit-primitive matrix (see [Jeffery]) A_{9,3} = [0,0,0,1; 0,0,1,1; 0,1,1,1; 1,1,1,1].

Extensions

a(24) and changed title from Hugo Pfoertner, Jan 05 2020

A205494 Conjectured row or column n=4 of array A205497.

Original entry on oeis.org

1, 26, 334, 2951, 20641, 123216, 656683, 3217526, 14786816, 64657546, 271838823, 1107586989, 4399926007, 17122243560, 65514790830, 247212893755, 922136438698, 3406871213836, 12486569116765, 45459575562313, 164578100859837, 593025025473647, 2128399709975819, 7613495897772440
Offset: 0

Views

Author

L. Edson Jeffery, Jan 28 2012

Keywords

Comments

The denominator of the generating function for this sequence is a polynomial of degree 35. Terms corresponding to n=0,...,23 are shown above, with those for n=24,...,40 as follows: {27157723973468595, 96643368020414337, 343226612286408932, 1216901732483780905, 4308339945395597755, 15234940157670046379, 53818220864065451564, 189952299613455045068, 669953408386151161398, 2361449534293944339096, 8319329987059336296021, 29296032314800671782284, 103126374236214419873734, 362907786820798388773987, 1276761054260676178577043, 4490840947292979020061377, 15793032895427304036405557}.
See A205497 regarding association of this sequence with generating functions for the rows of the tabular form of A050446.

Links

Crossrefs

Cf. A205497, A050446, A050447.

Formula

G.f.: (1+4*x-31*x^2 - 67*x^3 + 348*x^4 + 418*x^5 - 1893*x^6 - 1084*x^7 + 4326*x^8 + 4295*x^9 - 7680*x^10 - 9172*x^11 + 9104*x^12 + 11627*x^13 - 5483*x^14 - 10773*x^15 + 1108*x^16 + 7255*x^17 + 315*x^18 - 3085*x^19 - 228*x^20 + 669*x^21 + 102*x^22 - 23*x^23 - 45*x^24 - 16*x^25 + 11*x^26 + 2*x^27 - x^28) / ((1-x)^5 * (1-x-x^2)^4 * (1-2*x-x^2+x^3)^3 * (1-2*x-3*x^2+x^3+x^4)^2 * (1-3*x-3*x^2+4*x^3+x^4-x^5)).
CONJECTURE 1. a(n) = M_{n,4} = M_{4,n}, where M = A205497.
CONJECTURE 2. Let w=2*cos(Pi/11). Then lim_{n->oo} a(n+1)/a(n) = w^4-3*w^2+1 = spectral radius of the 5 X 5 unit-primitive matrix (see [Jeffery]) A_{11,4} = [0,0,0,0,1; 0,0,0,1,1; 0,0,1,1,1; 0,1,1,1,1; 1,1,1,1,1].

A373567 Expansion of x + 1/(-x - 1/(-x - 1/(-x + 1))).

Original entry on oeis.org

1, 4, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, 8997, 20216, 45425, 102069, 229347, 515338, 1157954, 2601899, 5846414, 13136773, 29518061, 66326481, 149034250, 334876920, 752461609, 1690765888, 3799116465, 8536537209, 19181424995, 43100270734, 96845429254
Offset: 0

Views

Author

Peter Luschny, Jun 10 2024

Keywords

Comments

a(n) is the number of up-down words of length n over an alphabet of size 4. - Sela Fried, Apr 08 2025

References

  • L. Carlitz and R. Scoville, Up-down sequences, Duke Math. J. (39) (1972), 583-598.

Links

Crossrefs

Essentially the same as A006356.
Cf. A050446.

Programs

  • Mathematica
    CoefficientList[Series[x + 1/(-x - 1/(-x - 1/(-x + 1))), {x, 0, 31}], x] (* Michael De Vlieger, Jun 10 2024 *)

Formula

a(n) = [x^n] (x^4 - x^3 - 3*x^2 + 2*x + 1) / (x^3 - x^2 - 2*x + 1).
Previous Showing 21-23 of 23 results.

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