A057960 -id:A057960 - cp's OEIS frontend

A117294 Number of sequences of length n starting with 1,2 which satisfy a recurrence a(k+1) = floor(c*a(k)) for some constant c.

1, 2, 5, 14, 37, 102, 279, 756, 2070, 5609, 15198, 41530, 114049, 315447, 876513, 2446326, 6861432, 19315953, 54556553, 154591186, 439307113, 1251678183, 3574777087, 10231666185, 29343549576, 84309936418, 242651784699, 699476361863, 2019289119525, 5837355573611, 16896103820563, 48963682959055, 142051622347551

Offset: 2

Franklin T. Adams-Watters, Apr 26 2006

nonn

It appears that a(n+1)/a(n) may be converging slowly to 3, but even that it converges is not obvious.
From Martin Fuller, Apr 05 2025: (Start)
Each finite sequence corresponds to a range of c, e.g. (1,2,4,9) has c in [9/4,5/2).
Let x=(maximum element)*(range width of c), e.g. x(1,2,4,9)=9*(5/2 - 9/4)=9/4.
Then the number of extensions of this sequence in a(n+1) is either floor(x)+1 or ceil(x)+1.  Non-rigorously, the expected number is x+1.
Also x is bounded 0Does X have a distribution for large n and approximate c? Can this be used to work out the growth rate of a(n)? (End)

			a(4) = 5: length 4 sequences are 1,2,4,8; 1,2,4,9; 1,2,5,12; 1,2,5,13; and 1,2,5,14.

Martin Fuller, Graph of sequences counted by A117294
Martin Fuller, C++ program

Some (infinite) examples of such sequences: A000079, A007051, A076883, A001519, A024537, A024576, A057960.

(define (A117294 n) (local ((define (get-ratios seq add?) (cond [(empty? (rest seq)) empty] [else (cons (/ (cond [add? (add1 (first seq))] [else (first seq)]) (second seq)) (get-ratios (rest seq) add?))])) (define (extend-one seq) (local ((define startnext (floor (* (apply max (get-ratios seq false)) (first seq)))) (define endnext (ceiling (* (apply min (get-ratios seq true )) (first seq)))) (define ltodo (build-list (- endnext startnext) (lambda (n) (cons (+ startnext n) seq))))) (cond [(>= (length seq) (sub1 n)) (length ltodo)] [else (apply + (map extend-one ltodo))])))) (extend-one (list 2 1)))) ;; Joshua Zucker, Jun 05 2006

More terms from Joshua Zucker, Jun 05 2006
Comment edited by Franklin T. Adams-Watters, May 14 2010
Ambiguous terms a(0), a(1) removed by Max Alekseyev, Jan 18 2012
a(21)-a(24) from Jinyuan Wang, Mar 20 2025
a(25)-a(34) from Martin Fuller, Apr 05 2025

A208718 Number of n-bead necklaces labeled with numbers 1..5 allowing reversal, with no adjacent beads differing by more than 1.

Original entry on oeis.org

5, 9, 13, 24, 38, 78, 140, 306, 634, 1464, 3326, 8066, 19454, 48534, 121294, 308154, 785222, 2018548, 5203634, 13482426, 35019010, 91251438, 238278314, 623629333, 1635062126, 4294493670, 11296419934, 29757590061, 78489973742, 207281830814

Offset: 1

R. H. Hardin, Mar 01 2012

nonn

			All solutions for n=3:
..3....1....2....4....5....1....2....2....3....4....1....4....3
..3....1....2....5....5....1....3....2....4....4....2....4....3
..4....1....2....5....5....2....3....3....4....5....2....4....3

Andrew Howroyd, Table of n, a(n) for n = 1..100

Column 5 of A208721.

Cf. A208774, A057960.

a(2n+1) = (1/2) * (A208774(2n+1) + r(n+1)) where r(n) = A057960(n+1). - Andrew Howroyd, Mar 03 2017

a(2n) = (1/2) * A208774(2n) + (1/4) * (r(n) + r(n+1)) where r(n) = A057960(n+1). - Andrew Howroyd, Mar 03 2017

a(25)-a(30) from Andrew Howroyd, Mar 03 2017

A296449 Triangle I(m,n) read by rows: number of perfect lattice paths on the m*n board.

Original entry on oeis.org

1, 2, 4, 3, 7, 17, 4, 10, 26, 68, 5, 13, 35, 95, 259, 6, 16, 44, 122, 340, 950, 7, 19, 53, 149, 421, 1193, 3387, 8, 22, 62, 176, 502, 1436, 4116, 11814, 9, 25, 71, 203, 583, 1679, 4845, 14001, 40503, 10, 28, 80, 230, 664, 1922, 5574, 16188, 47064, 136946, 11, 31, 89, 257, 745, 2165, 6303, 18375, 53625, 156629, 457795

Offset: 1

R. J. Mathar, Dec 13 2017

			Triangle begins:
   1;
   2,  4;
   3,  7, 17;
   4, 10, 26,  68;
   5, 13, 35,  95, 259;
   6, 16, 44, 122, 340,  950;
   7, 19, 53, 149, 421, 1193, 3387;
   8, 22, 62, 176, 502, 1436, 4116, 11814;
   9, 25, 71, 203, 583, 1679, 4845, 14001, 40503;
  10, 28, 80, 230, 664, 1922, 5574, 16188, 47064, 136946;

D. Yaqubi, M. Farrokhi D. G., and H. Ghasemian Zoeram, Lattice paths inside a table, I, arXiv:1612.08697 [math.CO], 2016-2017, array I(m,n).

Cf. A081113 (diagonal), A000079 (2nd row), A001333 (3rd row), A126358, A057960, A126360, A002714, A126362, A188866.

Inm := proc(n,m)
    if m >= n then
        (n+2)*3^(n-2)+(m-n)*add(A005773(i)*A005773(n-i),i=0..n-1)
            +2*add((n-k-2)*3^(n-k-3)*A001006(k),k=0..n-3) ;
    else
        0 ;
    end if;
end proc:
for m from 1 to 13 do
for n from 1 to m do
    printf("%a,",Inm(n,m)) ;
end do:
printf("\n") ;
end do:
# Second program:
A296449row := proc(n) local gf, ser;
gf := n -> 1 + (x*(n - (3*n + 2)*x) + (2*x^2)*(1 +
ChebyshevU(n - 1, (1 - x)/(2*x))) / ChebyshevU(n, (1 - x)/(2*x)))/(1 - 3*x)^2;
ser := n -> series(expand(gf(n)), x, n + 1);
seq(coeff(ser(n), x, k), k = 1..n) end:
for n from 0 to 11 do A296449row(n) od; # Peter Luschny, Sep 07 2021

(* b = A005773 *) b[0] = 1; b[n_] := Sum[k/n*Sum[Binomial[n, j] * Binomial[j, 2*j - n - k], {j, 0, n}], {k, 1, n}];
(* c = A001006 *) c[0] = 1; c[n_] := c[n] = c[n-1] + Sum[c[k] * c[n-2-k], {k, 0, n-2}];
Inm[n_, m_] := If[m >= n, (n + 2)*3^(n - 2) + (m - n)*Sum[b[i]*b[n - i], {i, 0, n - 1}] + 2*Sum[(n - k - 2)*3^(n - k - 3)*c[k], {k, 0, n-3}], 0];
Table[Inm[n, m], {m, 1, 13}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jan 23 2018, adapted from first Maple program. *)

I(m,n) = (n+2)*3^(n-2) + (m-n)*Sum_{i=0..n-1} A005773(i)*A005773(n-i) + 2*Sum_{k=0..n-3} (n-k-2)*3^(n-k-3)*A001006(k). [Yaqubi Corr. 2.10]

I(m,n) = A188866(m-1,n) for m > 1. - Pontus von Brömssen, Sep 06 2021

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A117294 Number of sequences of length n starting with 1,2 which satisfy a recurrence a(k+1) = floor(c*a(k)) for some constant c.

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A208718 Number of n-bead necklaces labeled with numbers 1..5 allowing reversal, with no adjacent beads differing by more than 1.

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A296449 Triangle I(m,n) read by rows: number of perfect lattice paths on the m*n board.

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