A247049 -id:A247049 - cp's OEIS frontend

A247301 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,0) to (n,k), where 0 >= k <= 2, consisting of segments given by the vectors (1,1), (2,1), (1,-1).

1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 4, 3, 4, 6, 6, 6, 12, 10, 12, 20, 18, 20, 36, 32, 36, 64, 56, 64, 112, 100, 112, 200, 176, 200, 352, 312, 352, 624, 552, 624, 1104, 976, 1104, 1952, 1728, 1952, 3456, 3056, 3456, 6112, 5408, 6112, 10816, 9568, 10816

Offset: 0

Clark Kimberling, Sep 11 2014

			First 10 columns:
0 .. 0 .. 1 .. 2 .. 3 .. 6 .. 10 .. 18 .. 32 .. 56
0 .. 1 .. 1 .. 2 .. 4 .. 6 .. 12 .. 20 .. 36 .. 64
1 .. 0 .. 1 .. 1 .. 2 .. 4 .. 6 ... 12 .. 20 .. 36
T(4,1) counts these 4 paths, given as vector sums applied to (0,0):
(1,1) + (1,-1) + (2,1);
(2,1) + (1,-1) + (1,-1);
(2,1) + (1,1) + (1,-1);
(1,1) + (2,1) + (1,-1).

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A247049, A247302.

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0;
t[1, 0] = 0; t[1, 1] = 1; t[1, 2] = 0;
t[2, 0] = 1; t[2, 1] = 1; t[2, 2] = 1; t[n_, 0] := t[n, 0] = t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2] + t[n - 2, 0];
t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 2, 1];
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]]
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]]   (* A247301 *)

(row 0, the bottom row):  r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 1, r(1) = 0, r(2) = 1, r(3) = 1;
(row 1, the middle row):  r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 0, r(1) = 2, r(2) = 1, r(3) = 2;
(row 2, the top row):  r(n) = 2*r(n-2) + 2*r(n-3), with r(0) = 0, r(1) = 0, r(2) = 1, r(3) = 2.
From Chai Wah Wu, Jan 24 2020: (Start)
a(n) = 2*a(n-6) + 2*a(n-9) for n > 14.
G.f.: (-x^14 - 2*x^11 + x^9 - x^8 - x^7 + x^6 - x^4 - 1)/(2*x^9 + 2*x^6 - 1). (End)

A247311 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 2, 2, 1, 4, 5, 3, 9, 12, 8, 21, 29, 20, 50, 70, 49, 120, 169, 119, 289, 408, 288, 697, 985, 696, 1682, 2378, 1681, 4060, 5741, 4059, 9801, 13860, 9800, 23661, 33461, 23660, 57122, 80782, 57121, 137904, 195025, 137903, 332929, 470832, 332928

Offset: 0

Clark Kimberling, Sep 12 2014

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, and for 0 < i <= n, s(i) is in {0,1,2}, and s(i) - s(i-1) is in {-1,0,1}.
(row 0, the bottom row): A024537;
(row 1, the middle row): A000129;
(row 2, the top row): A048739;
(n-th column sum): A000129.

			First 10 columns:
  0 .. 0 .. 1 .. 3 .. 8 ... 20 .. 49 .. 119 .. 288 .. 696
  0 .. 1 .. 2 .. 5 .. 12 .. 29 .. 70 .. 169 .. 408 .. 985
  1 .. 1 .. 2 .. 4 .. 9 ... 21 .. 50 .. 120 .. 289 .. 697
T(3,2) counts these 3 paths, given as vector sums applied to (0,0):
  (1,1) + (1,1) + (1,0); (1,1) + (1,0) + (1,1); (1,0) + (1,1) + (1,1).

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A247049, A247309, A247310, A000129, A024537, A048739.

t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[1, 2] = 0;
t[n_, 0] := t[n, 0] = t[n - 1, 0] + t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 1] + t[n - 1, 2]
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 2}]]]] (*  array *)
Flatten[Table[t[n, k], {n, 0, 20}, {k, 0, 2}]] (* A247311 *)

A242763 a(n) = 1 for n <= 7; a(n) = a(n-5) + a(n-7) for n>7.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 7, 7, 8, 9, 9, 12, 12, 15, 16, 17, 21, 21, 27, 28, 32, 37, 38, 48, 49, 59, 65, 70, 85, 87, 107, 114, 129, 150, 157, 192, 201, 236, 264, 286, 342, 358, 428, 465, 522, 606, 644, 770, 823, 950, 1071, 1166, 1376

Offset: 1

Vicente Jesús Maniega Granado, Oct 03 2016

Generalized Fibonacci growth sequence using i = 2 as maturity period, j = 5 as conception period, and k = 2 as growth factor.

Maturity period is the number of periods that a Fibonacci tree node needs for being able to start developing branches. Conception period is the number of periods in a Fibonacci tree node needed to develop new branches since its maturity. Growth factor is the number of additional branches developed by a Fibonacci tree node, plus 1, and equals the base of the exponential series related to the given tree if maturity factor would be zero. Standard Fibonacci would use 1 as maturity period, 1 as conception period, and 2 as growth factor as the series becomes equal to 2^n with a maturity period of 0. Related to Lucas sequences.

			For n = 13 the a(13) = a(8) + a(6) = 2 + 1 = 3.

Colin Barker, Table of n, a(n) for n = 1..1000
Julia Collins, Fibonacci Tree
Fractal Foundation, Fibonacci Fractals
D. H. Lehmer, An extended theory of Lucas' functions, Annals of Mathematics, Second Series, Vol. 31, No. 3 (Jul., 1930), pp. 419-448.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,0,1).

Cf. A000079 (i = 0, j = 1, k = 2), A000244 (i = 0, j = 1, k = 3), A000302 (i = 0, j = 1, k = 4), A000351 (i = 0, j = 1, k = 5), A000400 (i = 0, j = 1, k = 6), A000420 (i = 0, j = 1, k = 7), A001018 (i = 0, j = 1, k = 8), A001019 (i = 0, j = 1, k = 9), A011557 (i = 0, j = 1, k = 10), A001020 (i = 0, j = 1, k = 11), A001021 (i = 0, j = 1, k = 12), A016116 (i = 0, j = 2, k = 2), A108411 (i = 0, j = 2, k = 3), A213173 (i = 0, j = 2, k = 4), A074872 (i = 0, j = 2, k = 5), A173862 (i = 0, j = 3, k = 2), A127975 (i = 0, j = 3, k = 3), A200675 (i = 0, j = 4, k = 2), A111575 (i = 0, j = 4, k = 3), A000045 (i = 1, j = 1, k = 2), A001045 (i = 1, j = 1, k = 3), A006130 (i = 1, j = 1, k = 4), A006131 (i = 1, j = 1, k = 5), A015440 (i = 1, j = 1, k = 6), A015441 (i = 1, j = 1, k = 7), A015442 (i = 1, j = 1, k = 8), A015443 (i = 1, j = 1, k = 9), A015445 (i = 1, j = 1, k = 10), A015446 (i = 1, j = 1, k = 11), A015447 (i = 1, j = 1, k = 12), A000931 (i = 1, j = 2, k = 2), A159284 (i = 1, j = 2, k = 3), A238389 (i = 1, j = 2, k = 4), A097041 (i = 1, j = 2, k = 10), A079398 (i = 1, j = 3, k = 2), A103372 (i = 1, j = 4, k = 2), A103373 (i = 1, j = 5, k = 2), A103374 (i = 1, j = 6, k = 2), A000930 (i = 2, j = 1, k = 2), A077949 (i = 2, j = 1, k = 3), A084386 (i = 2, j = 1, k = 4), A089977 (i = 2, j = 1, k = 5), A178205 (i = 2, j = 1, k = 11), A103609 (i = 2, j = 2, k = 2), A077953 (i = 2, j = 2, k = 3), A226503 (i = 2, j = 3, k = 2), A122521 (i = 2, j = 6, k = 2), A003269 (i = 3, j = 1, k = 2), A052942 (i = 3, j = 1, k = 3), A005686 (i = 3, j = 2, k = 2), A237714 (i = 3, j = 2, k = 3), A238391 (i = 3, j = 2, k = 4), A247049 (i = 3, j = 3, k = 2), A077886 (i = 3, j = 3, k = 3), A003520 (i = 4, j = 1, k = 2), A108104 (i = 4, j = 2, k = 2), A005708 (i = 5, j = 1, k = 2), A237716 (i = 5, j = 2, k = 3), A005709 (i = 6, j = 1, k = 2), A122522 (i = 6, j = 2, k = 2), A005710 (i = 7, j = 1, k = 2), A237718 (i = 7, j = 2, k = 3), A017903 (i = 8, j = 1, k = 2).

[n le 7 select 1 else Self(n-5)+Self(n-7): n in [1..70]]; // Vincenzo Librandi, Nov 30 2016

LinearRecurrence[{0, 0, 0, 0, 1, 0, 1}, {1, 1, 1, 1, 1, 1, 1}, 70] (*  or *)
CoefficientList[ Series[(1+x+x^2+x^3+x^4)/(1-x^5-x^7), {x, 0, 70}], x] (* Robert G. Wilson v, Nov 25 2016 *)
nxt[{a_,b_,c_,d_,e_,f_,g_}]:={b,c,d,e,f,g,a+c}; NestList[nxt,{1,1,1,1,1,1,1},70][[;;,1]] (* Harvey P. Dale, Oct 22 2024 *)

Vec(x*(1+x+x^2+x^3+x^4)/((1-x+x^2)*(1+x-x^3-x^4-x^5)) + O(x^100)) \\ Colin Barker, Oct 27 2016

@CachedFunction # a = A242763
def a(n): return 1 if n<8 else a(n-5) +a(n-7)
[a(n) for n in range(1,76)] # G. C. Greubel, Oct 23 2024

Generic a(n) = 1 for n <= i+j; a(n) = a(n-j) + (k-1)*a(n-(i+j)) for n>i+j where i = maturity period, j = conception period, k = growth factor.
G.f.: x*(1+x+x^2+x^3+x^4) / ((1-x+x^2)*(1+x-x^3-x^4-x^5)). - Colin Barker, Oct 09 2016
Generic g.f.: x*(Sum_{l=0..j-1} x^l) / (1-x^j-(k-1)*x^(i+j)), with i > 0, j > 0 and k > 1.

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A247301 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,0) to (n,k), where 0 >= k <= 2, consisting of segments given by the vectors (1,1), (2,1), (1,-1).

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A247311 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 2, consisting of segments given by the vectors (1,1), (1,0), (1,-1).

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A242763 a(n) = 1 for n <= 7; a(n) = a(n-5) + a(n-7) for n>7.

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