A108411 -id:A108411 - cp's OEIS frontend

A218540 Reduced third-order Patalan numbers.

1, 1, 1, 5, 10, 66, 154, 1122, 2805, 21505, 55913, 442221, 1179256, 9524760, 25852920, 211993944, 582983346, 4835332458, 13431479050, 112400272050, 314720761740, 2652646420380, 7475639911980, 63380425340700, 179577871798650, 1530003467724498

Offset: 0

R. J. Mathar, Nov 01 2012

Obtained by removing powers of 3 in a systematic manner from the Patalan numbers A025748.

Richard J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (4.19).

Cf. A025748, A073006, A108411.

A218540 := proc(n)
    option remember;
    if n <=2 then
        1;
    elif n = 3 then
        5 ;
    else
        (n-1)*(n-2)*(n+4)*procname(n-1)-3*(3*n-4)*(3*n-7)*(n+2)*procname(n-2)-3*(3*n-10)*(n+4)*(3*n-7)*procname(n-3) ;
        -%/n/(n+2)/(n-1) ;
    end if;
end proc:

a[n_] := 3^(2*n-2-Floor[n/2]) * Pochhammer[2/3, n-1]/n!; a[0] = 1; Array[a, 26, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n) = A025748(n)/A108411(n).
D-finite with recurrence n*(n+2)*(n-1)*a(n) + (n-1)*(n-2)*(n+4)*a(n-1) - 3*(3*n-4)*(3*n-7)*(n+2)*a(n-2) - 3*(3*n-10)*(n+4)*(3*n-7)*a(n-3) = 0, n >= 4.
a(n) ~ 3^(2*n-2-floor(n/2)) / (Gamma(2/3) * n^(4/3)). - Amiram Eldar, Aug 20 2025

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.

A287479 Expansion of g.f. (x + x^2)/(1 + 3*x^2).

Original entry on oeis.org

0, 1, 1, -3, -3, 9, 9, -27, -27, 81, 81, -243, -243, 729, 729, -2187, -2187, 6561, 6561, -19683, -19683, 59049, 59049, -177147, -177147, 531441, 531441, -1594323, -1594323, 4782969, 4782969, -14348907, -14348907, 43046721, 43046721, -129140163, -129140163, 387420489

Offset: 0

Jean-François Alcover and Paul Curtz, Jul 19 2017

This is the inverse binomial transform of A157241.
Successive differences of A157241 begin:
0,   1,   3,   3,   -5,  -21,  -21,    43,   171,   171, ... = A157241
1,   2,   0,  -8,  -16,    0,   64,   128,     0,  -512, ... = A088138
1,  -2,  -8,  -8,   16,   64,   64,  -128,  -512,  -512, ... = A138230
-3, -6,   0,  24,   48,    0, -192,  -384,     0,  1536, ...
-3,  6,  24,  24,  -48, -192, -192,   384,  1536,  1536, ...
9,  18,   0, -72, -144,    0,  576,  1152,     0, -4608, ...
9, -18, -72  -72,  144,  576,  576, -1152, -4608, -4608, ...
...
a(n) is the n-th term of the first column.
Successive differences of a(n) begin:
0,     1,    1,   -3,   -3,    9,     9,   -27,   -27,    81, ...
1,     0,   -4,    0,   12,    0,   -36,     0,   108,     0, ...
-1,   -4,    4,   12,  -12,  -36,    36,   108,  -108,  -324, ...
-3,    8,    8,  -24,  -24,   72,    72,  -216,  -216,   648, ...
11,    0,  -32,    0,   96,    0,  -288,     0,   864,     0, ...
-11, -32,   32,   96,  -96, -288,   288,   864,  -864, -2592, ...
-21,  64,   64, -192, -192,  576,   576, -1728, -1728,  5184, ...
85,    0, -256,    0,  768,    0, -2304,     0,  6912,     0, ...
...
First column appears to be a subsequence of Jacobsthal numbers A001045 (the trisection A082311 is missing), second column is A104538, and third column is A137717.
a(n) = A128019(n-2) for n > 2. - Georg Fischer, Oct 23 2018

Index entries for linear recurrences with constant coefficients, signature (0, -3).

Cf. A001045, A082311, A088138, A104538, A108411, A128019, A137717, A138230, A140429, A157241.

Join[{0}, LinearRecurrence[{0, -3}, {1, 1}, 40]]
(* or, computation from b = A157241 : *)
b[n_] := (Switch[Mod[n, 3], 0, (-1)^((n + 3)/3), 1, (-1)^((n + 5)/3), 2, (-1)^((n + 4)/3)*2]*2^n + 1)/3; tb = Table[b[n], {n, 0, 40}]; Table[ Differences[tb, n], {n, 0, 40}][[All, 1]]

concat([0], Vec((x + x^2)/(1 + 3*x^2) + O(x^40))) \\ Felix Fröhlich, Oct 23 2018

a(n) = -3*a(n-2) for n > 2.

E.g.f.: (1 - cos(sqrt(3)*x) + sqrt(3)*sin(sqrt(3)*x))/3. - Stefano Spezia, Jul 15 2024

A376424 Nonnegative numbers m such that the run lengths in binary expansion of m, say (r_1, ..., r_k), satisfy r_1 + ... + r_i <> r_j + ... + r_k for any i in the interval 1..k-1 and j in the interval 2..k.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 8, 14, 15, 16, 19, 23, 24, 25, 28, 29, 30, 31, 32, 35, 47, 48, 49, 60, 61, 62, 63, 64, 67, 68, 71, 76, 79, 80, 88, 95, 96, 97, 102, 103, 110, 111, 112, 113, 114, 115, 120, 121, 122, 123, 124, 125, 126, 127, 128, 131, 132, 135, 156, 159, 160

Offset: 1

Rémy Sigrist, Sep 22 2024

Visually, if we consider a row of bricks of widths r_1, ..., r_k (in that order) above a row of widths r_k, ..., r_1 (in that order), we never have 4 bricks whose corners meet.

There are A108411(k) terms with k binary digits (ignoring leading zeros).

			The binary expansion of 35 is "100011", the corresponding run lengths are (1, 3, 2); the sums 1, 1+3 are distinct from the sums 3+2, 2, so 35 is a term. Visually, if we consider a row of bricks of widths 1, 3, 2 (in that order) above a row of widths 2, 3, 1 (in that order), we never have 4 bricks whose corners meet:
    .-.-----.---.
    | |     |   |
    .-.-.---.-.-.
    |   |     | |
    .---.-----.-.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A101211, A108411.

toruns(n) = { my (r = []); while (n, my (v = valuation(n+n%2, 2)); n \= 2^v; r = concat(v, r)); r }
is(n) = { if (n, my (r = toruns(n)); setintersect(vector(#r, k, vecsum(r[1..k])), vector(#r, k, vecsum(r[#r+1-k..#r])))==[vecsum(r)], 1); }

A107904 Expansion of (1+6x)/(1-12x^2).

Original entry on oeis.org

1, 6, 12, 72, 144, 864, 1728, 10368, 20736, 124416, 248832, 1492992, 2985984, 17915904, 35831808, 214990848, 429981696, 2579890176, 5159780352, 30958682112, 61917364224, 371504185344, 743008370688, 4458050224128, 8916100448256, 53496602689536, 106993205379072

Offset: 0

Paul Barry, May 27 2005

Fourth binomial transform is A107903.

Index entries for linear recurrences with constant coefficients, signature (0,12).

Cf. A094015, A107903, A108411.

LinearRecurrence[{0,12},{1,6},30] (* Harvey P. Dale, Sep 22 2014 *)

a(n) = ((1+sqrt(3))*(2*sqrt(3))^n + (1-sqrt(3))*(-2*sqrt(3))^n)/2.
a(2n) = 12^n, a(2n+1) = 6*12^n.
a(n) = 2^n*A108411(n+1). - R. J. Mathar, Aug 15 2023
From Amiram Eldar, Dec 06 2024: (Start)
Sum_{n>=0} 1/a(n) = 14/11.
Sum_{n>=0} (-1)^n/a(n) = 10/11. (End)

A227104 a(0)=-1, a(1)=3; a(n+2) = a(n+1) + a(n) + 2*A057078(n+1).

Original entry on oeis.org

-1, 3, 2, 3, 7, 10, 15, 27, 42, 67, 111, 178, 287, 467, 754, 1219, 1975, 3194, 5167, 8363, 13530, 21891, 35423, 57314, 92735, 150051, 242786, 392835, 635623, 1028458, 1664079, 2692539, 4356618, 7049155, 11405775, 18454930, 29860703, 48315635, 78176338, 126491971

Offset: 0

Paul Curtz, Jul 01 2013

a(n+1)/a(n) tends to A001622 (the golden ratio) as n -> infinity.
a(n) and its differences:
.  -1,   3,   2,   3,   7,  10,  15,  27,  42,
.   4,  -1,   1,   4,   3,   5,  12,  15,  25,
.  -5,   2,   3,  -1,   2,   7,   3,  10,  19,
.   7,   1,  -4,   3,   5,  -4,   7,   9,   4,
.  -6,  -5,   7,   2,  -9,  11,   2,  -5,  15,
.   1,  12,  -5, -11,  20,  -9,  -7,  20,  -5,
.  11, -17,  -6,  31, -29,   2,  27, -25,   2,
. -28,  11,  37, -60,  31,  25, -52,  27,  29,
.  39,  26, -97,  91,  -6, -77,  79,   2, -81.
Main diagonal: -(-1)^floor(n/2)*A108411(n).

			a(6) = 2*F(6)-1 = 2*8-1 = 15; a(7) = 2*F(7)+1 = 2*13+1 = 27; a(8) = 2*F(8) = 2*21 = 42.

Index entries for linear recurrences with constant coefficients, signature (0,1,2,1).

Cf. A000045.

a[n_] := (m = Mod[n, 3]; 2*Fibonacci[n] - (3*m - 1)*(m - 2)/2); Table[a[n], {n, 0, 39}]  (* Jean-François Alcover, Jul 02 2013 *)

a(3n) = 2*F(3n)-1, a(3n+1) = 2*F(3n+1)+1, a(3n+2) = 2*F(3n+2), where F=A000045.
a(n+3) = a(n) + 4*F(n+1).
a(n) = A226328(n) + 1 for n>1.
a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) and many others by telescoping the fundamental recurrence.
G.f.: -(1-3*x-3*x^2-2*x^3) / ( (1-x-x^2)*(1+x+x^2) ). [Bruno Berselli, Jul 02 2013]
a(n) = a(n-2) + 2*a(n-3) - a(n-4). [Bruno Berselli, Jul 02 2013]

Edited by Bruno Berselli, Jul 02 2013

cp's OEIS Frontend

A218540 Reduced third-order Patalan numbers.

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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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A287479 Expansion of g.f. (x + x^2)/(1 + 3*x^2).

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A376424 Nonnegative numbers m such that the run lengths in binary expansion of m, say (r_1, ..., r_k), satisfy r_1 + ... + r_i <> r_j + ... + r_k for any i in the interval 1..k-1 and j in the interval 2..k.

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A107904 Expansion of (1+6x)/(1-12x^2).

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A227104 a(0)=-1, a(1)=3; a(n+2) = a(n+1) + a(n) + 2*A057078(n+1).

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