A159284 -id:A159284 - cp's OEIS frontend

A237718 9-distance Pell numbers.

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 15, 17, 25, 27, 39, 41, 57, 59, 79, 89, 113, 139, 167, 217, 249, 331, 367, 489, 545, 715, 823, 1049, 1257, 1547, 1919, 2281, 2897, 3371, 4327, 5017, 6425, 7531, 9519

Offset: 0

Sergio Falcon, Feb 12 2014

			a(9)=2a(0)+a(7)=3; a(10)=2a(1)+a(8)=3; a(11)=2a(2)+a(9)=5.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,0,0,0,2).

Cf. A000129, A122522, A159284, A237714, A237716, A237717.

For[j = 0, j < 9, j++, a[j] = 1]
For[j = 9, j < 51, j++, a[j] = 2 a[j - 9] + a[j - 2]]
Table[a[j], {j, 0, 50}]
CoefficientList[Series[(1 + x)/(1 - x^2 - 2 x^9), {x,0,50}], x] (* G. C. Greubel, May 01 2017 *)

Vec((1+x)/(1-x^2-2*x^9)+O(x^99)) \\ Charles R Greathouse IV, Mar 06 2014

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=1, a(5)=1, a(6)=1, a(7)=1, a(8)=1; a(n) = 2*a(n-9) + a(n-2) for n>=9.
G.f. (1+x)/(1-x^2-2x^9).
a(2*n) = Sum_{j=0..n/9} Binomial[n-7j, 2j]*2^{2j} + Sum_{j=0..(n-5)/9} Binomial[n-4-7j, 2j+1]*2^{2j+1}.
a(2*n+1) = Sum_{j=0..n/9} Binomial[n-7j, 2j]*2^{2j} + Sum_{j=0..(n-4)/9} Binomial[n-3-7j, 2j+1]*2^{2j+1}.

A238389 Expansion of (1+x)/(1-x^2-3*x^3).

Original entry on oeis.org

1, 1, 1, 4, 4, 7, 16, 19, 37, 67, 94, 178, 295, 460, 829, 1345, 2209, 3832, 6244, 10459, 17740, 29191, 49117, 82411, 136690, 229762, 383923, 639832, 1073209, 1791601, 2992705, 5011228, 8367508, 13989343, 23401192, 39091867, 65369221, 109295443

Offset: 0

Sergio Falcon, Feb 26 2014

			a(3) = 3*a(0)+a(1) = 4; a(4) = 3*a(1)+a(2) = 4; a(5) = 3*a(2)+a(3) = 7.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,3).

Cf. A006190, A134816, A159284, A213713.

[n le 3 select 1 else Self(n-2) +3*Self(n-3): n in [1..41]]; // G. C. Greubel, May 09 2021

a:= n-> (<<0|1|0>, <0|0|1>, <3|1|0>>^n.<<(1$3)>>)[(1$2)]:
seq(a(n), n=0..44);  # Alois P. Heinz, May 09 2021

(* First program *)
For[j=0, j<3, j++, a[j] = 1]
For[j=3, j<51, j++, a[j] = 3a[j-3] + a[j-2]]
Table[a[j], {j, 0, 50}]
(* Second program *)
CoefficientList[Series[(1+x)/(1-x^2-3x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 16 2014 *)
LinearRecurrence[{0,1,3},{1,1,1},40] (* Harvey P. Dale, Feb 28 2023 *)

Vec((1+x)/(1-x^2-3*x^3)+O(x^99)) \\ Charles R Greathouse IV, Mar 06 2014

def A238389_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)/(1-x^2-3*x^3) ).list()
A238389_list(40) # G. C. Greubel, May 09 2021

a(0)=1, a(1)=1, a(2)=1; for n>2, a(n) = a(n-2) + 3*a(n-3).
a(2n) = Sum_{j=0}^{n/3} binomial(n-j,2j)*3^(2j) + Sum_{j=0}^{(n-2)/3} binomial(n-1-j,2j+1)*3^(2j+1).
a(2n+1) = Sum_{j=0}^{n/3} binomial(n-j,2j)*3^(2j) + Sum_{j=0}^{(n-1)/3} binomial(n-j,2j+1)*3^(2j+1).
a(n) = |A106855(n)| + |A106855(n-1)| . - R. J. Mathar, Mar 13 2014

Terms corrected by Charles R Greathouse IV, Mar 06 2014

A107854 G.f. x(x^2+1)(x^3-x-1)/((2x^3+x^2-1)(x^4+1)).

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 5, 8, 11, 19, 29, 42, 67, 99, 149, 232, 347, 531, 813, 1226, 1875, 2851, 4325, 6600, 10027, 15251, 23229, 35306, 53731, 81763, 124341, 189224, 287867, 437907, 666317, 1013642, 1542131, 2346275, 3569413, 5430536, 8261963, 12569363

Offset: 0

Creighton Dement, May 25 2005

The sequence A078028 is given by 1em[I* ]forzapseq and is from the same "batch" (i.e., corresponding to the same floretion and symmetry settings) as A107849, A107850, A107851, A107852, A107853 and (a(n)).

Floretion Algebra Multiplication Program, FAMP Code: 1dia[I]forzapseq[(.5i' + .5j' + .5'ki' + .5'kj')*(.5'i + .5'j + .5'ik' + .5'jk')], 1vesforzap = A000004

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

Cf. A107849, A107850, A107851, A107852, A107853, A078028.

CoefficientList[Series[x(x^2+1)(x^3-x-1)/((2x^3+x^2-1)(x^4+1)),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,2,-1,0,1,2},{0,1,1,2,3,3,5},50] (* Harvey P. Dale, Jun 21 2022 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 2,1,0,-1,2,1,0]^n*[0;1;1;2;3;3;5])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = A159284(n) + A014017(n+5).

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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A237718 9-distance Pell numbers.

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A238389 Expansion of (1+x)/(1-x^2-3*x^3).

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A107854 G.f. x*(x^2+1)*(x^3-x-1)/((2*x^3+x^2-1)*(x^4+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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A387267 Number of dissections of a convex n-gon into quadrilaterals and pentagons by strictly disjoint diagonals.

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A107854 G.f. x(x^2+1)(x^3-x-1)/((2x^3+x^2-1)(x^4+1)).