A000288 -id:A000288 - cp's OEIS frontend

A214830 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 8.

1, 8, 8, 17, 33, 58, 108, 199, 365, 672, 1236, 2273, 4181, 7690, 14144, 26015, 47849, 88008, 161872, 297729, 547609, 1007210, 1852548, 3407367, 6267125, 11527040, 21201532, 38995697, 71724269, 131921498, 242641464, 446287231, 820850193, 1509778888

Offset: 0

Abel Amene, Aug 07 2012

See comments in A214727.

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

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,8,8];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 24 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+7*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 24 2019

CoefficientList[Series[(x^2-7*x-1)/(x^3+x^2+x-1), {x, 0, 40}], x] (* Wesley Ivan Hurt, Jun 18 2014 *)
LinearRecurrence[{1,1,1}, {1,8,8}, 40] (* G. C. Greubel, Apr 24 2019 *)

my(x='x+O('x^40)); Vec((1+7*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 24 2019

((1+7*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 24 2019

G.f.: (1+7*x-x^2)/(1-x-x^2-x^3).

a(n) = -A000073(n) + 7*A000073(n+1) + A000073(n+2). - G. C. Greubel, Apr 24 2019

A247192 Indices of primes in the hexanacci numbers sequence A000383.

Original entry on oeis.org

7, 9, 30, 31, 33, 46, 52, 54, 82, 102, 109, 124, 210, 301, 351, 365, 369, 1045, 2044, 2125, 2143, 2815, 4377, 4754, 4893, 7310, 11558, 17602, 17929, 28389, 32100, 44298, 106725, 151678, 197953

Offset: 1

Robert Price, Dec 03 2014

a(36) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A001631, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523, A214825, A235862, A000288, A000322, A000383, A249413.

a={1,1,1,1,1}; For[n=5, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[n]]; a=RotateLeft[a]; a[[5]]=sum]

A248920 Indices of primes in the pentanacci numbers sequence A000322.

Original entry on oeis.org

5, 7, 13, 18, 19, 34, 35, 38, 43, 48, 188, 286, 450, 501, 759, 1446, 2021, 2419, 2997, 3715, 5677, 13566, 46303, 57174, 108844, 117145, 166683, 178863

Offset: 1

Robert Price, Oct 16 2014

a(29) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A001631, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523, A214825, A235862, A000288, A000322.

a={1,1,1,1,1}; For[n=5, n<=1000, n++, sum=Plus@@a; If[PrimeQ[sum], Print[n]]; a=RotateLeft[a]; a[[5]]=sum]

A251654 4-step Fibonacci sequence starting with 0, 1, 1, 0.

Original entry on oeis.org

0, 1, 1, 0, 2, 4, 7, 13, 26, 50, 96, 185, 357, 688, 1326, 2556, 4927, 9497, 18306, 35286, 68016, 131105, 252713, 487120, 938954, 1809892, 3488679, 6724645, 12962170, 24985386, 48160880, 92833081, 178941517, 344920864, 664856342, 1281551804

Offset: 0

Arie Bos, Dec 06 2014

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

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251655, A251656, A251672, A251703, A251704, A251705.

NB. see A251655 for the program and apply it to 0,1,1,0.

LinearRecurrence[Table[1, {4}], {0, 1, 1, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: x*(-1+2*x^2)/(-1+x+x^2+x^3+x^4). - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+2)-2*A000078(n). - R. J. Mathar, Mar 28 2025

A251655 4-step Fibonacci sequence starting with 0, 1, 1, 1.

Original entry on oeis.org

0, 1, 1, 1, 3, 6, 11, 21, 41, 79, 152, 293, 565, 1089, 2099, 4046, 7799, 15033, 28977, 55855, 107664, 207529, 400025, 771073, 1486291, 2864918, 5522307, 10644589, 20518105, 39549919, 76234920, 146947533, 283250477, 545982849, 1052415779, 2028596638

Offset: 0

Arie Bos, Dec 06 2014

Tamás Lengyel and Diego Marques, The 2-adic Order of Some Generalized Fibonacci Numbers, INTEGERS, 17, 2017, A5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251656, A251672, A251703, A251704, A251705.

(see www.jsoftware.com) First construct the generating matrix
   [M=: (#.@}: + {:)\"1&.|: <:/~i.4
1 1 1 1
1 2 2 2
2 3 4 4
4 6 7 8
Given that matrix, one can produce the first 4*250 numbers with
, M(+/ . *)^:(i.250) 0 1 1 1x

LinearRecurrence[Table[1, {4}], {0, 1, 1, 1}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: x*(x-1)*(1+x)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+2)-A000078(n). - R. J. Mathar, Mar 28 2025

A251704 4-step Fibonacci sequence starting with 1, 1, 0, 1.

Original entry on oeis.org

1, 1, 0, 1, 3, 5, 9, 18, 35, 67, 129, 249, 480, 925, 1783, 3437, 6625, 12770, 24615, 47447, 91457, 176289, 339808, 655001, 1262555, 2433653, 4691017, 9042226, 17429451, 33596347, 64759041, 124827065, 240611904, 463794357, 893992367, 1723225693

Offset: 0

Arie Bos, Dec 07 2014

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

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251656, A251703, A251705.

NB. see A251655 for the program and apply it to 1,1,0,1.

LinearRecurrence[Table[1, {4}], {1, 1, 0, 1}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: (1+x)*(x^2+x-1)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A001630(n-2)+A001630(n-1), n>2. - R. J. Mathar, Mar 28 2025

A251705 4-step Fibonacci sequence starting with 1, 1, 1, 0.

Original entry on oeis.org

1, 1, 1, 0, 3, 5, 9, 17, 34, 65, 125, 241, 465, 896, 1727, 3329, 6417, 12369, 23842, 45957, 88585, 170753, 329137, 634432, 1222907, 2357229, 4543705, 8758273, 16882114, 32541321, 62725413, 120907121, 233055969, 449229824, 865918327, 1669111241

Offset: 0

Arie Bos, Dec 07 2014

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

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251656, A251703, A251704.

NB. see A251655 for the program and apply it to 1,1,1,0.

LinearRecurrence[Table[1, {4}], {1, 1, 1, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).

G.f.: (-1+3*x^3+x^2)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025

A253318 Indices of primes in the 7th-order Fibonacci number sequence, A060455.

Original entry on oeis.org

7, 8, 11, 12, 14, 15, 16, 17, 18, 19, 21, 23, 26, 32, 33, 36, 42, 44, 71, 72, 137, 180, 193, 285, 679, 955, 1018, 1155, 1176, 1191, 2149, 2590, 2757, 3364, 4233, 6243, 6364, 7443, 10194, 11254, 13318, 18995, 20478, 22647, 29711, 34769, 61815, 71993, 107494, 135942, 148831

Offset: 1

Robert Price, Dec 30 2014

nonn

a(52) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4

Cf. A001590, A001631, A100683, A231574, A231575, A232542, A214899, A230607, A020992, A232498, A214727, A081172, A214752, A141523, A214825, A235862, A000288, A000322, A000383, A249413, A060455.

a={1,1,1,1,1,1,1}; step=7; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[7]]=sum]; lst

A061451 Array T(n,k) of k-th order Fibonacci numbers read by antidiagonals in up-direction.

Original entry on oeis.org

2, 3, 3, 4, 5, 5, 5, 7, 9, 8, 6, 9, 13, 17, 13, 7, 11, 17, 25, 31, 21, 8, 13, 21, 33, 49, 57, 34, 9, 15, 25, 41, 65, 94, 105, 55, 10, 17, 29, 49, 81, 129, 181, 193, 89, 11, 19, 33, 57, 97, 161, 253, 349, 355, 144, 12, 21, 37, 65, 113, 193, 321, 497, 673, 653, 233

Offset: 1

Frank Ellermann, Jun 11 2001

			2, 3, 5, 8 ... first order, a(1)=2, a(3)=3, a(6)=5, a(10)=8, ...
3, 5, 9,17 ... 2nd order, a(2)=3, a(5)=5, a(9)=9, ...
4, 7,13,25 ... 3rd order, a(4)=4, a(8)=7, ...
5, 9,17,33 ... 4th order, a(7)=5, ...

N. Wirth, Algorithmen und Datenstrukturen, 1975, table 2.15 (ch. 2.3.4)

T. D. Noe, Rows n=1..50 of triangle, flattened

Rows 1..6: A000045, A000213, A000288, A000322, A000383, A060455.

max = 12; Clear[f]; f[k_, n_] /; n > k := f[k, n] = Sum[f[k, n - j], {j, 1, k + 1}]; f[k_, n_] = 1; t = Table[ Table[ f[k, n], {n, k + 1, max}], {k, 1, max}]; Table[ t[[k - n + 1, n]], {k, 1, max - 1}, {n, 1, k}] // Flatten (* Jean-François Alcover, Apr 10 2013 *)

A207539 Dodecanacci numbers (12th-order Fibonacci sequence): a(n) = a(n-1) +...+ a(n-12) with a(0)=...=a(11)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 23, 45, 89, 177, 353, 705, 1409, 2817, 5633, 11265, 22529, 45057, 90102, 180181, 360317, 720545, 1440913, 2881473, 5762241, 11523073, 23043329, 46081025, 92150785, 184279041, 368513025, 736935948, 1473691715

Offset: 0

Michael Burkhart, Feb 18 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1,1).

Cf. A000045, A000213, A000288, A127624, A168083.

f12:=proc(n) option remember: if n<=12 then 1: else add(f12(n-i),i=1..12): fi: end:

LinearRecurrence[Table[1, {12}], Table[1, {12}], 100]

x='x+O('x^50); Vec((1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10-x^11 +10*x^12)/(1-2*x+x^13)) \\ G. C. Greubel, Jul 28 2017

G.f.: (1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10-x^11 +10*x^12)/(1 -2*x +x^13).

cp's OEIS Frontend

A214830 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 8.

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A247192 Indices of primes in the hexanacci numbers sequence A000383.

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A248920 Indices of primes in the pentanacci numbers sequence A000322.

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A251654 4-step Fibonacci sequence starting with 0, 1, 1, 0.

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J

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A251655 4-step Fibonacci sequence starting with 0, 1, 1, 1.

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J

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A251704 4-step Fibonacci sequence starting with 1, 1, 0, 1.

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J

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A251705 4-step Fibonacci sequence starting with 1, 1, 1, 0.

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J

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A253318 Indices of primes in the 7th-order Fibonacci number sequence, A060455.

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