A127193 -id:A127193 - cp's OEIS frontend

A256498 Indices of primes in the 9th-order Fibonacci number sequence, A127193.

11, 15, 25, 31, 36, 37, 42, 45, 48, 75, 149, 156, 160, 182, 268, 444, 581, 1025, 1125, 2504, 6900, 10924, 11807, 25262, 26774, 28739, 29367, 34902, 43345, 53878, 74473, 107070, 170300, 178994

Offset: 1

Robert Price, Mar 31 2015

a(35) > 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.
OEIS Wiki, Index of Fibonacci Numbers and Variants
OEIS Wiki, Index to OEIS Section Fi

Cf. A127193, A256499.

a={1,1,1,1,1,1,1,1,1}; step=9; lst={}; For[n=step+1,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst

A256499 Primes in the 9th-order Fibonacci numbers A127193.

Original entry on oeis.org

17, 257, 260609, 16580657, 527972353, 1054904321, 33590968001, 267934222337, 2137144350721, 279308966066204560897, 4904838477959792746889087209953222309396481, 623502124433801536413569315448615191583313921, 9936775914719167257001281976859570231260282873

Offset: 1

Robert Price, Mar 31 2015

nonn

a(14) is too large to display here. It has 53 digits and is the 182nd term in A127193.

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.
OEIS Wiki, Index of Prime Fibonacci Numbers and Variants
OEIS Wiki, Index to OEIS Section Fi

Cf. A127193, A256498.

a={1,1,1,1,1,1,1,1,1}; step=9; offset=1; lst={}; For[n=step+offset,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,sum]]; a=RotateLeft[a]; a[[step]]=sum]; lst

A166444 a(0) = 0, a(1) = 1 and for n > 1, a(n) = sum of all previous terms.

Original entry on oeis.org

0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 0

Robert G. Wilson v, Oct 13 2009

Essentially a duplicate of A000079. - N. J. A. Sloane, Oct 15 2009
a(n) is the number of compositions of n into an odd number of parts.
Also 0 together with A011782. - Omar E. Pol, Oct 28 2013
Inverse INVERT transform of A001519. - R. J. Mathar, Dec 08 2022

			G.f. = x + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 16*x^6 + 32*x^7 + 64*x^8 + 128*x^9 + ...

Indranil Ghosh, Table of n, a(n) for n = 0..3317
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000045, A000079, A000213, A000288, A000322, A000383, A001519.
Cf. A011782, A034008, A060455, A123526, A127193, A127194, A127624.
Cf. A131577, A163551.

[n le 1 select n else 2^(n-2): n in [0..40]]; // G. C. Greubel, Jul 27 2024

a:= n-> `if`(n<2, n, 2^(n-2)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 02 2021

a[0] = 0; a[1] = 1; a[n_] := a[n] = Plus @@ Array[a, n - 1]; Array[a, 35, 0]

[(2^n +2*int(n==1) -int(n==0))/4 for n in range(41)] # G. C. Greubel, Jul 27 2024

a(n) = A000079(n-1) for n > 0.
O.g.f.: x*(1 - x) / (1 - 2*x) = x / (1 - x / (1 - x)).
a(n) = (1-n) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
E.g.f.: (exp(2*x) + 2*x - 1)/4. - Stefano Spezia, Aug 07 2022

A125950 a(0)=a(1)=...=a(9)=1; a(n) = - a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) - a(n-9) - a(n-10).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 16, 18, 22, 25, 30, 35, 41, 49, 57, 67, 79, 93, 109, 129, 151, 178, 209, 246, 290, 340, 401, 471, 554, 652, 767, 902, 1061, 1248, 1468, 1727, 2031, 2390, 2810, 3306, 3889, 4574, 5381, 6329

Offset: 0

Luis A Restrepo (luisiii(AT)mac.com), Feb 04 2007

a(n) = O(n^c), where c is the larger real root of x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, 1.176280818..., the smallest known Salem constant.

Wolfram, S., A New Kind of Science. Champaign, IL: Wolfram Media, pp. 82-92, 2002.

E. Ghate and E. Hironaka, The Arithmetic And Geometry Of Salem Numbers, Bull. Amer. Math. Soc. 38 (2001), 293-314.
Eric Weisstein's World of Mathematics, MathWorld: Salem Constants
Eric Weisstein's World of Mathematics, MathWorld: Substitution System
Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1,1,1,1,0,-1,-1). [From _R. J. Mathar_, Jun 30 2010]

Cf. A070178, A029826, A107480, A127193, A127194, A127624.

LinearRecurrence[{-1,0,1,1,1,1,1,0,-1,-1},{1,1,1,1,1,1,1,1,1,1},70] (* Harvey P. Dale, May 31 2013 *)

G.f.: ( 1+2*x+2*x^2+x^3-x^5-2*x^6-3*x^7-3*x^8-2*x^9 ) / ( 1+x-x^3-x^4-x^5-x^6-x^7+x^9+x^10 ). [R. J. Mathar, Jun 30 2010]

Edited by Don Reble, Mar 09 2007

A127194 A 10th-order Fibonacci sequence.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 19, 37, 73, 145, 289, 577, 1153, 2305, 4609, 9217, 18424, 36829, 73621, 147169, 294193, 588097, 1175617, 2350081, 4697857, 9391105, 18772993, 37527562, 75018295, 149962969, 299778769, 599263345, 1197938593

Offset: 1

Luis A Restrepo (luisiii(AT)hotmail.com), Jan 11 2007

10th-order Fibonacci constant = 1.999018633...

Robert Price, Table of n, a(n) for n = 1..1000
E. S. Croot, Notes on Linear Recurrence Sequences
M. A. Lerma, Recurrence Relations
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1).

Cf. Fibonacci numbers A000045, tribonacci numbers A000213, tetranacci numbers A000288, pentanacci numbers A000322, hexanacci numbers A000383, heptanacci numbers A060455, octanacci numbers A123526, 9th-order Fibonacci sequence A127193.

Cf. A257073, A257074.

With[{t=Table[1,{10}]},LinearRecurrence[t,t,40]] (* Harvey P. Dale, Nov 12 2013 *)

a(n)=([0,1,0,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0,0; 0,0,0,0,1,0,0,0,0,0; 0,0,0,0,0,1,0,0,0,0; 0,0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,0,1; 1,1,1,1,1,1,1,1,1,1]^(n-1)*[1;1;1;1;1;1;1;1;1;1])[1,1] \\ Charles R Greathouse IV, Jun 15 2015

O.g.f.: x*(-1+x^2+2*x^3+3*x^4+4*x^5+5*x^6+6*x^7+7*x^8+8*x^9) / (-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10). - R. J. Mathar, Nov 23 2007

A127624 An 11th-order Fibonacci sequence: a(n) = a(n-1) + ... + a(n-11).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 21, 41, 81, 161, 321, 641, 1281, 2561, 5121, 10241, 20481, 40951, 81881, 163721, 327361, 654561, 1308801, 2616961, 5232641, 10462721, 20920321, 41830401, 83640321, 167239691, 334397501, 668631281

Offset: 1

Luis A Restrepo (Luisiii(AT)mac.com), Jan 19 2007

The ratio a(n+1)/a(n) approaches the unique real root of r^11 = r^10 + ... + r + 1; r is about 1.99951040197828549144.

All terms have last digit 1.

Robert Price, Table of n, a(n) for n = 1..1000
E. S. Croot, Notes on Linear Recurrence Sequences
M. A. Lerma, Recurrence Relations
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1,1,1).

Cf. Fibonacci numbers A000045, tribonacci numbers A000213, tetranacci numbers A000288, pentanacci numbers A000322, hexanacci numbers A000383, heptanacci numbers A060455, octanacci numbers A123526, 9th-order Fibonacci sequence A127193, 10th-order Fibonacci sequence A127194.

Cf. A257966 (indices of primes in a), A257967 (primes in a).

Module[{nn=11,lr},lr=PadRight[{},nn,1];LinearRecurrence[lr,lr,20]] (* Harvey P. Dale, Feb 04 2015 *)

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

O.g.f: x*(-1+x^2+2*x^3+3*x^4+4*x^5+5*x^6+6*x^7+7*x^8+8*x^9+9*x^10) / (-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11). - R. J. Mathar, Dec 02 2007

Edited by Dean Hickerson, Mar 09 2007

A251746 9-step Fibonacci sequence starting with 0,0,0,0,0,0,0,1,0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 4, 8, 16, 32, 64, 128, 255, 510, 1019, 2036, 4068, 8128, 16240, 32448, 64832, 129536, 258817, 517124, 1033229, 2064422, 4124776, 8241424, 16466608, 32900768, 65736704, 131343872, 262428927, 524340730, 1047648231, 2093232040

Offset: 0

Arie Bos, Dec 07 2014

a(n+9) equals the number of n-length binary words avoiding runs of zeros of lengths 9i+8, (i=0,1,2,...). - Milan Janjic, Feb 26 2015

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

Other 9-step Fibonacci sequences are A104144, A105755, A127193, A251747, A251748, A251749, A251750, A251751, A251752.

Cf. A255530 (Indices of primes in this sequence).

LinearRecurrence[Table[1, {9}], {0, 0, 0, 0, 0, 0, 0, 1, 0}, 44] (* Michael De Vlieger, Dec 09 2014 *)

a(n+9) = a(n)+a(n+1)+a(n+2)+a(n+3)+a(n+4)+a(n+5)+a(n+6)+a(n+7)+a(n+8).
G.f.: x^7*(x-1)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9) . - R. J. Mathar, Mar 28 2025
a(n) = A104144(n+1)-A104144(n). - R. J. Mathar, Mar 28 2025

A251747 9-step Fibonacci sequence starting with 0,0,0,0,0,0,1,0,0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1015, 2028, 4052, 8096, 16176, 32320, 64576, 129025, 257796, 515084, 1029153, 2056278, 4108504, 8208912, 16401648, 32770976, 65477376, 130825727, 261393658, 522272232, 1043515311, 2084974344

Offset: 0

Arie Bos, Dec 07 2014

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

Other 9-step Fibonacci sequences are A104144, A105755, A127193, A251746, A251748, A251749, A251750, A251751, A251752.

Cf. A255531 (Indices of primes in this sequence).

LinearRecurrence[Table[1, {9}], {0, 0, 0, 0, 0, 0, 1, 0, 0}, 44] (* Michael De Vlieger, Dec 09 2014 *)

a(n+9) = a(n)+a(n+1)+a(n+2)+a(n+3)+a(n+4)+a(n+5)+a(n+6)+a(n+7)+a(n+8).
G.f.: x^6*(-1+x+x^2)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9) . - R. J. Mathar, Mar 28 2025
a(n) = A104144(n+2)-A104144(n+1)-A104144(n). - R. J. Mathar, Mar 28 2025

A251749 9-step Fibonacci sequence starting with 0,0,0,0,1,0,0,0,0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 4, 8, 16, 31, 62, 124, 248, 496, 991, 1980, 3956, 7904, 15792, 31553, 63044, 125964, 251680, 502864, 1004737, 2007494, 4011032, 8014160, 16012528, 31993503, 63923962, 127721960, 255192240, 509881616, 1018758495, 2035509496

Offset: 0

Arie Bos, Dec 07 2014

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

Other 9-step Fibonacci sequences are A104144, A105755, A127193, A251746, A251747, A251748, A251750, A251751, A251752.

Cf. A255532 (Indices of primes in this sequence).

LinearRecurrence[Table[1, {9}], {0, 0, 0, 0, 1, 0, 0, 0, 0}, 44] (* Michael De Vlieger, Dec 09 2014 *)

a(n+9) = a(n)+a(n+1)+a(n+2)+a(n+3)+a(n+4)+a(n+5)+a(n+6)+a(n+7)+a(n+8).

G.f.: x^4*(-1+x+x^2+x^3+x^4)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9) . - R. J. Mathar, Mar 28 2025

A251750 9-step Fibonacci sequence starting with 0,0,0,1,0,0,0,0,0.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 4, 8, 15, 30, 60, 120, 240, 480, 959, 1916, 3828, 7648, 15281, 30532, 61004, 121888, 243536, 486592, 972225, 1942534, 3881240, 7754832, 15494383, 30958234, 61855464, 123589040, 246934544, 493382496, 985792767, 1969643000

Offset: 0

Arie Bos, Dec 07 2014

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

Other 9-step Fibonacci sequences are A104144, A105755, A127193, A251746, A251747, A251748, A251749, A251751, A251752.

Cf. A255533 (Indices of primes in this sequence).

LinearRecurrence[Table[1, {9}], {0, 0, 0, 1, 0, 0, 0, 0, 0}, 44] (* Michael De Vlieger, Dec 09 2014 *)

a(n+9) = a(n)+a(n+1)+a(n+2)+a(n+3)+a(n+4)+a(n+5)+a(n+6)+a(n+7)+a(n+8).

G.f.: x^3*(-1+x+x^2+x^3+x^4+x^5)/(-1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9) . - R. J. Mathar, Mar 28 2025

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A256498 Indices of primes in the 9th-order Fibonacci number sequence, A127193.

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A256499 Primes in the 9th-order Fibonacci numbers A127193.

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A166444 a(0) = 0, a(1) = 1 and for n > 1, a(n) = sum of all previous terms.

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A125950 a(0)=a(1)=...=a(9)=1; a(n) = - a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7) - a(n-9) - a(n-10).

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A127194 A 10th-order Fibonacci sequence.

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A127624 An 11th-order Fibonacci sequence: a(n) = a(n-1) + ... + a(n-11).

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A251746 9-step Fibonacci sequence starting with 0,0,0,0,0,0,0,1,0.

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A251747 9-step Fibonacci sequence starting with 0,0,0,0,0,0,1,0,0.

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