A104144 -id:A104144 - cp's OEIS frontend

A255529 Indices of primes in the 9th-order Fibonacci number sequence, A104144.

10, 19, 878

Offset: 1

Robert Price, Feb 24 2015

a(4) > 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, A079262, A104144.

a={0,0,0,0,0,0,0,0,1}; step=9; lst={}; For[n=step,n<=1000,n++, sum=Plus@@a; If[PrimeQ[sum], AppendTo[lst,n]]; a=RotateLeft[a]; a[[step]]=sum]; lst

a104144(n) = polcoeff(x^8/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9) + O(x^(n+1)), n);
lista(nn) = {for (n=1, nn, if (isprime(a104144(n)), print1(n, ", ")););} \\ Michel Marcus, Feb 27 2015

A122265 10th-order Fibonacci numbers: a(n+1) = a(n)+...+a(n-9) with a(0) = ... = a(8) = 0, a(9) = 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4088, 8172, 16336, 32656, 65280, 130496, 260864, 521472, 1042432, 2083841, 4165637, 8327186, 16646200, 33276064, 66519472, 132973664, 265816832, 531372800, 1062224128

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 18 2006

nonn

The (1,10)-entry of the matrix M^n, where M is the 10 X 10 matrix {{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}}.

Robert Price, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
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).

Cf. A000322, A001591, A001592, A079262.
Cf. A001591, A001592, A122189, A079262, A104144, A168082, A168083, A168084. - Richard Choulet, Feb 22 2010
Cf. A257227, A257228 for primes in this sequence.

with(linalg): p:=-1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9+x^10: M[1]:=transpose(companion(p,x)): for n from 2 to 40 do M[n]:=multiply(M[n-1],M[1]) od: seq(M[n][1,10],n=1..40);
k:=10:for n from 0 to 50 do l(n):=sum((-1)^i*binomial(n-k+1-k*i,i)*2^(n-k+1-(k+1)*i),i=0..floor((n-k+1)/(k+1)))-sum((-1)^i*binomial(n-k-k*i,i)*2^(n-k-(k+1)*i),i=0..floor((n-k)/(k+1))):od:seq(l(n),n=0..50);k:=10:a:=taylor((z^(k-1)-z^(k))/(1-2*z+z^(k+1)),z=0,51);for p from 0 to 50 do j(p):=coeff(a,z,p):od :seq(j(p),p=0..50); # Richard Choulet, Feb 22 2010

M = {{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}}; v[1] = {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}; v[n_] := v[n] = M.v[n - 1]; a = Table[Floor[v[n][[1]]], {n, 1, 50}]
a={1,0,0,0,0,0,0,0,0,0};Flatten[Prepend[Table[s=Plus@@a;a=RotateLeft[a];a[[ -1]]=s,{n,60}],Table[0,{m,Length[a]-1}]]] (* Vladimir Joseph Stephan Orlovsky, Nov 18 2009 *)
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, 50]  (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
With[{nn=10},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* Harvey P. Dale, Aug 17 2013 *)

a(n) = Sum_{j=1..10} a(n-j) for n>=10; a(n) = 0 for 0<=n<=8, a(9) = 1 (follows from the minimal polynomial of M; a Maple program based on this recurrence relation is much slower than the given Maple program, based on the definition).
G.f.: -x^9/(-1+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
Another form of the g.f. f: f(z)=(z^(k-1)-z^(k))/(1-2*z+z^(k+1)) with k=10. Then a(n)=sum((-1)^i*binomial(n-k+1-k*i,i)*2^(n-k+1-(k+1)*i),i=0..floor((n-k+1)/(k+1)))-sum((-1)^i*binomial(n-k-k*i,i)*2^(n-k-(k+1)*i),i=0..floor((n-k)/(k+1))) with k=10 and sum(alpha(i),i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010

Edited by N. J. A. Sloane, Oct 29 2006 and Mar 05 2011

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

A251751 9-step Fibonacci sequence starting with 0,0,1,0,0,0,0,0,0.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 4, 7, 14, 28, 56, 112, 224, 448, 895, 1788, 3572, 7137, 14260, 28492, 56928, 113744, 227264, 454080, 907265, 1812742, 3621912, 7236687, 14459114, 28889736, 57722544, 115331344, 230435424, 460416768, 919926271, 1838039800

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, A251750, A251752.

Cf. A255534 (Indices of primes in this sequence).

LinearRecurrence[Table[1, {9}], {0, 0, 1, 0, 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^2*(-1+x+x^2+x^3+x^4+x^5+x^6)/(-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) = A172319(n-2)-2*A172319(n-3)+A172319(n-9). - R. J. Mathar, Mar 28 2025

A251752 9-step Fibonacci sequence starting with 0,1,0,0,0,0,0,0,0.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 6, 12, 24, 48, 96, 192, 384, 767, 1532, 3061, 6116, 12220, 24416, 48784, 97472, 194752, 389120, 777473, 1553414, 3103767, 6201418, 12390616, 24756816, 49464848, 98832224, 197469696, 394550272, 788323071, 1575092728

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, A251750, A251751.

Cf. A255536 (Indices of primes in this sequence).

LinearRecurrence[Table[1, {9}], {0, 1, 0, 0, 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*(-1+x+x^2+x^3+x^4+x^5+x^6+x^7)/(-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) = A172319(n-1)-2*A172319(n-2)+A172319(n-9). - R. J. Mathar, Mar 28 2025

A105753 Lexicographically earliest sequence of positive integers with the property that a(a(n)) = a(1)+a(2)+...+a(n).

Original entry on oeis.org

1, 3, 4, 8, 6, 22, 9, 16, 53, 11, 133, 13, 279, 15, 573, 69, 18, 1233, 20, 2486, 23, 44, 4995, 25, 10059, 27, 20145, 29, 40319, 31, 80669, 33, 161371, 35, 322777, 37, 645591, 39, 1291221, 41, 2582483, 43, 5165009, 5039, 46, 10335103, 48

Offset: 1

Eric Angelini, Aug 13 2006

nonn

The Fibonacci 9-step numbers referenced in the Noe-Post paper are in A104144. - T. D. Noe, Oct 27 2008

			Sequence reads from the beginning:
- at position a(1)=1 we see the sum of all previously written terms [indeed, nil + 1=1]
- at position a(2)=3 we see the sum of all previously written terms [indeed, 1+ 3=4]
- at position a(3)=4 we see the sum of all previously written terms [indeed, 1+3+4=8]
- at position a(4)=8 we see the sum of all previously written terms [indeed, 1+3+4+8=16]
- at position a(5)=6 we see the sum of all previously written terms [indeed, 1+3+4+8+6=22]
- at position a(6)=22 we see the sum of all previously written terms [indeed, 1+3+4+8+6+22=44 and 44 is the 22nd term of S]
etc.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences 8 (2005), Article 05.4.4

Cf. A121053, A121173, A121174, A121175, A104144.

More terms from Max Alekseyev, Aug 14 2006

Edited by Max Alekseyev, Mar 08 2015

A251748 9-step Fibonacci sequence starting with 0,0,0,0,0,1,0,0,0.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 4, 8, 16, 32, 63, 126, 252, 504, 1007, 2012, 4020, 8032, 16048, 32064, 64065, 128004, 255756, 511008, 1021009, 2040006, 4075992, 8143952, 16271856, 32511648, 64959231, 129790458, 259325160, 518139312, 1035257615, 2068475224

Offset: 0

Arie Bos, Dec 07 2014

The only primes in this sequence whose indices are less than 2*10^5 are 2 and 65865769729, which correspond to indices of 10 and 45. - Robert Price, Feb 24 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, A251746, A251747, A251749, A251750, A251751, A251752.

LinearRecurrence[Table[1, {9}], {0, 0, 0, 0, 0, 1, 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^5*(-1+x+x^2+x^3)/(-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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A255529 Indices of primes in the 9th-order Fibonacci number sequence, A104144.

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A122265 10th-order Fibonacci numbers: a(n+1) = a(n)+...+a(n-9) with a(0) = ... = a(8) = 0, a(9) = 1.

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

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

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A251751 9-step Fibonacci sequence starting with 0,0,1,0,0,0,0,0,0.

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A251752 9-step Fibonacci sequence starting with 0,1,0,0,0,0,0,0,0.

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A105753 Lexicographically earliest sequence of positive integers with the property that a(a(n)) = a(1)+a(2)+...+a(n).

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