A001592 -id:A001592 - cp's OEIS frontend

A251707 6-step Fibonacci sequence starting with (0,0,0,1,0,0).

0, 0, 0, 1, 0, 0, 1, 2, 4, 8, 15, 30, 60, 119, 236, 468, 928, 1841, 3652, 7244, 14369, 28502, 56536, 112144, 222447, 441242, 875240, 1736111, 3443720, 6830904, 13549664, 26876881, 53312520, 105749800, 209763489, 416083258, 825335612, 1637121560, 3247366239

Offset: 0

Arie Bos, Dec 07 2014

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

Other 6-step Fibonacci sequences are A001592, A074584, A251706, A251708, A251709.

LinearRecurrence[Table[1, {6}], {0, 0, 0, 1, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

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

G.f.: x^3*(-1+x+x^2)/(-1+x+x^2+x^3+x^4+x^5+x^6) . - R. J. Mathar, Feb 27 2023

A251708 6-step Fibonacci sequence starting with (0,0,1,0,0,0).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 2, 4, 7, 14, 28, 56, 111, 220, 436, 865, 1716, 3404, 6752, 13393, 26566, 52696, 104527, 207338, 411272, 815792, 1618191, 3209816, 6366936, 12629345, 25051352, 49691432, 98567072, 195515953, 387822090, 769277244, 1525925143, 3026798934

Offset: 0

Arie Bos, Dec 07 2014

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

Other 6-step Fibonacci sequences are A001592, A074584, A251706, A251707, A251709.

LinearRecurrence[Table[1, {6}], {0, 0, 1, 0, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

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

G.f.: x^2*(-1+x+x^2+x^3)/(-1+x+x^2+x^3+x^4+x^5+x^6) . - R. J. Mathar, Feb 27 2023

A108566 a(0) = 0, a(1) = a(2) = 1, a(3) = 2, a(4) = 4, a(5) = 8, for n>4: a(n+1) = SORT[ a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5)], where SORT places digits in ascending order and deletes 0's.

Original entry on oeis.org

0, 1, 1, 2, 4, 8, 16, 23, 45, 89, 158, 339, 67, 127, 258, 138, 178, 117, 588, 146, 1245, 1224, 3489, 689, 1378, 1178, 239, 1789, 2678, 1579, 1488, 1589, 2369, 11249, 2259, 2335, 12289, 239, 347, 12788, 2357, 3355, 13357, 23344, 45558, 1579, 5589

Offset: 0

Jonathan Vos Post, Jun 10 2005

Extended by T. D. Noe, who also found that verified that the maximum is attained at a(48968063)=12336789999. The periodic part of the sequence begins with a(4847516) and has length 156501072. So the maximum is in the periodic part. Primes include: a(3) = 2, a(7) = 23, a(9) = 89, a(12) = 67, a(13) = 127, a(27) = 1789, a(29) = 1579, a(36) = 12289, a(37) = a(26) = 239, a(38) = 347, a(40) = 2357, a(45) = 1579, a(58) = 25579, a(59) = 23459. Semiprimes include: a(4) = 4 = 2^2, a(10) = 158 = 2 * 79, a(11) = 339 = 3 * 113, a(16) = 178 = 2 * 89, a(19) = 146 = 2 * 73, a(22) = 3489 = 3 * 1163, a(23) = 689 = 13 * 53, a(31) = 1589 = 7 * 227, a(32) = 2369 = 23 * 103, a(33) = 11249 = 7 * 1607, a(35) = 2335 = 5 * 467, a(47) = 22789 = 13 * 1753, a(50) = 178999 = 19 * 9421, a(54) = 14567 = 7 * 2081, a(55) = 23469 = 3 * 7823, a(57) = 22467 = 3 * 7489, a(60) = 12499 = 29 * 431, a(63) = 1477 = 7 * 211, a(66) = 799 = 17 * 47.

			a(7) = SORT[a(2) + a(3) + a(4) + a(5) + a(6) + a(7)] = SORT[1 + 1 + 2 + 4 + 8 + 16] = SORT[32] = 23.

Richard I. Hess, Problem 920: sorted Fibonacci sequence, Pi Mu Epsilon Journal, Vol. 10 (Fall 1998) No. 9, pp. 754-755.

Cf. A001592, A069638, A107281, A108564, A108565, 108567-108573.

nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,FromDigits[Sort[IntegerDigits[Total[{a,b,c,d,e,f}]]]]}; NestList[nxt,{0,1,1,2,4,8},50][[All,1]] (* Harvey P. Dale, May 05 2022 *)

Sorted hexanacci numbers, a.k.a. sorted Fibonacci 6-step sequence.

A247506 Generalized Fibonacci numbers: square array A(n,k) read by ascending antidiagonals, A(n,k) = [x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0, k>=0).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 4, 5, 1, 0, 1, 1, 2, 4, 7, 8, 1, 0, 1, 1, 2, 4, 8, 13, 13, 1, 0, 1, 1, 2, 4, 8, 15, 24, 21, 1, 0, 1, 1, 2, 4, 8, 16, 29, 44, 34, 1, 0, 1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1, 0

Offset: 0

Peter Luschny, Nov 02 2014

			[n\k] [0][1][2][3][4] [5] [6] [7]  [8]  [9] [10]  [11]  [12]
   [0] 1, 0, 0, 0, 0,  0,  0,  0,   0,   0,   0,    0,    0
   [1] 1, 1, 1, 1, 1,  1,  1,  1,   1,   1,   1,    1,    1
   [2] 1, 1, 2, 3, 5,  8, 13, 21,  34,  55,  89,  144,  233  [A000045]
   [3] 1, 1, 2, 4, 7, 13, 24, 44,  81, 149, 274,  504,  927  [A000073]
   [4] 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401,  773, 1490  [A000078]
   [5] 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464,  912, 1793  [A001591]
   [6] 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492,  976, 1936  [A001592]
   [7] 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000  [A066178]
   [8] 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028  [A079262]
   [.] .  .  .  .  .   .   .   .    .    .    .     .     .
  [oo] 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048  [A011782]
.
As a triangular array, starts:
  1,
  1, 0,
  1, 1, 0,
  1, 1, 1, 0,
  1, 1, 2, 1, 0,
  1, 1, 2, 3, 1, 0,
  1, 1, 2, 4, 5, 1, 0,
  1, 1, 2, 4, 7, 8, 1, 0,
  1, 1, 2, 4, 8, 13, 13, 1, 0,
  1, 1, 2, 4, 8, 15, 24, 21, 1, 0,
  ...

Stefano Spezia, First 140 antidiagonals of the array, flattened
Harold R. Parks and Dean C. Wills, Sum of k-bonacci Numbers, arXiv:2208.01224 [math.CO], 2022. See p. 5.

Cf. A247505, A011782, A000045, A000073, A000078, A001591, A001592, A066178, A079262.

Cf. A048887, A092921, A144406.

A := (n,k) -> coeff(series((1-add(x^j, j=1..n))^(-1),x,k+2),x,k):
seq(print(seq(A(n,k), k=0..12)), n=0..9);

A[n_, k_] := A[n, k] = If[k<0, 0, If[k==0, 1, Sum[A[n, j], {j, k-n, k-1}]]]; Table[A[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 08 2019 *)

A(n, k) = Sum_{j=0..floor(k/(n+1))} (-1)^j*((k - j*n) + j + delta(k,0))/(2*(k - j*n) + delta(k,0))*binomial(k - j*n, j)*2^(k-j*(n+1)), where delta denotes the Kronecker delta (see Corollary 3.2 in Parks and Wills). - Stefano Spezia, Aug 06 2022

A251706 6-step Fibonacci sequence starting with (0,0,0,0,1,0).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 2, 4, 8, 16, 31, 62, 123, 244, 484, 960, 1904, 3777, 7492, 14861, 29478, 58472, 115984, 230064, 456351, 905210, 1795559, 3561640, 7064808, 14013632, 27797200, 55138049, 109370888, 216946217, 430330794, 853596780, 1693179928, 3358562656

Offset: 0

Arie Bos, Dec 07 2014

a(n+6) equals the number of n-length binary words avoiding runs of zeros of lengths 6i+5, (i=0,1,2,...). - Milan Janjic, Feb 26 2015

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

Other 6-step Fibonacci sequences are A001592, A074584, A251707, A251708, A251709.

LinearRecurrence[Table[1, {6}], {0, 0, 0, 0, 1, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

a(n+6) = a(n) + a(n+1) + a(n+2) + a(n+3) + a(n+4) + a(n+5).
G.f.: x^4*(x-1)/(-1+x+x^2+x^3+x^4+x^5+x^6) . - R. J. Mathar, Mar 28 2025
a(n) = A001592(n+1)-A001592(n). - R. J. Mathar, Mar 28 2025

A251709 6-step Fibonacci sequence starting with (0,1,0,0,0,0).

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 1, 2, 3, 6, 12, 24, 48, 95, 188, 373, 740, 1468, 2912, 5776, 11457, 22726, 45079, 89418, 177368, 351824, 697872, 1384287, 2745848, 5446617, 10803816, 21430264, 42508704, 84319536, 167254785, 331763722, 658080827, 1305357838, 2589285412

Offset: 0

Arie Bos, Dec 07 2014

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

Other 6-step Fibonacci sequences are A001592, A074584, A251706, A251707, A251708.

LinearRecurrence[Table[1, {6}], {0, 1, 0, 0, 0, 0}, 40] (* Michael De Vlieger, Dec 09 2014 *)

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

G.f.: x*(-1+x+x^2+x^3+x^4)/(-1+x+x^2+x^3+x^4+x^5+x^6) . - R. J. Mathar, Feb 27 2023

A144406 Rectangular array A read by upward antidiagonals: entry A(n,k) in row n and column k gives the number of compositions of k in which no part exceeds n, n>=1, k>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 5, 1, 1, 1, 2, 4, 7, 8, 1, 1, 1, 2, 4, 8, 13, 13, 1, 1, 1, 2, 4, 8, 15, 24, 21, 1, 1, 1, 2, 4, 8, 16, 29, 44, 34, 1, 1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1, 1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1, 1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 29 2008

Polynomial expansion as antidiagonal of p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n)). Based on the Pisot general polynomial type q(x,n) = x^n - (x^n-1)/(x-1) (the original name of the sequence).
Row sums are 1, 2, 3, 5, 8, 14, ... (A079500).
Conjecture: Since the array row sequences successively tend to A000079, the absolute values of nonzero differences between two successive row sequences tend to A045623 = {1,2,5,12,28,64,144,320,704,1536,...}, as k -> infinity. - L. Edson Jeffery, Dec 26 2013

			Array A begins:
  {1, 1, 1, 1, 1,  1,  1,  1,   1,   1,   1, ...}
  {1, 1, 2, 3, 5,  8, 13, 21,  34,  55,  89, ...}
  {1, 1, 2, 4, 7, 13, 24, 44,  81, 149, 274, ...}
  {1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, ...}
  {1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, ...}
  {1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, ...}
  ... - _L. Edson Jeffery_, Dec 26 2013
As a triangle:
  {1},
  {1, 1},
  {1, 1, 1},
  {1, 1, 2, 1},
  {1, 1, 2, 3, 1},
  {1, 1, 2, 4, 5, 1},
  {1, 1, 2, 4, 7, 8, 1},
  {1, 1, 2, 4, 8, 13, 13, 1},
  {1, 1, 2, 4, 8, 15, 24, 21, 1},
  {1, 1, 2, 4, 8, 16, 29, 44, 34, 1},
  {1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1},
  {1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1},
  {1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1},
  {1, 1, 2, 4, 8, 16, 32, 64, 125, 236, 401, 504, 233, 1},
  {1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 464, 773, 927, 377, 1}

Cf. A000079, A045623, A092921, A175331.
Same as A048887 but with a column of 1's added on the left (the number of compositions of 0 is defined to be equal to 1).
Array rows (with appropriate offsets) are A000012, A000045, A000073, A000078, A001591, A001592, etc.

g[x_, n_] = x^(n) - (x^n - 1)/(x - 1);
h[x_, n_] = FullSimplify[ExpandAll[x^(n)*g[1/x, n]]];
f[t_, n_] := 1/h[t, n];
a = Table[CoefficientList[Series[f[t, m], {t, 0, 30}], t], {m, 1, 31}];
b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}];
Flatten[b] (* Triangle version *)
Grid[Table[CoefficientList[Series[(1 - x)/(1 - 2 x + x^(n + 1)), {x, 0, 10}], x], {n, 1, 10}]] (* Array version - L. Edson Jeffery, Jul 18 2014 *)

t(n,m) = antidiagonal_expansion of p(x,n) where p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n)).

G.f. for array A: (1-x)/(1 - 2*x + x^(n+1)), n>=1. - L. Edson Jeffery, Dec 26 2013

Definition changed by L. Edson Jeffery, Jul 18 2014

A227880 Primes in the union of all n-Fibonacci sequences.

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 31, 61, 89, 127, 149, 233, 401, 509, 773, 1021, 1597, 4093, 8191, 16381, 28657, 31489, 128257, 131071, 514229, 524287, 1048573, 4194301, 5976577, 16777213, 433494437, 536870909, 2147483647, 2971215073, 4293722117, 5350220959, 13435170943

Offset: 1

Robert Price, Oct 25 2013

nonn

T. D. Noe, Table of n, a(n) for n = 1..127 (first 62 terms from Robert Price)
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. A124168, A000045, A000073, A000078, A001591, A001592, A124257.

plst = {};plimit=10^39; For[n = 2, n ≤ 1 + Log[2, plimit], n++,flst = {};For[i = 1, i < n, i++, AppendTo[flst, 0]];AppendTo[flst, 1];For[k = 2, k ≤ 1 + Log[GoldenRatio, plimit*Sqrt[5] + 0.5], k++,sum = 0;For[j = 0, j < n, j++, sum = sum + flst[[j + k - 1]]];AppendTo[flst, sum];If[sum ≤ plimit && PrimeQ[sum], AppendTo[plst, sum]]]];Union[plst]

Primes in A124168.

A302990 a(n) = index of first odd prime number in the (n-th)-order Fibonacci sequence Fn, or 0 if no such index exists.

Original entry on oeis.org

0, 0, 4, 6, 9, 10, 40, 14, 17, 19, 361, 23, 90, 26, 373, 47, 288, 34, 75, 38, 251, 43, 67, 47, 74, 310, 511, 151534, 57, 20608, 1146, 62, 197, 94246, 9974, 287, 271172, 758

Offset: 0

Jacques Tramu, Apr 17 2018

Fn is defined by: Fn(0) = Fn(1) = ... = Fn(n-2) = 0, Fn(n-1) = 1, and Fn(k+1) = Fn(k) + Fn(k-1) + ... + Fn(k-n+1).
In general, Fn(k) is odd iff k == -1 or -2 (mod n+1), therefore a(n) = k*(n+1) - (1 or 2) for all n. Since Fn(n-1) = F(n) = 1, we must have a(n) >= 2n. Since Fn(k) = 2^(k-n) for n <= k < 2n, Fn(2n) = 2^n-1, so a(n) = 2n exactly for the Mersenne prime exponents A000043, while a(n) = 2n+1 when n is not in A000043 but n+1 is in A050414. - M. F. Hasler, Apr 18 2018
Further terms of the sequence: a(38) > 62000, a(39) > 72000, a(40) = 285, a(41) > 178000, a(42) = 558, a(44) = 19529, a(46) = 33369, a(47) = 239, a(48) = 6368, a(53) = 2860, a(54) = 2418, a(58) = 176, a(59) = 18418, a(60) = 1463, a(61) = 122, a(62) = 8755, a(63) = 5118,  a(64) = 25089, a(65) = 988, a(66) = 333, a(67) = 406, a(70) = 1632, a(74) = 374, a(76) = 13704, a(77) = 4991, a(86) = 347, a(89) = 178, a(92) = 1114, a(93) = 187, a(98) = 395, a(100) > 80000; a(n) > 10^4 for all other n up to 100. - Jacques Tramu and M. F. Hasler, Apr 18 2018

			a(2) = 4 because F2 (Fibonacci) = 0, 1, 1, 2, 3, 5, 8, ... and F2(4) = 3 is prime.
a(3) = 6 because F3 (tribonacci) = 0, 0, 1, 1, 2, 4, 7, 13, ... and F3(6) = 7 is prime.
a(4) = 9 because F4 (tetranacci) = 0, 0, 0, 1, 1, 2, 4, 8, 15, 29, 56, ...  and F4(9) = 29 is prime.
From _M. F. Hasler_, Apr 18 2018: (Start)
We see that Fn(k) = 2^(k-n) for n <= k < 2n and thus Fn(2n) = 2^n-1, so a(n) = 2n exactly for the Mersenne prime exponents A000043.
a(n) = 2n + 1 when 2^(n+1) - 3 is prime (n+1 in A050414) but 2^n-1 is not, i.e., n = 4, 8, 9, 11, 21, 23, 28, 93, 115, 121, 149, 173, 212, 220, 232, 265, 335, 451, 544, 688, 693, 849, 1735, ...
For other primes we have: a(29) = 687*30 - 2, a(37) = 20*38 - 2, a(41) > 10^4, a(43) > 10^4, a(47) = 5*48 - 1, a(53) = 53*54 - 2, a(59) = 307*60 - 2, a(67) = 6*67 - 1. (End)

Cf. A000045 (F2), A000073 (F3), A000078 (F4), A001591 (F5), A001592 (F6), A122189(F7), A079262 (F8), A104144 (F9), A122265 (F10).
(According to the definition, F0 = A000004 and F1 = A000012.)
Cf. A001605 (indices of prime numbers in F2).
Cf. A000043, A050414.

A302990(n,L=oo,a=vector(n+1,i,if(i1 && for(i=-2+2*n+=1,L, ispseudoprime(a[i%n+1]=2*a[(i-1)%n+1]-a[i%n+1]) && return(i))} \\ Testing primality only for i%n>n-3 is not faster, even for large n. - M. F. Hasler, Apr 17 2018; improved Apr 18 2018

a(n) == -1 or -2 (mod n+1). a(n) >= 2n, with equality iff n is in A000043. a(n) <= 2n+1 for n+1 in A050414. - M. F. Hasler, Apr 18 2018

a(29) from Jacques Tramu, Apr 19 2018
a(33) from Daniel Suteu, Apr 20 2018
a(36) from Jacques Tramu, Apr 25 2018

A303264 Indices of primes in tetranacci sequence A000078.

Original entry on oeis.org

5, 9, 13, 14, 38, 58, 403, 2709, 8419, 14098, 31563, 50698, 53194, 155184

Offset: 1

M. F. Hasler, Apr 18 2018

T = A000078 is defined by T(n) = Sum_{k=1..4} T(n-k), T(3) = 1, T(n) = 0 for n < 3.

The largest terms correspond to unproven probable primes T(a(n)).

Cf. A000045, A000073, A000078, A001591, A001592, A122189 (or A066178), ... (Fibonacci, tribonacci, tetranacci numbers).
Cf. A005478, A092836, A104535, A105757, A105759, A105761, ... (primes in Fibonacci numbers and above generalizations).
Cf. A001605, A303263, A303264, A248757, A249635, ... (indices of primes in A000045, A000073, A000078, ...).
Cf. A247027: Indices of primes in the tetranacci sequence A001631 (starting 0, 0, 1, 0...), A104534 (a variant: a(n) - 2), A105756 (= A248757 - 3), A105758 (= A249635 - 4).

a(n,N=5,S=vector(N,i,i>N-2))={for(i=N,oo,ispseudoprime(S[i%N+1]=2*S[(i-1)%N+1]-S[i%N+1])&&!n--&&return(i))}

a(n) = A104534(n) + 2.

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A251707 6-step Fibonacci sequence starting with (0,0,0,1,0,0).

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A251708 6-step Fibonacci sequence starting with (0,0,1,0,0,0).

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A108566 a(0) = 0, a(1) = a(2) = 1, a(3) = 2, a(4) = 4, a(5) = 8, for n>4: a(n+1) = SORT[ a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5)], where SORT places digits in ascending order and deletes 0's.

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A247506 Generalized Fibonacci numbers: square array A(n,k) read by ascending antidiagonals, A(n,k) = [x^k]((1-Sum_{j=1..n} x^j)^(-1)), (n>=0, k>=0).

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A251706 6-step Fibonacci sequence starting with (0,0,0,0,1,0).

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A251709 6-step Fibonacci sequence starting with (0,1,0,0,0,0).

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A144406 Rectangular array A read by upward antidiagonals: entry A(n,k) in row n and column k gives the number of compositions of k in which no part exceeds n, n>=1, k>=0.

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A227880 Primes in the union of all n-Fibonacci sequences.

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