A001631 -id:A001631 - cp's OEIS frontend

A248920 Indices of primes in the pentanacci numbers sequence A000322.

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

A232563 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 4*x are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 16, 12, 9, 32, 6, 20, 17, 64, 13, 48, 10, 36, 33, 128, 7, 24, 21, 80, 18, 68, 65, 256, 14, 52, 49, 192, 11, 40, 37, 144, 34, 132, 129, 512, 28, 25, 96, 22, 84, 81, 320, 19, 72, 69, 272, 66, 260, 257, 1024, 15, 56, 53, 208, 50, 196, 193, 768

Offset: 1

Clark Kimberling, Nov 26 2013

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 4*x are in S. Then S is the set of all positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,4), g(3) = (3,8,5,16), g(4) = (12,9,32,6,20,17,64), etc. Concatenating these gives A232563, a permutation of the positive integers. The number of numbers in g(n) is A001631(n), the n-th tetranacci number. It is helpful to show the results as a tree with the terms of S as nodes and edges from x to x + 1 if x + 1 has not already occurred, and an edge from x to 4*x if 4*x has not already occurred.

			Each x begets x + 1 and 4*x, but if either has already occurred it is deleted.  Thus, 1 begets 2 and 4; in the next generation, 2 begets 3 and 8, and 4 begets 5 and 16.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A232559, A232564, A001631.

z = 8; g[1] = {1}; g[2] = {2, 4}; g[n_] := Riffle[g[n - 1] + 1, 4 g[n - 1]]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* A232563 *)
Table[Length[g1[n]], {n, 1, z}]  (* A001631 *)
t1 = Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232564 *)

A233324 Triangle read by rows: T(n,k) = number of palindromic compositions of n in which no part exceeds k, 1 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 3, 3, 4, 1, 2, 3, 3, 4, 1, 5, 6, 7, 7, 8, 1, 3, 6, 6, 7, 7, 8, 1, 8, 11, 14, 14, 15, 15, 16, 1, 5, 11, 12, 14, 14, 15, 15, 16, 1, 13, 20, 27, 28, 30, 30, 31, 31, 32, 1, 8, 20, 23, 28, 28, 30, 30, 31, 31, 32, 1, 21, 37, 52, 55, 60, 60, 62, 62, 63, 63, 64

Offset: 1

L. Edson Jeffery, Dec 11 2013

A palindromic composition of a natural number m is an ordered partition of m into N+1 natural numbers (or parts), p_0, p_1, ..., p_N, of the form m = p_0 + p_1 + ... + p_N such that p_j = p_{N-j}, for each j in {0,...,N}. Two palindromic compositions, sum_{j=0..N} p_j and sum_{j=0..N} q_j (say), are identical if and only if p_j = q_j, j = 0,...,N; otherwise they are taken to be distinct.
Partial sums of rows of A233323.
T(n,k) is defined for n,k >= 0.  T(n,k) = T(n,n) = A016116(n) for k>= 0. - Alois P. Heinz, Dec 11 2013

			Triangle T(n,k) begins:
1;
1,  2;
1,  1,  2;
1,  3,  3,  4;
1,  2,  3,  3,  4;
1,  5,  6,  7,  7,  8;
1,  3,  6,  6,  7,  7,  8;
1,  8, 11, 14, 14, 15, 15, 16;
1,  5, 11, 12, 14, 14, 15, 15, 16;
1, 13, 20, 27, 28, 30, 30, 31, 31, 32;

Alois P. Heinz, Rows n = 1..141, flattened
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Cf. A233323.

T(n,2) = A053602(n+1) = A123231(n). T(2n,3) = A001590(n+3). T(2n,4) = A001631(n+4). - Alois P. Heinz, Dec 11 2013

T:= proc(n, k) option remember; `if`(n<=k, 1, 0)+
      add(T(n-2*j, k), j=1..min(k, iquo(n, 2)))
    end:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Dec 11 2013

T[n_, k_] := T[n, k] = If[n <= k, 1, 0] + Sum[T[n-2*j, k], {j, 1, Min[k, Quotient[ n, 2]]}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

T(n,k)=if(n<1,return(n==0));sum(i=1,k,T(n-2*i,k))+(n<=k) \\ Charles R Greathouse IV, Dec 11 2013

A243573 Irregular triangular array generated as in Comments; contains every positive integer exactly once.

Original entry on oeis.org

1, 2, 4, 3, 5, 8, 16, 6, 9, 12, 17, 20, 32, 64, 7, 10, 13, 18, 21, 24, 33, 36, 48, 65, 68, 80, 128, 256, 11, 14, 19, 22, 25, 28, 34, 37, 40, 49, 52, 66, 69, 72, 81, 84, 96, 129, 132, 144, 192, 257, 260, 272, 320, 512, 1024, 15, 23, 26, 29, 35, 38, 41, 44, 50

Offset: 1

Clark Kimberling, Jun 07 2014

Decree that (row 1) = (1), (row 2) = (2, 4), (row 3) = (3,5,8,16), (row 4) = (6,9,12,17,20,32,64). Let r(n) = A001563(n+3), so that r(r) = r(n-1) + r(n-2) + r(n-3) + r(n-4) with r(1) =1, r(2) = 2, r(3) = 4, r(4) = 7. Row n of the array, for n >= 5, consists of the numbers, in increasing order, defined as follows: all 4*x from x in row n-1, together with all 1 + 4*x from x in row n-2, together with all 2 + 4*x from x in row n-3, together with all 3 + 4*x for x in row n-4. Thus, the number of numbers in row n is r(n), a tetranacci number. Every positive integer occurs exactly once in the array, so that the resulting sequence is a permutation of the positive integers.

			First 5 rows of the array:
1
2 .. 4
3 .. 5 .. 8 .. 16
6 .. 9 .. 12 . 17 . 20 . 32 . 64
7 .. 10 . 13 . 18 . 21 . 24 . 33 . 36 . 48 . 65 . 68 . 80 . 128 . 256

Clark Kimberling, Table of n, a(n) for n = 1..2500

Cf. A243571, A243572, A001631, A232563.

z = 8; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 4 x; h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
u = Table[g[n], {n, 1, z}]; v = Flatten[u] (* A243573 *)

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

A105758 Indices of prime hexanacci (or Fibonacci 6-step) numbers A001592 (using offset -4).

Original entry on oeis.org

3, 36, 37, 92, 660, 6091, 8415, 11467, 13686, 38831, 49828, 97148

Offset: 1

T. D. Noe, Apr 22 2005

No other n < 30000.
This sequence uses the convention of the Noe and Post reference. Their indexing scheme differs by 4 from the indices in A001592. Sequence A249635 lists the indices of the same primes (A105759) using the indexing scheme as defined in A001592. - Robert Price, Nov 02 2014 [Edited by M. F. Hasler, Apr 22 2018]
a(13) > 3*10^5. - Robert Price, Nov 02 2014

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.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A105759 (prime Fibonacci 6-step numbers), A249635 (= a(n) + 4), A001592.
Cf. A000045, A000073, A000078 (and A001631), A001591, A122189 (or A066178),  A079262, A104144,  A122265, A168082, A168083 (Fibonacci, tribonacci, tetranacci numbers and other generalizations).
Cf. A005478, A092836, A104535, A105757, A105761, ... (primes in these sequence).
Cf. A001605, A303263, A303264 (and A104534 and A247027), A248757 (and A105756), ... (indices of primes in A000045, A000073, A000078, ...).

a={1, 0, 0, 0, 0, 0}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, n]], {n, 30000}]; lst

a(n) = A249635(n) - 4. A105759(n) = A001592(A249635(n)) = A001592(a(n) + 4). - M. F. Hasler, Apr 22 2018

a(10)-a(12) from Robert Price, Nov 02 2014

Edited by M. F. Hasler, Apr 22 2018

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

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

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A232563 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 4*x are in S, and duplicates are deleted as they occur.

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A233324 Triangle read by rows: T(n,k) = number of palindromic compositions of n in which no part exceeds k, 1 <= k <= n.

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A243573 Irregular triangular array generated as in Comments; contains every positive integer exactly once.

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