A066178 -id:A066178 - cp's OEIS frontend

A104414 Number of prime factors, with multiplicity, of the heptanacci numbers A066178.

0, 0, 1, 2, 3, 4, 5, 6, 1, 2, 6, 3, 7, 6, 9, 8, 1, 4, 2, 5, 6, 5, 8, 9, 2, 3, 10, 6, 7, 7, 16, 10, 4, 2, 7, 5, 6, 9, 12, 10, 4, 3, 6, 4, 9, 8, 14, 12, 2, 3, 7, 6, 11, 8, 7, 10, 5, 5, 12, 6, 7, 9, 12, 11, 3, 4, 3, 6, 7, 5, 6, 11, 4, 2, 9, 4, 7, 9, 14, 8, 4, 3

Offset: 0

Jonathan Vos Post, Mar 06 2005

Prime heptanacci numbers: a(2) = 2, a(8) = 127, a(16) = 31489, ... Semiprime heptanacci numbers: a(4) = 4 = 2^2, a(9) = 253 = 11 * 23, a(18) = 124946 = 2 * 62473, a(24) = 7805695 = 5 * 1561139.

			a(0)=a(1)=0 because the first two nonzero heptanacci numbers are both 1, which has zero prime divisors.
a(2)=1 because the 3rd nonzero heptanacci number is 2, a prime, with only one prime divisor.
a(3)=2 because the 4th nonzero pentanacci number is 4 = 2^2 which has (with multiplicity) 2 prime divisors (which happen to be equal).
a(4)=3 because the 5th nonzero heptanacci number is 8 = 2^3.
a(12)= 7 because A066178(12) = 2000 = 2^4 * 5^3 which has seven prime factors (four of the 2, three of them 5).

Cf. A001222, A066178, A104411, A104412, A104413.

PrimeOmega[#]&/@LinearRecurrence[{1,1,1,1,1,1,1},{1,1,2,4,8,16,32},100] (* Harvey P. Dale, Oct 08 2015 *)

a(n) = A001222(A066178(n)). a(n) = bigomega(A066178(n)).

More terms from Harvey P. Dale, Oct 08 2015

A105761 Prime Fibonacci 7-step numbers, A066178.

Original entry on oeis.org

2, 127, 31489

Offset: 1

T. D. Noe, Apr 22 2005, May 13 2008

The next term is the 30673-digit probable prime 59456197869789581...206057955817 that is the 102490th term of the heptanacci sequence (using the definition in Noe-Post and Weisstein). This is too large to include here.

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

a={1, 0, 0, 0, 0, 0, 0}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, s]], {n, 100}]; lst

A079262 Octanacci numbers: a(0)=a(1)=...=a(6)=0, a(7)=1; for n >= 8, a(n) = Sum_{i=1..8} a(n-i).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1016, 2028, 4048, 8080, 16128, 32192, 64256, 128257, 256005, 510994, 1019960, 2035872, 4063664, 8111200, 16190208, 32316160, 64504063, 128752121, 256993248, 512966536, 1023897200, 2043730736

Offset: 0

Michael Joseph Halm, Feb 04 2003

a(n+7) is the number of compositions of n into parts <= 8. - Joerg Arndt, Sep 24 2020

			a(16) = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255.

T. D. Noe, Table of n, a(n) for n=0..207
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.
Spiros D. Dafnis, Andreas N. Philippou, and Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
Tian-Xiao He, Impulse Response Sequences and Construction of Number Sequence Identities, J. Int. Seq. 16 (2013) #13.8.2.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
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.
Fred J. Rispoli, Fibonacci Polytopes and Their Applications, Fib. Q., 43,3 (2005), 227-233.
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).

Cf. A066178, A001592, A001591, A001630, A000073, A000045.
Row 8 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
Cf. A253706, A253705. Primes and indices of primes in this sequence.

for j from 0 to 6 do a[j]:=0 od: a[7]:=1: for n from 8 to 45 do a[n]:=sum(a[n-i],i=1..8) od:seq(a[n],n=0..45); # Emeric Deutsch, Apr 16 2005

LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
With[{nn=8},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* Harvey P. Dale, Aug 17 2013 *)

G.f.: x^7/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8). - Emeric Deutsch, Apr 16 2005
a(1)..a(9) = 1, 1, 2, 4, 8, 16, 32, 64, 128. a(10) and following are given by 63*2^(n-8)+(1/2+sqrt(5/4))^(n-6)/sqrt(5)-(1/2-sqrt(5/4))^(n-6)/sqrt(5). Offset 10. a(10)=255. - Al Hakanson (hawkuu(AT)gmail.com), Feb 14 2009
Another form of the g.f.: f(z) = (z^7 - z^8)/(1 - 2*z + z^9), then a(n) = Sum_{i=0..floor((n-7)/9)} (-1)^i*binomial(n-7-8*i,i)*2^(n-7-9*i) - Sum_{i=0..floor((n-8)/9)} (-1)^i*binomial(n-8-8*i,i)*2^(n-8-9*i) with Sum_{i=m..n} alpha(i) = 0 for m>n. - Richard Choulet, Feb 22 2010
Sum_{k=0..7*n} a(k+b)*A171890(n,k) = a(8*n+b), b>=0.
a(n) = 2*a(n-1) - a(n-9). - Vincenzo Librandi, Dec 20 2010

Corrected by Joao B. Oliveira (oliveira(AT)inf.pucrs.br), Nov 25 2004

A092921 Array F(k, n) read by descending antidiagonals: k-generalized Fibonacci numbers in row k >= 1, starting (0, 1, 1, ...), for column n >= 0.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 3, 2, 1, 1, 0, 1, 5, 4, 2, 1, 1, 0, 1, 8, 7, 4, 2, 1, 1, 0, 1, 13, 13, 8, 4, 2, 1, 1, 0, 1, 21, 24, 15, 8, 4, 2, 1, 1, 0, 1, 34, 44, 29, 16, 8, 4, 2, 1, 1, 0, 1, 55, 81, 56, 31, 16, 8, 4, 2, 1, 1, 0, 1, 89, 149, 108, 61, 32, 16, 8, 4, 2, 1, 1, 0

Offset: 0

Ralf Stephan, Apr 17 2004

For all k >= 1, the k-generalized Fibonacci number F(k,n) satisfies the recurrence obtained by adding more terms to the recurrence of the Fibonacci numbers.
The number of tilings of an 1 X n rectangle with tiles of size 1 X 1, 1 X 2, ..., 1 X k is F(k,n).
T(k,n) is the number of 0-balanced ordered trees with n edges and height k (height is the number of edges from root to a leaf). - Emeric Deutsch, Jan 19 2007
Brlek et al. (2006) call this table "number of psp-polyominoes with flat bottom". - N. J. A. Sloane, Oct 30 2018

			From _Peter Luschny_, Apr 03 2021: (Start)
Array begins:
                n = 0  1  2  3  4  5   6   7   8    9   10
  -------------------------------------------------------------
  [k=1, mononacci ] 0, 1, 1, 1, 1, 1,  1,  1,  1,   1,   1, ...
  [k=2, Fibonacci ] 0, 1, 1, 2, 3, 5,  8, 13, 21,  34,  55, ...
  [k=3, tribonacci] 0, 1, 1, 2, 4, 7, 13, 24, 44,  81, 149, ...
  [k=4, tetranacci] 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ...
  [k=5, pentanacci] 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, ...
  [k=6]             0, 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, ...
  [k=7]             0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, ...
  [k=8]             0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, ...
  [k=9]             0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...
Note that the first parameter in F(k, n) refers to rows, and the second parameter refers to columns. This is always the case. Only the usual naming convention for the indices is not adhered to because it is common to call the row sequences k-bonacci numbers. (End)
.
From _Peter Luschny_, Aug 12 2015: (Start)
As a triangle counting compositions of n with largest part k:
  [n\k]| [0][1] [2] [3] [4][5][6][7][8][9]
   [0] | [0]
   [1] | [0, 1]
   [2] | [0, 1,  1]
   [3] | [0, 1,  1,  1]
   [4] | [0, 1,  2,  1,  1]
   [5] | [0, 1,  3,  2,  1, 1]
   [6] | [0, 1,  5,  4,  2, 1, 1]
   [7] | [0, 1,  8,  7,  4, 2, 1, 1]
   [8] | [0, 1, 13, 13,  8, 4, 2, 1, 1]
   [9] | [0, 1, 21, 24, 15, 8, 4, 2, 1, 1]
For example for n=7 and k=3 we have the 7 compositions [3, 3, 1], [3, 2, 2], [3, 2, 1, 1], [3, 1, 3], [3, 1, 2, 1], [3, 1, 1, 2], [3, 1, 1, 1, 1]. (End)

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Srecko Brlek, Andrea Frosini, Simone Rinaldi, and Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol. 13, (2006). Table 1 is essentially this array. - _N. J. A. Sloane_, Jul 20 2014
Eric S. Egge, Restricted permutations related to Fibonacci numbers and k-generalized Fibonacci numbers, arXiv:math/0109219 [math.CO], 2001.
Eric S. Egge, Restricted 3412-Avoiding Involutions, arXiv:math/0307050 [math.CO], 2003.
Eric S. Egge and Toufik Mansour, Restricted permutations, Fibonacci numbers and k-generalized Fibonacci numbers, arXiv:math/0203226 [math.CO], 2002.
Eric S. Egge and Toufik Mansour, 231-avoiding involutions and Fibonacci numbers, arXiv:math/0209255 [math.CO], 2002.
Nathaniel D. Emerson, A Family of Meta-Fibonacci Sequences Defined by Variable-Order Recursions, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.8.
Abraham Flaxman, Aram W. Harrow, and Gregory B. Sorkin, Strings with Maximally Many Distinct Subsequences and Substrings, Electronic J. Combinatorics 11 (1) (2004), Paper R8.
Ivan Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
H. Gabai, Generalized Fibonacci k-sequences, Fib. Quart., 8 (1970), 31-38.
R. Kemp, Balanced ordered trees, Random Structures and Alg., 5 (1994), pp. 99-121.
E. P. Miles jr., Generalized Fibonacci numbers and associated matrices, The Amer. Math. Monthly, 67 (1960) 745-752.
M. D. Miller, On generalized Fibonacci numbers, The Amer. Math. Monthly, 78 (1971) 1108-1109.
Michel Mollard, p-th order generalized Fibonacci cubes and maximal cubes in Fibonacci p-cubes, arXiv:2507.16387 [math.CO], 2025. See p. 3.
Harold R. Parks and Dean C. Wills, Sum of k-bonacci Numbers, arXiv:2208.01224 [math.CO], 2022. See p. 5.
Hsin-Po Wang and Chi-Wei Chin, On Counting Subsequences and Higher-Order Fibonacci Numbers, arXiv:2405.17499 [cs.IT], 2024. See p. 2.

Columns converge to A166444: each column n converges to A166444(n) = 2^(n-2).
Rows 1-8 are (shifted) A057427, A000045, A000073, A000078, A001591, A001592, A066178, A079262.
Essentially a reflected version of A048887.
See A048004 and A126198 for closely related arrays.
Cf. A066099.

F:= proc(k, n) option remember; `if`(n<2, n,
      add(F(k, n-j), j=1..min(k,n)))
    end:
seq(seq(F(k, d+1-k), k=1..d+1), d=0..12);  # Alois P. Heinz, Nov 02 2016
# Based on the above function:
Arow := (k, len) -> seq(F(k, j), j = 0..len):
seq(lprint(Arow(k, 14)), k = 1..10); # Peter Luschny, Apr 03 2021

F[k_, n_] := F[k, n] = If[n<2, n, Sum[F[k, n-j], {j, 1, Min[k, n]}]];
Table[F[k, d+1-k], {d, 0, 12}, {k, 1, d+1}] // Flatten (* Jean-François Alcover, Jan 11 2017, translated from Maple *)

F(k,n)=if(n<2,if(n<1,0,1),sum(i=1,k,F(k,n-i)))

T(m,n)=!!n*(matrix(m,m,i,j,j==i+1||i==m)^(n+m-2))[1,m] \\ M. F. Hasler, Apr 20 2018

F(k,n) = if(n==0,0, polcoeff(lift(Mod('x, Pol(vector(k+1,i, if(i==1,1,-1))))^(n+k-2)), k-1)); \\ Kevin Ryde, Jun 05 2020

# As a triangle of compositions of n with largest part k.
C = lambda n,k: Compositions(n, max_part=k, inner=[k]).cardinality()
for n in (0..9): [C(n,k) for k in (0..n)] # Peter Luschny, Aug 12 2015

F(k,n) = F(k,n-1) + F(k,n-2) + ... + F(k,n-k); F(k,1) = 1 and F(k,n) = 0 for n <= 0.
G.f.: x/(1-Sum_{i=1..k} x^i).
F(k,n) = 2^(n-2) for 1 < n <= k+1. - M. F. Hasler, Apr 20 2018
F(k,n) = Sum_{j=0..floor(n/(k+1))} (-1)^j*((n - j*k) + j + delta(n,0))/(2*(n - j*k) + delta(n,0))*binomial(n - j*k, j)*2^(n-j*(k+1)), where delta denotes the Kronecker delta (see Corollary 3.2 in Parks and Wills). - Stefano Spezia, Aug 06 2022

A104144 a(n) = Sum_{k=1..9} a(n-k); a(8) = 1, a(n) = 0 for n < 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729, 2097364960

Offset: 0

Jean Lefort (jlefort.apmep(AT)wanadoo.fr), Mar 07 2005

Sometimes called the Fibonacci 9-step numbers.
For n >= 8, this gives the number of integers written without 0 in base ten, the sum of digits of which is equal to n-7. E.g., a(11) = 8 because we have the 8 numbers: 4, 13, 22, 31, 112, 121, 211, 1111.
The offset for this sequence is fairly arbitrary. - N. J. A. Sloane, Feb 27 2009

T. D. Noe, Table of n, a(n) for n = 0..208
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.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1,1,1).

Cf. A000045, A000073, A000078, A001591, A001592, A066178, A079262 (Fibonacci n-step numbers).

Cf. A255529 (Indices of primes in this sequence).

for n from 0 to 50 do k(n):=sum((-1)^i*binomial(n-8-9*i,i)*2^(n-8-10*i),i=0..floor((n-8)/10))-sum((-1)^i*binomial(n-9-9*i,i)*2^(n-9-10*i),i=0..floor((n-9)/10)):od:seq(k(n),n=0..50);a:=taylor((z^8-z^9)/(1-2*z+z^(10)),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

a={1, 0, 0, 0, 0, 0, 0, 0, 0}; Table[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s, {n, 50}]
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
With[{nn=9},LinearRecurrence[Table[1,{nn}],Join[Table[0,{nn-1}],{1}],50]] (* Harvey P. Dale, Aug 17 2013 *)

a(n)=([0,1,0,0,0,0,0,0,0; 0,0,1,0,0,0,0,0,0; 0,0,0,1,0,0,0,0,0; 0,0,0,0,1,0,0,0,0; 0,0,0,0,0,1,0,0,0; 0,0,0,0,0,0,1,0,0; 0,0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,0,1; 1,1,1,1,1,1,1,1,1]^n*[0;0;0;0;0;0;0;0;1])[1,1] \\ Charles R Greathouse IV, Jun 16 2015

A104144(n,m=9)=(matrix(m,m,i,j,j==i+1||i==m)^n)[1,m] \\ M. F. Hasler, Apr 22 2018

a(n) = Sum_{k=1..9} a(n-k) for n > 8, a(8) = 1, a(n) = 0 for n=0..7.
G.f.: x^8/(1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9). - N. J. A. Sloane, Dec 04 2011
Another form of the g.f. f: f(z) = (z^8-z^9)/(1-2*z+z^(10)), then a(n) = Sum_((-1)^i*binomial(n-8-9*i,i)*2^(n-8-10*i), i=0..floor((n-8)/10))-Sum_((-1)^i*binomial(n-9-9*i,i)*2^(n-9-10*i), i=0..floor((n-9)/10)) with Sum_(alpha(i), i=m..n)=0 for m>n. - Richard Choulet, Feb 22 2010
From N. J. A. Sloane, Dec 04 2011: (Start)
Let b be the smallest root (in magnitude) of g(x) := 1-x-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9, b = 0.50049311828655225605926845999420216157202861343888...
Let c = -b^8/g'(b) = 0.00099310812055463178382193226558248643030626601288701...
Then a(n) is the nearest integer to c/b^n. (End)

Edited by N. J. A. Sloane, Aug 15 2006 and Nov 11 2006
Incorrect formula deleted by N. J. A. Sloane, Dec 04 2011
Name edited by M. F. Hasler, Apr 22 2018

A104621 Heptanacci-Lucas numbers.

Original entry on oeis.org

7, 1, 3, 7, 15, 31, 63, 127, 247, 493, 983, 1959, 3903, 7775, 15487, 30847, 61447, 122401, 243819, 485679, 967455, 1927135, 3838783, 7646719, 15231991, 30341581, 60439343, 120393007, 239818559, 477709983, 951581183, 1895515647, 3775799303, 7521257025

Offset: 0

Jonathan Vos Post, Mar 17 2005

This 7th-order linear recurrence is a generalization of the Lucas sequence A000032. Mario Catalani would refer to this is a generalized heptanacci sequence, had he not stopped his series of sequences after A001644 "generalized tribonacci", A073817 "generalized tetranacci", A074048 "generalized pentanacci", A074584 "generalized hexanacci." T. D. Noe and I have noted that each of these has many more primes than the corresponding tribonacci A000073 (see A104576), tetranacci A000288 (see A104577), pentanacci, hexanacci and heptanacci (see A104414). For primes in Heptanacci-Lucas numbers, see A104622. For semiprimes in Heptanacci-Lucas numbers, see A104623.

G. C. Greubel, Table of n, a(n) for n = 0..3300 (terms 0..200 from T. D. Noe)
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.
Spiros D. Dafnis, Andreas N. Philippou, Ioannis E. Livieris, An Alternating Sum of Fibonacci and Lucas Numbers of Order k, Mathematics (2020) Vol. 9, 1487.
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
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Cf. A000032, A001644, A073817, A074048, A074584, A104414, A104576, A104577.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (-7+6*x+ 5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6+x^7) )); // G. C. Greubel, Apr 22 2019

A104621 := proc(n)
    option remember;
    if n <=6 then
        op(n+1,[7, 1, 3, 7, 15, 31, 63])
    else
        add(procname(n-i),i=1..7) ;
    end if;
end proc: # R. J. Mathar, Mar 26 2015

a[0]=7; a[1]=1; a[2]=3; a[3]=7; a[4]=15; a[5]=31; a[6]=63; a[n_]:= a[n]= a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]+a[n-7]; Table[a[n], {n,0,40}] (* Robert G. Wilson v, Mar 17 2005 *)
LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {7, 1, 3, 7, 15, 31, 63}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

my(x='x+O('x^40)); Vec((-7+6*x+5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6+x^7)) \\ G. C. Greubel, Dec 18 2017

polsym(polrecip(1-x-x^2-x^3-x^4-x^5-x^6-x^7), 40) \\ G. C. Greubel, Apr 22 2019

((-7+6*x+5*x^2+4*x^3+3*x^4+2*x^5+x^6)/(-1+x +x^2+x^3+x^4+x^5+x^6 +x^7)).series(x, 41).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7); a(0) = 7, a(1) = 1, a(2) = 3, a(3) = 7, a(4) = 15, a(5) = 31, a(6) = 63.
From R. J. Mathar, Nov 16 2007: (Start)
G.f.: (7 - 6*x - 5*x^2 - 4*x^3 - 3*x^4 - 2*x^5 - x^6)/(1 - x - x^2 - x^3 - x^4 - x^5 - x^6 - x^7).
a(n) = 7*A066178(n) - 6*A066178(n-1) - 5*A066178(n-2) - ... - 2*A066178(n-5) - A066178(n-6) if n >= 6. (End)

A122189 Heptanacci numbers: each term is the sum of the preceding 7 terms, with a(0),...,a(6) = 0,0,0,0,0,0,1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, 1004, 2000, 3984, 7936, 15808, 31489, 62725, 124946, 248888, 495776, 987568, 1967200, 3918592, 7805695, 15548665, 30972384, 61695880, 122895984, 244804400, 487641600, 971364608, 1934923521

Offset: 0

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

See A066178 (essentially the same sequence) for more about the heptanacci numbers and other generalizations of the Fibonacci numbers (A000045).

Robert Price, Table of n, a(n) for n = 0..1000
Tomás Aguilar-Fraga, Jennifer Elder, Rebecca E. Garcia, Kimberly P. Hadaway, Pamela E. Harris, Kimberly J. Harry, Imhotep B. Hogan, Jakeyl Johnson, Jan Kretschmann, Kobe Lawson-Chavanu, J. Carlos Martínez Mori, Casandra D. Monroe, Daniel Quiñonez, Dirk Tolson III, and Dwight Anderson Williams II, Interval and L-interval Rational Parking Functions, arXiv:2311.14055 [math.CO], 2023. See p. 14.
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.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
Tian-Xiao He, Impulse Response Sequences and Construction of Number Sequence Identities, J. Int. Seq. 16 (2013) #13.8.2.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
Benjamin E. Merkel, Probabilities of Consecutive Events in Coin Flipping, Master's Thesis, Univ. Cincinatti, May 11 2011.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1,1,1).

Cf. A000045 (k=2, Fibonacci numbers), A000073 (k=3, tribonacci) A000078 (k=4, tetranacci) A001591 (k=5, pentanacci) A001592 (k=6, hexanacci), A122189 (k=7, heptanacci).

Cf. A066178, A000322, A248700.

for n from 0 to 50 do k(n):=sum((-1)^i*binomial(n-6-7*i,i)*2^(n-6-8*i),i=0..floor((n-6)/8))-sum((-1)^i*binomial(n-7-7*i,i)*2^(n-7-8*i),i=0..floor((n-7)/8)):od:seq(k(n),n=0..50); a:=taylor((z^6-z^7)/(1-2*z+z^8),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

LinearRecurrence[{1, 1, 1, 1, 1, 1, 1}, {0, 0, 0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
a={0,0,0,0,0,0,1} For[n=7, n≤100, n++, sum=Plus@@a; Print[sum]; a=RotateLeft[a]; a[[7]]=sum] (* Robert Price, Dec 04 2014 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 1,1,1,1,1,1,1]^n*[0;0;0;0;0;0;1])[1,1] \\ Charles R Greathouse IV, Jun 20 2015

G.f.: x^6/(1-x-x^2-x^3-x^4-x^5-x^6-x^7). - R. J. Mathar, Feb 13 2009
G.f.: Sum_{n >= 0} x^(n+5) * [ Product_{k = 1..n} (k + k*x + k*x^2 + k*x^3 + k*x^4 + k*x^5 + x^6)/(1 + k*x + k*x^2 + k*x^3 + k*x^4 + k*x^5 + k*x^6) ]. - Peter Bala, Jan 04 2015
Another form of the g.f.: f(z) = (z^6-z^7)/(1-2*z+z^8), then a(n) = Sum_{i=0..floor((n-6)/8)} (-1)^i*binomial(n-6-7*i,i)*2^(n-6-8*i) - Sum_{i=0..floor((n-7)/8)} (-1)^i*binomial(n-7-7*i,i)*2^(n-7-8*i) with Sum_{i=m..n} alpha(i) = 0 for m>n. - Richard Choulet, Feb 22 2010
Sum_{k=0..6*n} a(k+b)*A063265(n,k) = a(7*n+b), b>=0.
a(n) = 2*a(n-1) - a(n-8). - Joerg Arndt, Sep 24 2020

Edited by N. J. A. Sloane, Nov 20 2007

Wrong Binet-type formula removed by R. J. Mathar, Feb 13 2009

A107245 Sum of squares of heptanacci numbers (Fibonacci 7-step numbers).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21591, 85600, 339616, 1347632, 5347632, 21219888, 84199984, 334092848, 1325649969, 5260075594, 20871578510, 82816815054, 328610657230, 1303901211854, 5173777051854, 20529140314318

Offset: 0

Jonathan Vos Post, May 20 2005

Primes include: a(7) = 2. Semiprimes include a(8) = 6 = 2 * 3, a(9) = 22 = 2 * 11, a(10) = 86 = 2 * 43, a(12) = 1366 = 2 * 683, a(13) = 5462 = 2 * 2731.

			a(0) = 0 = 0^2
a(1) = 0 = 0^2 + 0^2
a(2) = 0 = 0^2 + 0^2 + 0^2
a(3) = 0 = 0^2 + 0^2 + 0^2 + 0^2
a(4) = 0 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2
a(5) = 0 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2
a(6) = 1 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2
a(7) = 2 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2
a(8) = 6 = 0^2 + 0^2 + 0^2+ 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2
a(9) = 22 = 0^2 + 0^2 +0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2 = 2*11
a(10) = 86 = 8^2 + 22
a(11) = 342 = 16^2 + 86

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 4, 7, 15, 30, 60, -131, -9, -8, 28, -11, -25, -32, 68, 5, 5, -10, 0, 5, 9, -14, 0, -1, 1, 0, 0, -1, 1).

Cf. A066178, A107239-A107244, A107246-A107248.

LinearRecurrence[{3, 2, 4, 7, 15, 30, 60, -131, -9, -8, 28, -11, -25, -32, 68, 5, 5, -10, 0, 5, 9, -14, 0, -1, 1, 0, 0, -1, 1},{0, 0, 0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1366, 5462, 21591, 85600, 339616, 1347632, 5347632, 21219888, 84199984, 334092848, 1325649969, 5260075594, 20871578510, 82816815054, 328610657230, 1303901211854, 5173777051854},30] (* Ray Chandler, Aug 02 2015 *)
Accumulate[LinearRecurrence[{1,1,1,1,1,1,1},{0,0,0,0,0,0,1},30]^2] (* Ray Chandler, Aug 02 2015 *)

a(n) = F_7(0)^2 + F_7(1)^2 + ... F_7(n)^2, note that F_7(n) = A066178(n) with corrected offset (from leading zeros). a(0) = 0, a(n+1) = a(n) + F_7(n)^2.

a(14) inserted by R. J. Mathar, Aug 11 2009

A118428 Decimal expansion of heptanacci constant.

Original entry on oeis.org

1, 9, 9, 1, 9, 6, 4, 1, 9, 6, 6, 0, 5, 0, 3, 5, 0, 2, 1, 0, 9, 7, 7, 4, 1, 7, 5, 4, 5, 8, 4, 3, 7, 4, 9, 6, 3, 4, 7, 9, 3, 1, 8, 9, 6, 0, 0, 5, 3, 1, 5, 7, 9, 9, 5, 2, 4, 4, 7, 8, 2, 1, 5, 3, 4, 0, 0, 9, 5, 1, 9, 8, 0, 3, 0, 9, 6, 2, 2, 1, 8, 3, 5, 6, 3, 1, 4, 1, 5, 7, 7, 0, 2, 2, 7, 1, 9, 0, 1, 7, 0, 9, 9, 1, 6

Offset: 1

Eric W. Weisstein, Apr 27 2006

Other roots of the equation x^7 - x^6 - ... - x - 1 see in A239566. For n>=7, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - Vladimir Shevelev, Mar 21 2014

Note that we have: c + c^(-7) = 2, and the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - Bernard Schott, May 07 2022

			1.9919641966050350210...

Martin Gardner, The Second Scientific American Book Of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, p. 101, Simon & Schuster, NY, 1961.

S. Litsyn and Vladimir Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
Vladimir Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein's World of Mathematics, Heptanacci Number
Eric Weisstein's World of Mathematics, Heptanacci Constant
Index entries for algebraic numbers, degree 7.

Cf. A066178, A239566.

k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), A086088 (tetranacci), A103814 (pentanacci), A118427 (hexanacci), this sequence (heptanacci).

RealDigits[x/.FindRoot[x^7+Total[-x^Range[0,6]]==0,{x,2}, WorkingPrecision-> 110]][[1]] (* Harvey P. Dale, Dec 13 2011 *)

polrootsreal(x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1)[1] \\ Charles R Greathouse IV, Feb 11 2025

A172317 8th column of A172119.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 128, 255, 508, 1012, 2016, 4016, 8000, 15936, 31744, 63233, 125958, 250904, 499792, 995568, 1983136, 3950336, 7868928, 15674623, 31223288, 62195672, 123891552, 246787536, 491591936, 979233536

Offset: 0

Richard Choulet, Jan 31 2010

			a(4) = binomial(4,4)*2^4 = 16.
a(9) = binomial(9,9)*2^9 - binomial(2,1)*2^1 = 512 - 4 = 508.

Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,0,0,-1).

Cf. A172316, A001949, A107066, A008937, A172119.

Partial sums of A066178.

k:=7:taylor(1/(1-2*z+z^(k+1)),z=0,30); for k from 0 to 20 do for n from 0 to 30 do b(n):=sum((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j),j=0..floor(n/(k+1))):od:k: seq(b(n),n=0..30):od;

The generating function is f such that: f(z)=1/(1-2*z+z^8). Recurrence relation: a(n+8)=2*a(n+7)-a(n). General term: a(n) = Sum_{j=0..floor(n/(k+1))} ((-1)^j*binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j)) with k=7.

cp's OEIS Frontend

A104414 Number of prime factors, with multiplicity, of the heptanacci numbers A066178.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A105761 Prime Fibonacci 7-step numbers, A066178.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

A079262 Octanacci numbers: a(0)=a(1)=...=a(6)=0, a(7)=1; for n >= 8, a(n) = Sum_{i=1..8} a(n-i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A092921 Array F(k, n) read by descending antidiagonals: k-generalized Fibonacci numbers in row k >= 1, starting (0, 1, 1, ...), for column n >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Sage

Formula

A104144 a(n) = Sum_{k=1..9} a(n-k); a(8) = 1, a(n) = 0 for n < 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A104621 Heptanacci-Lucas numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

A122189 Heptanacci numbers: each term is the sum of the preceding 7 terms, with a(0),...,a(6) = 0,0,0,0,0,0,1.

Views