A104576 -id:A104576 - cp's OEIS frontend

A104621 Heptanacci-Lucas numbers.

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)

A104577 Indices of prime generalized tetranacci numbers, A073817.

Original entry on oeis.org

2, 3, 8, 9, 16, 19, 24, 27, 46, 68, 71, 78, 107, 198, 309, 377, 477, 1057, 1631, 2419, 3974, 4293, 8247, 10513, 10709, 12011, 15042, 30543, 31607, 39664, 47552, 145858

Offset: 1

T. D. Noe, Mar 16 2005

nonn

The sequence of generalized tetranacci numbers is defined as beginning with 1, 3, 7, 15. Subsequent terms are the sum of the previous four terms. Note that the sequence of these generalized tetranacci numbers has many more primes than the tetranacci sequence A000078 (whose prime indices are in A104534).

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. A104576 (indices of prime generalized tribonacci numbers).

a={-1, -1, -1, 4}; Do[s=Plus@@a; a=RotateLeft[a]; a[[4]]=s; If[PrimeQ[s], Print[n]], {n, 30000}]

A104622 Indices of prime values of heptanacci-Lucas numbers A104621.

Original entry on oeis.org

0, 2, 3, 5, 7, 10, 17, 24, 25, 26, 28, 38, 40, 49, 62, 79, 89, 114, 140, 145, 182, 248, 353, 437, 654, 702, 784, 921, 931, 986, 1206, 2136, 2137, 3351, 5411, 13264, 13757, 16348, 27087, 27160

Offset: 1

Jonathan Vos Post, Mar 17 2005

The 7th-order linear recurrence A104622 (heptanacci-Lucas numbers) is a generalization of the Lucas sequence A000032. T. D. Noe and I have noted that the heptanacci-Lucas numbers have many more primes than the corresponding heptanacci (see A104414) which he found has only the first 3 primes that I identified through the first 5000 values, whereas these heptanacci-Lucas numbers have 17 primes among the first 100 values. For semiprimes in heptanacci-Lucas numbers, see A104623.

			A104621(0) = 7,
A104621(2) = 3,
A104621(3) = 7,
A104621(5) = 31,
A104621(7) = 127,
A104621(10) = 983,
A104621(17) = 122401,
A104621(24) = 15231991.

Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math.CO/0210201 v1, Oct 14 2002
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. A000032, A001644, A073817, A074048, A074584, A104414, A104576, A104577, A104621, A104623.

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]; Do[ If[ PrimeQ[ a[n]], Print[n]], {n, 5000}] (* Robert G. Wilson v, Mar 17 2005 *)
Flatten[Position[LinearRecurrence[{1,1,1,1,1,1,1},{7,1,3,7,15,31,63},28000],?PrimeQ]]-1 (* _Harvey P. Dale, Jan 02 2016 *)

Prime values of the heptanacci-Lucas numbers, which are defined by: a(0) = 7, a(1) = 1, a(2) = 3, a(3) = 7, a(4) = 15, a(5) = 31, a(6) = 63, for n > 6: a(n) = a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-5)+a(n-6)+a(n-7).

More terms from T. D. Noe and Robert G. Wilson v, Mar 17 2005

A073446 Product L(n)S(n), where L(n) are Lucas numbers and S(n) are Lucas 3-step numbers = A000032(n) A001644(n).

Original entry on oeis.org

6, 1, 9, 28, 77, 231, 702, 2059, 6157, 18316, 54489, 162185, 482678, 1436397, 4274853, 12722028, 37861085, 112675763, 335326230, 997940307, 2969899037, 8838503884, 26303639349, 78280380217, 232964641030, 693309407681

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 01 2002

a(n) is also the trace of the matrix R^n, where R is the Kronecker product of the Fibonacci matrix (Fibomatrix): first row (1,1), second row (1,0), times the Tribomatrix: first row (1,1,0), second row (1,0,1), third row (1,0,0).

a(n) is semiprime iff n is an element of A001606 (an index of a prime Lucas number) and an element of A104576 (an index of a prime Lucas 3-step number). The only known such are n = 2, 4, 7, 8 (through 67661). - Jonathan Vos Post, May 10 2005

Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Elia, Derived Sequences, The Tribonacci Recurrence and Cubic Forms, The Fibonacci Quarterly 39.2 (2001): 107-109.
F. T. Howard, A Tribonacci Identity, The Fibonacci Quarterly 39.4 (2001): 352-357.
Index entries for linear recurrences with constant coefficients, signature (1,4,5,2,-1,1).

Cf. A000032, A000040, A001358, A001606, A001644, A104576.

a:=[6,1,9,28,77,231];; for n in [7..40] do a[n]:=a[n-1]+4*a[n-2] +5*a[n-3]+2*a[n-4]-a[n-5]+a[n-6]; od; a; # G. C. Greubel, Feb 19 2019

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6) )); // G. C. Greubel, Feb 19 2019

CoefficientList[Series[(6-5x-16x^2-15x^3-4x^4+x^5)/(1-x-4x^2-5x^3-2x^4 +x^5-x^6), {x, 0, 50}], x]

my(x='x+O('x^40)); Vec((6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6)) \\ G. C. Greubel, Feb 19 2019

((6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 19 2019

a(n) = a(n-1)+4*a(n-2)+5*a(n-3)+2*a(n-4)-a(n-5)+a(n-6), a(0)=6, a(1)=1, a(2)=9, a(3)=28, a(4)=77, a(5)=231.

G.f.: (6-5*x-16*x^2-15*x^3-4*x^4+x^5)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6).

A105762 Primes in A001644 (the Lucas 3-step numbers).

Original entry on oeis.org

3, 7, 11, 71, 131, 241, 443, 1499, 196331, 86992799, 541292033, 292997064989357251, 129824812729295169371, 238785058551151434437, 5026368970977284897651, 105803877284856287511991

Offset: 1

T. D. Noe, Apr 22 2005

nonn

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, Lucas n-Step Number

Cf. A104576 (indices of prime Lucas 3-step numbers).

a={-1, -1, 3}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, s]], {n, 1000}]; lst

a(n) = A001644(A104576(n)). - Arthur O'Dwyer, Jul 31 2024

Name clarified by Arthur O'Dwyer, Jul 31 2024

A106627 Product L(n)*L_4(n), where L(n) are Lucas numbers and L_4(n) are Lucas 4-step numbers.

Original entry on oeis.org

8, 1, 9, 28, 105, 286, 918, 2871, 8977, 27892, 87084, 271635, 847182, 2641991, 8240325, 25700488, 80156033, 249994997, 779700654, 2431777739, 7584375260

Offset: 0

Jonathan Vos Post, May 11 2005

a(n) is semiprime iff n is an element of A001606 (an index of a prime Lucas number) and an element of A104577 (an index of a prime Lucas 4-step number). The only known such are n = 2, 8, 16, 19, 71, (through 145858).

			a(0) = 8 because L(0) * L_4(0) = 2 * 4.
a(1) = 1 because L(1) * L_4(1) = 1 * 1.
a(2) = 9 because L(2) * L_4(2) = 3 * 3.
a(3) = 28 because L(3) * L_4(3) = 4 * 7.
a(4) = 105 because L(4) * L_4(4) = 7 * 15.
a(5) = 286 because L(5) * L_4(5) = 11 * 26.
a(6) = 918 because L(6) * L_4(6) = 18 * 51.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,5,9,3,-2,1,-1).

Cf. A000032, A000040, A001358, A001606, A073446, A073817, A104576.

a:=[8,1,9,28,105,286,918,2871];; for n in [9..30] do a[n]:=a[n-1] +4*a[n-2]+5*a[n-3]+9*a[n-4]+3*a[n-5]-2*a[n-6]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Feb 19 2019

I:=[8,1,9,28,105,286,918,2871]; [n le 8 select I[n] else Self(n-1)+4*Self(n-2)+5*Self(n-3)+9*Self(n-4)+3*Self(n-5)-2*Self(n-6) + Self(n-7)-Self(n-8): n in [1..30]]; // G. C. Greubel, Feb 19 2019

LinearRecurrence[{1,4,5,9,3,-2,1,-1}, {8,1,9,28,105,286,918,2871}, 40] (* G. C. Greubel, Feb 19 2019 *)

my(x='x+O('x^40)); Vec((8-7*x-24*x^2-25*x^3-36*x^4-9*x^5+4*x^6 -x^7)/(1-x-4*x^2-5*x^3-9*x^4-3*x^5+2*x^6-x^7+x^8)) \\ G. C. Greubel, Feb 19 2019

((8-7*x-24*x^2-25*x^3-36*x^4-9*x^5+4*x^6-x^7)/(1-x-4*x^2-5*x^3 -9*x^4-3*x^5+2*x^6-x^7+x^8)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 19 2019

a(n) = A000032(n) * A073817(n).
a(n) = +a(n-1) +4*a(n-2) +5*a(n-3) +9*a(n-4) +3*a(n-5) -2*a(n-6) +a(n-7) -a(n-8). - R. J. Mathar, Dec 22 2010
G.f.: (8 -7*x -24*x^2 -25*x^3 -36*x^4 -9*x^5 +4*x^6 -x^7) / (1 -x -4*x^2 -5*x^3 -9*x^4 -3*x^5 +2*x^6 -x^7 +x^8). - Colin Barker, Jun 17 2012

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A104621 Heptanacci-Lucas numbers.

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A104577 Indices of prime generalized tetranacci numbers, A073817.

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A104622 Indices of prime values of heptanacci-Lucas numbers A104621.

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A073446 Product L(n)*S(n), where L(n) are Lucas numbers and S(n) are Lucas 3-step numbers = A000032(n) * A001644(n).

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A105762 Primes in A001644 (the Lucas 3-step numbers).

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A106627 Product L(n)*L_4(n), where L(n) are Lucas numbers and L_4(n) are Lucas 4-step numbers.

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A073446 Product L(n)S(n), where L(n) are Lucas numbers and S(n) are Lucas 3-step numbers = A000032(n) A001644(n).