A073817 -id:A073817

A001648

Tetranacci numbers A073817 without the leading term 4.
(Formerly M2648 N1055)

+20
18

1, 3, 7, 15, 26, 51, 99, 191, 367, 708, 1365, 2631, 5071, 9775, 18842, 36319, 70007, 134943, 260111, 501380, 966441, 1862875, 3590807, 6921503, 13341626, 25716811, 49570747, 95550687, 184179871, 355018116, 684319421, 1319068095, 2542585503, 4900991135

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Indranil Ghosh, Table of n, a(n) for n = 1..3501 (terms 1..200 from T. D. Noe)

Daniel C. Fielder, Special integer sequences controlled by three parameters, Fibonacci Quarterly 6, 1968, 64-70.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

S. Saito, T. Tanaka, N. Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values , J. Int. Seq. 14 (2011) # 11.2.4, Table 3.

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

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

FORMULA

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

a(n) = trace of M^n, where M = the 4 X 4 matrix [ 0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 1 1 1]. E.g., the trace (sum of diagonal terms) of M^12 = a(12) = 2631 = (108 + 316 + 717 + 1490). - Gary W. Adamson, Feb 22 2004

a(n) = n*Sum_{k=ceiling(n/5)..n} Sum_{i=0..(n-k)/4} (-1)^i*binomial(k,k-i)*binomial(n-i*4-1,k-1))/k), n>0. - Vladimir Kruchinin, Jan 20 2012

MAPLE

A001648:=-(1+2*z+3*z**2+4*z**3)/(-1+z+z**2+z**3+z**4); # conjectured by Simon Plouffe in his 1992 dissertation

MATHEMATICA

Rest@ CoefficientList[ Series[(4 - 3 x - 2 x^2 - x^3)/(1 - x - x^2 - x^3 - x^4), {x, 0, 40}], x] (* Or *)

a[0] = 4; a[1] = 1; a[2] = 3; a[3] = 7; a[4] = 15; a[n_] := 2*a[n - 1] - a[n - 5]; Array[a, 33] (* Robert G. Wilson v *)

LinearRecurrence[{1, 1, 1, 1}, {1, 3, 7, 15}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

PROG

(PARI) a(n)=if(n<0, 0, polcoeff(x*(1+2*x+3*x^2+4*x^3)/(1-x-x^2-x^3-x^4)+x*O(x^n), n))

(Maxima) a(n):=n*sum(sum((-1)^i*binomial(k, k-i)*binomial(n-i*4-1, k-1), i, 0, ((n-k)/4))/k, k, ceiling(n/5), n); /* Vladimir Kruchinin, Jan 20 2012 */

(Magma) I:=[1, 3, 7, 15]; [n le 4 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 18 2017

CROSSREFS

Cf. A000288, A073817.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A074058

Reflected tetranacci numbers A073817.

+20
9

4, -1, -1, -1, 7, -6, -1, -1, 15, -19, 4, -1, 31, -53, 27, -6, 63, -137, 107, -39, 132, -337, 351, -185, 303, -806, 1039, -721, 791, -1915, 2884, -2481, 2303, -4621, 7683, -7846, 7087, -11545, 19987, -23375, 22020, -30177, 51519, -66737, 67415, -82374, 133215, -184993, 201567, -232163, 348804

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

Also a(n) is the trace of A^(-n), where A is the 4 X 4 matrix ((1,1,0,0), (1,0,1,0), (1,0,0,1), (1,0,0,0)).

REFERENCES

R. L. Graham, D. E. Knuth and O. Patashnik, "Concrete Mathematics", Addison-Wesley, Reading, MA, 1998.

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..8998

A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.

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

FORMULA

a(n) = -a(n-1)-a(n-2)-a(n-3)+a(n-4), a(0)=4, a(1)=-1, a(2)=-1, a(3)=-1.

G.f.: (4+3x+2x^2+x^3)/(1+x+x^2+x^3-x^4).

From Peter Bala, Jan 19 2023: (Start)

a(n) = (-1)^n*A073937(n).

The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. (End)

MATHEMATICA

CoefficientList[Series[(4+3*x+2*x^2+x^3)/(1+x+x^2+x^3-x^4), {x, 0, 1}], x]

PROG

(PARI) polsym(polrecip(1+x+x^2+x^3-x^4), 55) \\ Joerg Arndt, Jan 21 2023

CROSSREFS

Cf. A073817, A073937.

KEYWORD

easy,sign

AUTHOR

Mario Catalani (mario.catalani(AT)unito.it), Aug 16 2002

STATUS

approved

A106295

Period of the Lucas 4-step sequence A073817 mod n.

+20
8

1, 5, 26, 10, 312, 130, 342, 20, 78, 1560, 120, 130, 84, 1710, 312, 40, 4912, 390, 6858, 1560, 4446, 120, 12166, 260, 1560, 420, 234, 1710, 280, 1560, 61568, 80, 1560, 24560, 17784, 390, 1368, 34290, 1092, 1560, 240, 22230, 162800, 120, 312, 60830, 103822

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This sequence is the same as the period of Fibonacci 4-step sequence (A000078) mod n for n<563 because the discriminant of the characteristic polynomial x^4-x^3-x^2-x-1 is -563. The two sequences differ only at n that are multiples of 563.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..1486

Marcellus E. Waddill, Some Properties of the Tetranacci Sequence Modulo m, The Fibonacci Quarterly, vol. 30, no. 3, 232-238 (1992).

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

FORMULA

Let the prime factorization of n be p1^e1...pk^ek. Then a(n) = lcm(a(p1^e1), ..., a(pk^ek)).

a(2^k) = 5*2^(k-1) for k > 0. If a(p) != a(p^2) for p prime, then a(p^k) = p^(k-1)*a(p) for k > 0 [Waddill, 1992]. - Chai Wah Wu, Feb 25 2022

MATHEMATICA

n=4; Table[p=i; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 60}]

PROG

(Python)

from itertools import count

def A106295(n):

a = b = (4%n, 1%n, 3%n, 7%n)

s = sum(b) % n

for m in count(1):

b, s = b[1:] + (s, ), (s+s-b[0]) % n

if a == b:

return m # Chai Wah Wu, Feb 22-27 2022

CROSSREFS

Cf. A000078, A073817, A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1).

KEYWORD

nonn

AUTHOR

T. D. Noe, May 02 2005

STATUS

approved

A075116

Binomial transform of A073817: a(n)=Sum(Binomial(n,k)*A073817(k),(k=0,..,n)).

+20
5

4, 5, 9, 23, 69, 210, 627, 1846, 5405, 15809, 46254, 135382, 396327, 1160294, 3396892, 9944688, 29113741, 85232259, 249522603, 730492701, 2138562494, 6260774221, 18328804756, 53658712275, 157089206159, 459888386910

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..25.

N. J. A. Sloane, Transforms

Index entries for linear recurrences with constant coefficients, signature (5,-8,6,-1).

FORMULA

a(n)=5a(n-1)-8a(n-2)+6a(n-3)-a(n-4), a(0)=4, a(1)=5, a(2)=9, a(3)=23. G.f.: (4-15*z+16*z^2-6*z^3)/(1-5*z+8*z^2-6*z^3+z^4).

MATHEMATICA

CoefficientList[Series[(4-15*z+16*z^2-6*z^3)/(1-5*z+8*z^2-6*z^3+z^4), {z, 0, 30}], z]

LinearRecurrence[{5, -8, 6, -1}, {4, 5, 9, 23}, 30] (* Harvey P. Dale, Jun 24 2017 *)

CROSSREFS

Cf. A073817.

KEYWORD

easy,nonn

AUTHOR

Mario Catalani (mario.catalani(AT)unito.it), Sep 02 2002

STATUS

approved

A104577

Indices of prime generalized tetranacci numbers, A073817.

+20
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..32.

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

MATHEMATICA

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

CROSSREFS

Cf. A104576 (indices of prime generalized tribonacci numbers).

KEYWORD

nonn

AUTHOR

T. D. Noe, Mar 16 2005

STATUS

approved

A106296

Period of the Lucas 4-step sequence A073817 mod prime(n).

+20
3

5, 26, 312, 342, 120, 84, 4912, 6858, 12166, 280, 61568, 1368, 240, 162800, 103822, 303480, 205378, 226980, 100254, 357910, 2664, 998720, 1157520, 9320, 368872, 1030300, 10608, 1225042, 2614040, 13874, 2048382, 4530768, 136, 772880, 3307948

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This sequence is the same as the period of Fibonacci 4-step sequence (A000078) mod prime(n) except for n=103, which corresponds to the prime 563 because the discriminant of the characteristic polynomial x^4-x^3-x^2-x-1 is -563. We have a(n) < prime(n) for primes 563 and A106280.

LINKS

Table of n, a(n) for n=1..35.

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

FORMULA

a(n) = A106295(prime(n)).

MATHEMATICA

n=4; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 60}]

CROSSREFS

Cf. A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1), A106280 (primes p such that x^4-x^3-x^2-x-1 mod p has 4 distinct zeros), A106295.

KEYWORD

nonn

AUTHOR

T. D. Noe, May 02 2005

STATUS

approved

A075128

Binomial transform of generalized tetranacci numbers A073817: a(n)=Sum((-1)^k Binomial(n,k)*A073817(k),(k=0,..,n)).

+20
2

4, 3, 5, 3, 5, 8, 23, 52, 109, 201, 350, 586, 983, 1680, 2952, 5288, 9549, 17207, 30803, 54761, 96910, 171223, 302736, 536225, 951487, 1690208, 3003408, 5335509, 9473756, 16814058, 29833868, 52932503, 93922925, 166678207, 295825369

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

Index entries for linear recurrences with constant coefficients, signature (3, -2, -2, 3).

FORMULA

a(n)=3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4), a(0)=4, a(1)=3, a(2)=5, a(3)=3. G.f.: (4 - 9*z + 4*z^2 + 2*z^3)/(1 - 3*z + 2*z^2 + 2*z^3 - 3*z^4).

MATHEMATICA

CoefficientList[Series[(4-9*z+4*z^2+2*z^3)/(1-3*z+2*z^2+2*z^3-3*z^4), {z, 0, 40}], z]

LinearRecurrence[{3, -2, -2, 3}, {4, 3, 5, 3}, 40] (* Harvey P. Dale, Jul 13 2023 *)

CROSSREFS

Cf. A073817.

KEYWORD

easy,nonn

AUTHOR

Mario Catalani (mario.catalani(AT)unito.it), Sep 03 2002

STATUS

approved

A106300

Primes that do not divide any term of the Lucas 4-step sequence A073817.

+20
2

2789, 3847, 4451, 4751, 5431, 6203, 8317, 9533, 9629, 9907, 10093, 11839, 13903, 13907, 14207, 15823, 16319, 16759, 19543, 20939, 21379, 21859, 25303, 26683, 29483, 30871, 31267, 31699, 32003, 32771, 33967, 34963, 36229, 37061, 39983

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

If a prime p divides a term a(k) of this sequence, then k must be less than the period of the sequence mod p. Hence these primes are found by computing A073817(k) mod p for increasing k and stopping when either A073817(k) mod p = 0 or the end of the period is reached. Interestingly, for all of these primes, the period of the sequence A073817(k) mod p appears to be (p-1)/d, where d is a small integer.

LINKS

Table of n, a(n) for n=1..35.

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

MATHEMATICA

n=4; lst={}; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; While[s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; !(a==a0 || s==0)]; If[s>0, AppendTo[lst, p]], {i, 10000}]; lst

CROSSREFS

Cf. A053028 (primes not dividing any Lucas number), A106299 (primes not dividing any Lucas 3-step number), A106301 (primes not dividing any Lucas 5-step number).

KEYWORD

nonn

AUTHOR

T. D. Noe, May 02 2005

STATUS

approved

A106791

Sum of two consecutive squares of Lucas 4-step numbers (A073817).

+20
1

17, 10, 58, 274, 901, 3277, 12402, 46282, 171170, 635953, 2364489, 8785386, 32637202, 121265666, 450571589, 1674090725, 6220049810, 23110593298, 85867345570, 319039636721, 1185390110881, 4404311472106, 16364198176874

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

A106729 is sum of two consecutive squares of Lucas numbers (A001254), for which L(n)^2 + L(n+1)^2 = 5*{F(n)^2 + F(n+1)^2} = 5*A001519(n). A106789 is sum of two consecutive squares of Lucas 3-step numbers (A001644). Sum of two consecutive squares of Lucas 4-step numbers can be expressed in terms of tetranacci numbers, but not quite as neatly.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,4,6,12,-4,-6,0,-2,0,1).

FORMULA

a(n) = A073817(n)^2 + A073817(n+1)^2.

a(n) = 5*A073817(n)^2 + 4*A073817(n)*A073817(n-4) + A073817(n-4)^2.

G.f.: (17-24*x-30*x^2+16*x^3-143*x^4-21*x^5+46*x^6-32*x^7+2*x^8+17*x^9)/( (1- 3*x-3*x^2+x^3+x^4)*(1+x+2*x^2+2*x^3-2*x^4+x^5-x^6)). - Colin Barker, Dec 17 2012

EXAMPLE

a(0) = A073817(0)^2 + A073817(1)^2 = 4^2 + 1^2 = 16 + 1 = 17.

a(1) = A073817(1)^2 + A073817(2)^2 = 1^2 + 3^2 = 1 + 9 = 10.

a(2) = A073817(2)^2 + A073817(3)^2 = 3^2 + 7^2 = 9 + 49 = 58.

a(3) = A073817(3)^2 + A073817(4)^2 = 7^2 + 15^2 = 49 + 225 = 274.

a(4) = A073817(4)^2 + A073817(5)^2 = 15^2 + 26^2 = 225 + 676 = 901 = 30^2 + 1.

a(5) = A073817(5)^2 + A073817(6)^2 = 26^2 + 51^2 = 676 + 2601 = 3277.

MATHEMATICA

LinearRecurrence[{2, 4, 6, 12, -4, -6, 0, -2, 0, 1}, {17, 10, 58, 274, 901, 3277, 12402, 46282, 171170, 635953}, 40] (* G. C. Greubel, Apr 23 2019 *)

PROG

(PARI) my(x='x+O('x^40)); Vec((17-24*x-30*x^2+16*x^3-143*x^4-21*x^5 +46*x^6-32*x^7+2*x^8+17*x^9)/(1-2*x-4*x^2-6*x^3-12*x^4+4*x^5+6*x^6+2*x^8 -x^10)) \\ G. C. Greubel, Apr 23 2019

(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (17-24*x-30*x^2+16*x^3-143*x^4-21*x^5 +46*x^6-32*x^7+2*x^8+17*x^9)/(1-2*x-4*x^2 -6*x^3-12*x^4+4*x^5+6*x^6+2*x^8 -x^10) )); // G. C. Greubel, Apr 23 2019

(Sage) ((17-24*x-30*x^2+16*x^3-143*x^4-21*x^5 +46*x^6-32*x^7+2*x^8+ 17*x^9)/(1-2*x-4*x^2-6*x^3-12*x^4+4*x^5+6*x^6+2*x^8 -x^10)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

(GAP) a:=[17, 10, 58, 274, 901, 3277, 12402, 46282, 171170, 635953];; for n in [11..40] do a[n]:=2*a[n-1]+4*a[n-2]+6*a[n-3]+12*a[n-4]-4*a[n-5] -6*a[n-6]-2*a[n-8]+a[n-10]; od; a; # G. C. Greubel, Apr 23 2019

CROSSREFS

Cf. A001254, A001519, A001644, A073817, A106729.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, May 16 2005

STATUS

approved

A075112

a(n)=Sum((-1)^(i+Floor(n/2))S(2i+e),(i=0,..,Floor(n/2))), where S(n) are generalized Tetranacci numbers (A073817) and e=(1/2)(1-(-1)^n).

+20
0

4, 1, -1, 6, 16, 20, 35, 79, 156, 288, 552, 1077, 2079, 3994, 7696, 14848, 28623, 55159, 106320, 204952, 395060, 761489, 1467815, 2829318, 5453688, 10512308, 20263123, 39058439, 75287564, 145121432, 279730552, 539197989, 1039337543, 2003387514, 3861653592, 7443576640, 14347955295

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

a(n) is the convolution of S(n) with the sequence (1,0,-1,0,1,0,-1,0,....) A056594.

LINKS

Table of n, a(n) for n=0..36.

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

FORMULA

a(n)=a(n-1)+2a(n-3)+2a(n-4)+a(n-5)+a(n-6), a(0)=4, a(1)=1, a(2)=-1, a(3)=6, a(4)=16, a(5)=20. G.f.: (4 - 3*x - 2*x^2 - x^3)/(1 - x - 2*x^3 - 2*x^4 - x^5 - x^6).

MATHEMATICA

CoefficientList[Series[(4 - 3*x - 2*x^2 - x^3)/(1 - x - 2*x^3 - 2*x^4 - x^5 - x^6), {x, 0, 40}], x]

LinearRecurrence[{1, 0, 2, 2, 1, 1}, {4, 1, -1, 6, 16, 20}, 40] (* Harvey P. Dale, Mar 09 2013 *)

CROSSREFS

Cf. A073817, A056594, A074662, A074677, A074678, A075111.

KEYWORD

easy,sign

AUTHOR

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

STATUS

approved