A001648 -id:A001648 - cp's OEIS frontend

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

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

A001645 A Fielder sequence.

Original entry on oeis.org

1, 3, 7, 11, 26, 45, 85, 163, 304, 578, 1090, 2057, 3888, 7339, 13862, 26179, 49437, 93366, 176321, 332986, 628852, 1187596, 2242800, 4235569, 7998951, 15106172, 28528288, 53876211, 101746240, 192149690, 362878313, 685302531, 1294206745, 2444133829

Offset: 1

N. J. A. Sloane

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).

T. D. Noe, Table of n, a(n) for n = 1..1000
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
Wikipedia, Companion matrix.
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 (1,1,1,0,1).

Cf. A001609, A001634 - A001636, A001638 - A001645, A001648, A001649.

I:=[1,3,7,11,26]; [n le 5 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017

A001645:=-(1+2*z+3*z**2+5*z**4)/(-1+z+z**2+z**3+z**5); [Conjectured by Simon Plouffe in his 1992 dissertation.]

LinearRecurrence[{1, 1, 1, 0, 1}, {1, 3, 7, 11, 26}, 50] (* T. D. Noe, Aug 09 2012 *)
CoefficientList[Series[x*(1+2*x+3*x^2+5*x^4)/(1-x-x^2-x^3-x^5), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)

a(n)=if(n<0,0,polcoeff(x*(1+2*x+3*x^2+5*x^4)/(1-x-x^2-x^3-x^5)+x*O(x^n),n))

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

a(n) = trace(M^n), where M = [0, 0, 0, 0, 1; 1, 0, 0, 0, 0; 0, 1, 0, 0, 1; 0, 0, 1, 0, 1; 0, 0, 0, 1, 1] is the 5 x 5 companion matrix to the monic polynomial x^5 - x^4 - x^3 - x^2 - 1. It follows that the sequence satisfies the Gauss congruences: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. - Peter Bala, Jan 09 2023

A127221 a(n) = 2^npentanacci(n) or (2^n)A023424(n-1).

Original entry on oeis.org

2, 12, 56, 240, 992, 3648, 14464, 57088, 224768, 883712, 3471360, 13651968, 53682176, 211075072, 829915136, 3263102976, 12830244864, 50447253504, 198353354752, 779904614400, 3066503888896, 12057176965120, 47407572189184, 186401664532480, 732912043425792

Offset: 1

Artur Jasinski, Jan 09 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,4,8,16,32).

Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127220, A127222.

Cf. A023424, A074048.

I:=[2, 12, 56, 240, 992]; [n le 5 select I[n] else 2*Self(n-1) + 4*Self(n-2) + 8*Self(n-3) + 16*Self(n-4) + 32*Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017

Table[Tr[MatrixPower[2*{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{2, 4, 8, 16, 32}, {2, 12, 56, 240, 992}, 50] (* G. C. Greubel, Dec 19 2017 *)

x='x+O('x^30); Vec(-2*x*(1 +4*x +12*x^2 +32*x^3 +80*x^4)/(-1 +2*x +4*x^2 +8*x^3 +16*x^4 +32*x^5)) \\ G. C. Greubel, Dec 19 2017

a(n) = Trace of matrix [({2,2,2,2,2},{2,0,0,0,0},{0,2,0,0,0},{0,0,2,0,0},{0,0,0,2,0})^n].
a(n) = 2^n * Trace of matrix [({1,1,1,1,1},{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0})^n].
G.f.: -2*x*(1 +4*x +12*x^2 +32*x^3 +80*x^4)/(-1 +2*x +4*x^2 +8*x^3 +16*x^4 +32*x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; corrected by R. J. Mathar, Sep 16 2009
a(n) = 2*a(n-1)+4*a(n-2)+8*a(n-3)+16*a(n-4)+32*a(n-5). - Colin Barker, Sep 02 2013

Definition corrected by R. J. Mathar, Sep 17 2009

More terms from Colin Barker, Sep 02 2013

A127222 a(n) = 3^npentanacci(n) or (3^n)A023424(n-1).

Original entry on oeis.org

3, 27, 189, 1215, 7533, 41553, 247131, 1463103, 8640837, 50959287, 300264165, 1771292853, 10447598619, 61618989627, 363414767589, 2143339285311, 12641143135581, 74555586323649, 439717218548643, 2593383067853775, 15295369041550269, 90209719910309895

Offset: 1

Artur Jasinski, Jan 09 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,9,27,81,243).

Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127220, A127221.

Cf. A023424, A074048.

I:=[3, 27, 189, 1215, 7533]; [n le 5 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3) + 81*Self(n-4) + 243*Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017

Table[Tr[MatrixPower[3*{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{3, 9, 27, 81, 243}, {3, 27, 189, 1215, 7533}, 50] (* G. C. Greubel, Dec 19 2017 *)

x='x+O('x^30); Vec(-3*x*(1 +6*x +27*x^2 +108*x^3 +405*x^4)/(-1 +3*x +9*x^2 +27*x^3 +81*x^4 +243*x^5)) \\ G. C. Greubel, Dec 19 2017

a(n) = Trace of matrix [({3,3,3,3,3},{3,0,0,0,0},{0,3,0,0,0},{0,0,3,0,0},{0,0,0,3,0})^n].
a(n) = 3^n * Trace of matrix [({1,1,1,1,1},{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0})^n].
G.f.: -3*x*(1 +6*x +27*x^2 +108*x^3 +405*x^4)/(-1 +3*x +9*x^2 +27*x^3 +81*x^4 +243*x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
a(n) = 3*a(n-1)+9*a(n-2)+27*a(n-3)+81*a(n-4)+243*a(n-5). - Colin Barker, Sep 02 2013

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009
Definition corrected by R. J. Mathar, Sep 17 2009
More terms from Colin Barker, Sep 02 2013

A001641 A Fielder sequence: a(n) = a(n-1) + a(n-2) + a(n-4).

Original entry on oeis.org

1, 3, 4, 11, 16, 30, 50, 91, 157, 278, 485, 854, 1496, 2628, 4609, 8091, 14196, 24915, 43720, 76726, 134642, 236283, 414645, 727654, 1276941, 2240878, 3932464, 6900996, 12110401, 21252275, 37295140, 65448411, 114853952, 201554638, 353703730, 620706779

Offset: 1

N. J. A. Sloane

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).

T. D. Noe, Table of n, a(n) for n = 1..1000
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.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Wikipedia, Companion matrix.
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 (1,1,0,1).

Cf. A001609, A001634 - A001636, A001638 - A001645, A001648, A001649, A060945.

I:=[1,3,4,11]; [n le 4 select I[n] else Self(n-1) + Self(n-2) + Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 09 2018

A001641:=-(1+2*z+4*z**3)/(z+1)/(z**3-z**2+2*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{1, 1, 0, 1}, {1, 3, 4, 11}, 50] (* T. D. Noe, Aug 09 2012 *)

a(n):=(sum(sum(binomial(j,n-4*k+3*j)*binomial(k,j),j,floor((4*k-n)/3),floor((4*k-n)/2))/k,k,1,n))*n; /* Vladimir Kruchinin, May 25 2011 */

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

G.f.: x*(1+2*x+4*x^3)/(1-x-x^2-x^4).
a(n) = n*Sum_{k=1..n} Sum_{j=floor((4*k-n)/3)..floor((4*k-n)/2)} binomial(j,n-4*k+3*j)*binomial(k,j)/k. - Vladimir Kruchinin, May 25 2011
a(n) = Trace(M^n), where M = [0, 0, 0, 1; 1, 0, 0, 0; 0, 1, 0, 1; 0, 0, 1, 1] is the companion matrix to the monic polynomial x^4 - x^3 - x^2 - 1. It follows that the sequence satisfies the Gauss congruences: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for positive integers n and r and all primes p. See Zarelua. - Peter Bala, Dec 31 2022

A125129 Partial sums of diagonals of array of k-step Lucas numbers as in A125127, read by antidiagonals.

Original entry on oeis.org

1, 1, 4, 1, 8, 11, 1, 12, 19, 26, 1, 19, 33, 45, 57, 1, 30, 58, 84, 102, 120, 1, 48, 101, 157, 197, 222, 247, 1, 77, 179, 292, 380, 436, 469, 502, 1, 124, 318, 546, 731, 855, 929, 971, 1013, 1, 200, 567, 1026, 1409, 1674, 1838, 1932, 1984, 2036

Offset: 1

Jonathan Vos Post, Nov 23 2006

Array of partial sums of diagonals of L(k,n) begins: 0.|.1...4..11...26...57..120..247..502.1013.2036.
|.1...8..19...45..102..222..469..971.1984.
|.1..12..33...84..197..436..929.1932.
|.1..19..58..157..380..855.1838.
|.1..30.101..292..731.1674.
|.1..48.179..546.1409.
|.1..77.318.1026.
|.1.124.567.
|.1.200.
|.1.

			Row 1 of the derived array is the partial sum of the diagonal above the main diagonal of array of k-step Lucas numbers as in A125127, hence the partial sums of: 1, 7, 11, 26, 57, 120, 247, 502, 103, ... are 1 = 1; 8 = 1 + 7; 19 = 1 + 7 + 11; 45 = 1 + 7 + 11 + 26; and so forth.

Cf. A000012, A000032, A000204, A001644, A001648, A048887, A048888, A074048, A074584, A092921, A104621, A105754, A105755, A125127, A000295.

Row 0 = SUM[i=1..n]L(i,i) = A127128 = partial sum of main diagonal of array of A125127. Row 1 = SUM[i=1..n]L(i,i+1) = partial sum of diagonal above main diagonal of array of A125127. Row 2 = SUM[i=1..n]L(i,i+2) = partial sum of diagonal 2 above main diagonal of array of A125127. .. Row m = SUM[i=1..n]L(i,i+m) = partial sum of diagonal 2 above main diagonal of array of A125127.

cp's OEIS Frontend

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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A001645 A Fielder sequence.

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A127221 a(n) = 2^n*pentanacci(n) or (2^n)*A023424(n-1).

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A127222 a(n) = 3^n*pentanacci(n) or (3^n)*A023424(n-1).

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A001641 A Fielder sequence: a(n) = a(n-1) + a(n-2) + a(n-4).

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A125129 Partial sums of diagonals of array of k-step Lucas numbers as in A125127, read by antidiagonals.

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A127221 a(n) = 2^npentanacci(n) or (2^n)A023424(n-1).

A127222 a(n) = 3^npentanacci(n) or (3^n)A023424(n-1).