A124789 -id:A124789 - cp's OEIS frontend

A103372 a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n-4) + a(n-5).

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 7, 8, 8, 9, 12, 15, 16, 17, 21, 27, 31, 33, 38, 48, 58, 64, 71, 86, 106, 122, 135, 157, 192, 228, 257, 292, 349, 420, 485, 549, 641, 769, 905, 1034, 1190, 1410, 1674, 1939, 2224, 2600, 3084, 3613, 4163, 4824, 5684, 6697, 7776

Offset: 1

Jonathan Vos Post, Feb 03 2005

k=4 case of the family of sequences whose k=1 case is the Fibonacci sequence A000045, k=2 case is the Padovan sequence A000931 (offset so as to begin 1,1,1) and k=3 case is A079398 (offset so as to begin 1,1,1,1).
The general case for integer k>1 is defined: a(1) = a(2) = ... = a(k+1) and for n>(k+1) a(n) = a(n-k) + a(n-[k+1]).
For this k=4 case, the ratio of successive terms a(n)/a(n-1) approaches the unique positive root of the irreducible characteristic polynomial: x^5 - x - 1 = 0, A160155.
The sequence of prime values in this k=4 case is A103382; The sequence of semiprime values in this k=4 case is A103392.

			a(14) = 5 because a(14) = a(14-4) + a(14-5) = a(10) + a(9) = 3 + 2 = 5.

Zanten, A. J. van, The golden ratio in the arts of painting, building and mathematics, Nieuw Archief voor Wiskunde, 4 (17) (1999) 229-245.

Indranil Ghosh, Table of n, a(n) for n = 1..14857
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [The hal link does not always work. - _N. J. A. Sloane_, Feb 19 2025]
J.-P. Allouche and T. Johnson, Narayana's cows and delayed morphisms, in G. Assayag, M. Chemillier, and C. Eloy, Troisièmes Journées d'Informatique Musicale, JIM '96, Île de Tatihou, France, 1996, pp. 2-7. [Local copy with annotations and a correction from _N. J. A. Sloane_, Feb 19 2025]
Richard Padovan, Dom Hans van der Laan and the Plastic Number.
E. S. Selmer, On the irreducibility of certain trinomials, Math. Scand., 4 (1956) 287-302.
J. Shallit, A generalization of automatic sequences, Theoretical Computer Science, 61 (1988), 1-16.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1).

Cf. A000931, A079398, A103373-A103380, A103382, A103392.

k = 4; Do[a[n] = 1, {n, k + 1}]; a[n_] := a[n] = a[n - k] + a[n - k - 1]; Array[a, 61]
LinearRecurrence[{0,0,0,1,1},{1,1,1,1,1},70] (* Harvey P. Dale, Apr 22 2015 *)

a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,1,0,0,0]^(n-1)*[1;1;1;1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f. -x*(1+x)*(1+x^2) / ( -1+x^4+x^5 ). - R. J. Mathar, Aug 26 2011

a(n) = A124789(n-2)+A124798(n-1). - R. J. Mathar, Jun 30 2020

Edited by Ray Chandler and Robert G. Wilson v, Feb 06 2005

A124788 Triangle read by rows: expansion of (1+xy)/(1-x^2y^2-x^3*y^2).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 0, 3, 3, 1, 0, 0, 0, 0, 0, 0, 1, 3, 4, 1, 0, 0, 0, 0, 0, 0, 0, 1, 6, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 4, 6, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 10, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 10, 6

Offset: 0

Paul Barry, Nov 07 2006

Row sums give A000931(n+5). Diagonal sums are A124789.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 0, 1, 1,
0, 0, 0, 1, 1,
0, 0, 0, 0, 2, 1,
0, 0, 0, 0, 1, 2, 1,
0, 0, 0, 0, 0, 1, 3, 1,
0, 0, 0, 0, 0, 0, 3, 3, 1,
0, 0, 0, 0, 0, 0, 1, 3, 4, 1,
0, 0, 0, 0, 0, 0, 0, 1, 6, 4, 1

Cf. A124745.

A124788 := proc(n,k) binomial(floor(k/2),n-k) ; end: for n from 0 to 20 do for k from 0 to n do printf("%d, ",A124788(n,k)) ; od ; od ; # R. J. Mathar, Feb 10 2007

Number triangle T(n,k) = binomial(floor(k/2),n-k).

Column k has g.f. x^k*(1+x)^floor(k/2). - Paul Barry, Feb 01 2007

More terms from R. J. Mathar, Feb 10 2007

A124746 Expansion of (1+x^2)/(1-x^4+x^5).

Original entry on oeis.org

1, 0, 1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 4, -4, 4, -5, 7, -8, 8, -9, 12, -15, 16, -17, 21, -27, 31, -33, 38, -48, 58, -64, 71, -86, 106, -122, 135, -157, 192, -228, 257, -292, 349, -420, 485, -549, 641, -769, 905, -1034, 1190, -1410, 1674, -1939, 2224, -2600

Offset: 0

Paul Barry, Nov 06 2006

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,-1).

Diagonal sums of A124744.

Cf. A124789.

LinearRecurrence[{0, 0, 0, 1, -1}, {1, 0, 1, 0, 1}, 100] (* Paolo Xausa, Aug 27 2024 *)

a(n) = Sum_{k=0..floor(n/2)} C(floor(k/2),n-2k)*(-1)^n.

a(n) = (-1)^n*A124789(n). - R. J. Mathar, Jun 30 2020

A376647 a(n) = Sum_{k=0..floor(n/3)} binomial(floor(k/2),n-3*k).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 3, 3, 2, 3, 3, 2, 4, 6, 5, 5, 6, 5, 6, 10, 11, 10, 11, 11, 11, 16, 21, 21, 21, 22, 22, 27, 37, 42, 42, 43, 44, 49, 64, 79, 84, 85, 87, 93, 113, 143, 163, 169, 172, 180, 206, 256, 306, 332, 341, 352, 386, 462, 562

Offset: 0

Seiichi Manyama, Oct 01 2024

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

Cf. A124789, A134816, A376648.

Cf. A017847.

a(n) = sum(k=0, n\3, binomial(k\2, n-3*k));

my(N=80, x='x+O('x^N)); Vec((1+x^3)/(1-x^6-x^7))

G.f.: (1-x^6)/((1-x^3) * (1-x^6-x^7)) = (1+x^3)/(1-x^6-x^7).
a(n) = a(n-6) + a(n-7).
a(n) = A017847(n) + A017847(n-3).

A376648 a(n) = Sum_{k=0..floor(n/4)} binomial(floor(k/2),n-4*k).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 1, 3, 3, 1, 1, 4, 6, 4, 2, 4, 6, 4, 2, 5, 10, 10, 6, 6, 10, 10, 6, 7, 15, 20, 16, 12, 16, 20, 16, 13, 22, 35, 36, 28, 28, 36, 36, 29, 35, 57, 71, 64, 56, 64, 72, 65, 64, 92, 128, 135

Offset: 0

Seiichi Manyama, Oct 01 2024

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

Cf. A124789, A134816, A376647.

Cf. A017867.

a(n) = sum(k=0, n\4, binomial(k\2, n-4*k));

my(N=80, x='x+O('x^N)); Vec((1+x^4)/(1-x^8-x^9))

G.f.: (1-x^8)/((1-x^4) * (1-x^8-x^9)) = (1+x^4)/(1-x^8-x^9).
a(n) = a(n-8) + a(n-9).
a(n) = A017867(n) + A017867(n-4).

cp's OEIS Frontend

A103372 a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n-4) + a(n-5).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A124788 Triangle read by rows: expansion of (1+x*y)/(1-x^2*y^2-x^3*y^2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A124746 Expansion of (1+x^2)/(1-x^4+x^5).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A376647 a(n) = Sum_{k=0..floor(n/3)} binomial(floor(k/2),n-3*k).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A376648 a(n) = Sum_{k=0..floor(n/4)} binomial(floor(k/2),n-4*k).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A124788 Triangle read by rows: expansion of (1+xy)/(1-x^2y^2-x^3*y^2).