A017847 -id:A017847 - cp's OEIS frontend

A373936 Number of compositions of 7*n-4 into parts 6 and 7.

0, 0, 0, 1, 5, 15, 35, 70, 126, 211, 342, 573, 1079, 2366, 5733, 14197, 34223, 78832, 173166, 364876, 745066, 1493990, 2985725, 6030652, 12428911, 26199706, 56231526, 121847272, 264270015, 570020037, 1218672066, 2581172411, 5424947523, 11347651254

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-6,1).

Cf. A369809, A373912, A373933, A373934, A373935, A373937.

Cf. A017847, A369807.

a(n) = sum(k=0, n\6, binomial(n+k, n-4-6*k));

a(n) = A017847(7*n-4).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-4-6*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7).
G.f.: x^4*(1-x)^2/((1-x)^7 - x^6).
a(n) = A373937(n+1)-A373937(n). - R. J. Mathar, Jun 24 2024

A373937 Number of compositions of 7*n-5 into parts 6 and 7.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 463, 805, 1378, 2457, 4823, 10556, 24753, 58976, 137808, 310974, 675850, 1420916, 2914906, 5900631, 11931283, 24360194, 50559900, 106791426, 228638698, 492908713, 1062928750, 2281600816, 4862773227, 10287720750

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-6,1).

Cf. A369809, A373912, A373933, A373934, A373935, A373936.

Cf. A017847, A369808.

a(n) = sum(k=0, n\6, binomial(n+k, n-5-6*k));

a(n) = A017847(7*n-5).
a(n) = Sum_{k=0..floor(n/6)} binomial(n+k,n-5-6*k).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 6*a(n-6) + a(n-7).
G.f.: x^5*(1-x)/((1-x)^7 - x^6).
a(n) = A369809(n+1)-A369809(n). - R. J. Mathar, Jun 24 2024

A017901 Expansion of 1/(1 - x^7 - x^8 - ...).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 13, 17, 22, 28, 35, 43, 53, 66, 83, 105, 133, 168, 211, 264, 330, 413, 518, 651, 819, 1030, 1294, 1624, 2037, 2555, 3206, 4025, 5055, 6349, 7973, 10010, 12565, 15771, 19796, 24851, 31200, 39173

Offset: 0

N. J. A. Sloane

A Lamé sequence of higher order.
a(n) = number of compositions of n in which each part is >= 7. - Milan Janjic, Jun 28 2010
a(n+7) equals the number of n-length binary words such that 0 appears only in a run length that is a multiple of 7. - Milan Janjic, Feb 17 2015
A017847(n) = |a(-n)| for n>=0. - Michael Somos, Oct 28 2018

			G.f. = 1 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + 2*x^14 + ... - _Michael Somos_, Oct 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1000
I. M. Gessel, Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1).

For Lamé sequences of orders 1 through 9 see A000045, A000930, A017898, A017899, A017900, A017901, A017902, A017903, A017904.

Cf. A005709, A017847.

f := proc(r) local t1,i; t1 := []; for i from 1 to r do t1 := [op(t1),0]; od: for i from 1 to r+1 do t1 := [op(t1),1]; od: for i from 2*r+2 to 50 do t1 := [op(t1),t1[i-1]+t1[i-1-r]]; od: t1; end; # set r = order
a := n -> (Matrix(7, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 0$5, 1][i] else 0 fi)^n)[7,7]: seq(a(n), n=0..50); # Alois P. Heinz, Aug 04 2008

LinearRecurrence[{1,0,0,0,0,0,1}, {1,0,0,0,0,0,0}, 60] (* Jean-François Alcover, Mar 28 2017 *)

Vec((x-1)/(x-1+x^7)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

{a(n) = if( n < 0, polcoeff( 1 / (1 + x^6 - x^7) + x * O(x^-n), -n), polcoeff( (1 - x) / (1 - x - x^7) + x * O(x^n), n))}; /* Michael Somos, Oct 28 2018 */

G.f.: (x-1)/(x-1+x^7). - Alois P. Heinz, Aug 04 2008
For positive integers n and k such that k <= n <= 7*k, and 6 divides n-k, define c(n,k) = binomial(k,(n-k)/6), and c(n,k) = 0, otherwise. Then, for n>=1, a(n+7) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011
a(n) = A005709(n) - A005709(n-1). - R. J. Mathar, Sep 07 2016
0 == a(n) + a(n+6) - a(n+7) for all n in Z. - Michael Somos, Oct 28 2018

A127841 a(1)=1, a(2)=...=a(7)=0, a(n) = a(n-7)+a(n-6) for n>7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 3, 1, 0, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 2, 7, 21, 35, 35, 21, 8, 9, 28, 56, 70, 56, 29, 17, 37, 84, 126, 126, 85, 46, 54, 121, 210, 252, 211

Offset: 1

Stephen Suter (sms5064(AT)psu.edu), Apr 02 2007

Part of the phi_k family of sequences defined by a(1)=1,a(2)=...=a(k)=0, a(n)=a(n-k)+a(n-k+1) for n>k. phi_2 is a shift of the Fibonacci sequence and phi_3 is a shift of the Padovan sequence.

Apart from offset same as A017847. - Georg Fischer, Oct 07 2018

S. Suter, Binet-like formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007. [Apparently unpublished as of May 2016]

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

a:=[1,0,0,0,0,0,0];;  for n in [8..80] do a[n]:=a[n-6]+a[n-7]; od; a; # Muniru A Asiru, Oct 07 2018

CoefficientList[Series[(1-x)*(1+x)*(1-x+x^2)*(1+x+x^2) / (1-x^6-x^7), {x, 0, 50}], x] (* or *)
LinearRecurrence[{0, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0, 0}, 50] (* Stefano Spezia, Oct 08 2018 *)

Vec(x*(1-x)*(1+x)*(1-x+x^2)*(1+x+x^2)/(1-x^6-x^7) + O(x^100)) \\ Colin Barker, May 30 2016

Binet-like formula: a(n) = Sum_{i=1..7} (r_i^n)/(6(r_i)^2+7(r_i)) where r_i is a root of x^7=x+1.

G.f.: x*(1-x)*(1+x)*(1-x+x^2)*(1+x+x^2) / (1-x^6-x^7). - Colin Barker, May 30 2016

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).

cp's OEIS Frontend

A373936 Number of compositions of 7*n-4 into parts 6 and 7.

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A373937 Number of compositions of 7*n-5 into parts 6 and 7.

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A017901 Expansion of 1/(1 - x^7 - x^8 - ...).

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A127841 a(1)=1, a(2)=...=a(7)=0, a(n) = a(n-7)+a(n-6) for n>7.

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

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