A017837 -id:A017837 - cp's OEIS frontend

A107025 Binomial transform of the expansion of 1/(1-x^5-x^6).

1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 464, 938, 1808, 3459, 6826, 14198, 30960, 69143, 154433, 340006, 734561, 1561313, 3286129, 6900097, 14542101, 30855957, 65908862, 141395972, 303745077, 651763377, 1395140215, 2978858672

Offset: 0

Paul Barry, May 09 2005

In general, the binomial transform of 1/(1-x^r-x^(r+1)) is given by (1-x)^r/((1-x)^(r+1)-x^r), with a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n+k,(r+1)k) = Sum_{k=0..floor((r+1)n/r)} binomial(k,(r+1)n-r*k).

Number of compositions of 6*n into parts 5 and 6. - Seiichi Manyama, Jun 22 2024

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,7,-1).

Cf. A017837, A099099, A099131.

G.f.: (1-x)^5/((1-x)^6-x^5).
a(n) = 6a(n-1)-15a(n-2)+20a(n-3)-15a(n-4)+7a(n-5)-a(n-6).
a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n+k, 6k).
a(n) = Sum_{k=0..floor(6n/5)} binomial(k, 6n-5k).
a(n) = A017837(6*n). - Seiichi Manyama, Jun 22 2024

A306713 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 1, 5, 1, 0, 0, 1, 1, 8, 1, 0, 0, 0, 1, 2, 13, 1, 0, 0, 0, 1, 0, 2, 21, 1, 0, 0, 0, 0, 1, 1, 3, 34, 1, 0, 0, 0, 0, 1, 0, 2, 4, 55, 1, 0, 0, 0, 0, 0, 1, 0, 1, 5, 89, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 7, 144, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 9, 233

Offset: 0

Seiichi Manyama, Mar 05 2019

A(n,k) is the number of compositions of n into parts k and k+1.

			Square array begins:
    1, 1, 1, 1, 1, 1, 1, 1, 1, ...
    1, 0, 0, 0, 0, 0, 0, 0, 0, ...
    2, 1, 0, 0, 0, 0, 0, 0, 0, ...
    3, 1, 1, 0, 0, 0, 0, 0, 0, ...
    5, 1, 1, 1, 0, 0, 0, 0, 0, ...
    8, 2, 0, 1, 1, 0, 0, 0, 0, ...
   13, 2, 1, 0, 1, 1, 0, 0, 0, ...
   21, 3, 2, 0, 0, 1, 1, 0, 0, ...
   34, 4, 1, 1, 0, 0, 1, 1, 0, ...
   55, 5, 1, 2, 0, 0, 0, 1, 1, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-10 give A000045(n+1), A182097, A017817, A017827, A017837, A017847, A017857, A017867, A017877, A017887.

Cf. A306646, A306680.

T[n_, k_] := Sum[Binomial[j, n-k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(j,n-k*j).

A369794 Expansion of 1/(1 - x^5/(1-x)^6).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 253, 474, 870, 1651, 3367, 7372, 16762, 38183, 85290, 185573, 394555, 826752, 1724816, 3613968, 7642004, 16313856, 35052905, 75487110, 162349105, 348018300, 743376838, 1583718457, 3370144462, 7173308802, 15285181447

Offset: 0

Seiichi Manyama, Feb 01 2024

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

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,7,-1).

Cf. A095263, A290998, A368475.

Cf. A000389, A099242, A017837, A107025.

my(N=40, x='x+O('x^N)); Vec(1/(1-x^5/(1-x)^6))

a(n) = A107025(n)-A107025(n-1). First differences of A107025.
a(n) = A017837(6*n-5) = Sum_{k=0..floor((6*n-5)/5)} binomial(k,6*n-5-5*k) for n > 0.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6) for n > 6.
a(n) = Sum_{k=0..floor(n/5)} binomial(n-1+k,n-5*k).

A099132 Quintisection of 1/(1-x^5-x^6).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 253, 464, 804, 1354, 2289, 4005, 7372, 14198, 28033, 55523, 108699, 208982, 394555, 734561, 1357136, 2504932, 4643816, 8671852, 16313856, 30855957, 58502733, 110882143, 209689343, 395358538, 743376838

Offset: 0

Paul Barry, Sep 29 2004

Seiichi Manyama, Table of n, a(n) for n = 0..3646
V. C. Harris, C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,1,5).
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1,1)

Cf. A005676, A017837.

LinearRecurrence[{5,-10,10,-5,1,1},{1,1,1,1,1,1},40] (* Harvey P. Dale, Aug 20 2012 *)

Vec((1-x)^4/((1-x)^5-x^6) + O(x^40)) \\ Michel Marcus, Sep 06 2017

G.f.: (1-x)^4/((1-x)^5-x^6);
a(n) = Sum_{k=0..n} binomial(k, 5(n-k));
a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+a(n-5)+a(n-6);
a(n) = A017837(5n).
a(n) = Sum_{k=0..floor(n/5)} binomial(n-k, 5k). - Paul Barry, May 09 2005

A339089 Number of compositions (ordered partitions) of n into distinct parts congruent to 5 mod 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 4, 1, 0, 0, 0, 6, 4, 1, 0, 0, 0, 6, 6, 1, 0, 0, 0, 12, 6, 1, 0, 0, 0, 18, 8, 1, 0, 0, 24, 24, 8, 1, 0, 0, 24, 30, 10, 1, 0, 0, 48, 42, 10, 1, 0, 0, 72, 48, 12, 1, 0, 0, 120, 60, 12, 1, 0, 120, 144

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(33) = 6 because we have [17, 11, 5], [17, 5, 11], [11, 17, 5], [11, 5, 17], [5, 17, 11] and [5, 11, 17].

Index entries for sequences related to compositions

Cf. A016969, A017837, A032020, A032021, A109702, A281244, A337547, A337548, A339059, A339060, A339086, A339087, A339088.

nmax = 86; CoefficientList[Series[Sum[k! x^(k (3 k + 2))/Product[1 - x^(6 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(3*k + 2)) / Product_{j=1..k} (1 - x^(6*j)).

A373961 Number of compositions of 6*n-1 into parts 5 and 6.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 15, 44, 129, 340, 804, 1742, 3550, 7009, 13835, 28033, 58993, 128136, 282569, 622575, 1357136, 2918449, 6204578, 13104675, 27646776, 58502733, 124411595, 265807567, 569552644, 1221316021, 2616456236, 5595314908, 11944318042, 25466629978

Offset: 1

Seiichi Manyama, Jun 23 2024

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,7,-1).

Cf. A107025, A369794, A373962, A373963, A373964.

Cf. A017837.

a(n) = sum(k=0, n\5, binomial(n+k, n-1-5*k));

a(n) = A017837(6*n-1).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-1-5*k).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6).
G.f.: x*(1-x)^4/((1-x)^6 - x^5).
a(n) = A373962(n+1) - A373962(n).

A373962 Number of compositions of 6*n-2 into parts 5 and 6.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 22, 37, 81, 210, 550, 1354, 3096, 6646, 13655, 27490, 55523, 114516, 242652, 525221, 1147796, 2504932, 5423381, 11627959, 24732634, 52379410, 110882143, 235293738, 501101305, 1070653949, 2291969970, 4908426206, 10503741114, 22448059156

Offset: 1

Seiichi Manyama, Jun 24 2024

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,7,-1).

Cf. A107025, A369794, A373961, A373963, A373964.

Cf. A017837.

LinearRecurrence[{6,-15,20,-15,7,-1},{0,1,3,6,10,15},40] (* Harvey P. Dale, Nov 16 2024 *)

a(n) = sum(k=0, n\5, binomial(n+k, n-2-5*k));

a(n) = A017837(6*n-2).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-2-5*k).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6).
G.f.: x^2*(1-x)^3/((1-x)^6 - x^5).
a(n) = A373963(n+1) - A373963(n).

A373963 Number of compositions of 6*n-3 into parts 5 and 6.

Original entry on oeis.org

0, 0, 1, 4, 10, 20, 35, 57, 94, 175, 385, 935, 2289, 5385, 12031, 25686, 53176, 108699, 223215, 465867, 991088, 2138884, 4643816, 10067197, 21695156, 46427790, 98807200, 209689343, 444983081, 946084386, 2016738335, 4308708305, 9217134511, 19720875625

Offset: 1

Seiichi Manyama, Jun 24 2024

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,7,-1).

Cf. A107025, A369794, A373961, A373962, A373964.

Cf. A017837.

a(n) = sum(k=0, n\5, binomial(n+k, n-3-5*k));

a(n) = A017837(6*n-3).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-3-5*k).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6).
G.f.: x^3*(1-x)^2/((1-x)^6 - x^5).
a(n) = A373964(n+1) - A373964(n).

A373964 Number of compositions of 6*n-4 into parts 5 and 6.

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 70, 127, 221, 396, 781, 1716, 4005, 9390, 21421, 47107, 100283, 208982, 432197, 898064, 1889152, 4028036, 8671852, 18739049, 40434205, 86861995, 185669195, 395358538, 840341619, 1786426005, 3803164340, 8111872645, 17329007156

Offset: 1

Seiichi Manyama, Jun 24 2024

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,7,-1).

Cf. A107025, A369794, A373961, A373962, A373963.

Cf. A017837.

a(n) = sum(k=0, n\5, binomial(n+k, n-4-5*k));

a(n) = A017837(6*n-4).
a(n) = Sum_{k=0..floor(n/5)} binomial(n+k,n-4-5*k).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 7*a(n-5) - a(n-6).
G.f.: x^4*(1-x)/((1-x)^6 - x^5).
a(n) = A369794(n+1) - A369794(n).

A127840 a(1)=1, a(2)=...=a(6)=0, a(n) = a(n-6)+a(n-5) for n>6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 2, 6, 15, 20, 15, 7, 8, 21, 35, 35, 22, 15, 29, 56, 70, 57, 37, 44, 85, 126, 127, 94, 81, 129, 211, 253, 221, 175, 210, 340, 464, 474, 396, 385, 550

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.

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
Sadjia Abbad and Hacène Belbachir, The r-Fibonacci polynomial and its companion sequences linked with some classical sequences, Integers (2025), Vol. 25, Art. No. A38. See p. 17.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1,1).

Vec(x*(1-x)*(1+x+x^2+x^3+x^4)/(1-x^5-x^6) + O(x^100)) \\ Colin Barker, May 30 2016

Binet-like formula: a(n) = Sum_{i=1..6} (r_i^n)/(5(r_i)^2+6(r_i)) where r_i is a root of x^6=x+1.
a(n) = A017837(n-6). - R. J. Mathar, Sep 20 2012
G.f.: x*(1-x)*(1+x+x^2+x^3+x^4) / (1-x^5-x^6). - Colin Barker, May 30 2016

cp's OEIS Frontend

A107025 Binomial transform of the expansion of 1/(1-x^5-x^6).

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A306713 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).

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A369794 Expansion of 1/(1 - x^5/(1-x)^6).

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A099132 Quintisection of 1/(1-x^5-x^6).

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A339089 Number of compositions (ordered partitions) of n into distinct parts congruent to 5 mod 6.

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A373961 Number of compositions of 6*n-1 into parts 5 and 6.

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A373962 Number of compositions of 6*n-2 into parts 5 and 6.

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A373963 Number of compositions of 6*n-3 into parts 5 and 6.

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A373964 Number of compositions of 6*n-4 into parts 5 and 6.

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