A092521 -id:A092521 - cp's OEIS frontend

A077828 Expansion of 1/(1-3x-3x^2-3*x^3).

1, 3, 12, 48, 189, 747, 2952, 11664, 46089, 182115, 719604, 2843424, 11235429, 44395371, 175422672, 693160416, 2738935377, 10822555395, 42763953564, 168976333008, 667688525901, 2638286437419, 10424853888984, 41192486556912, 162766880649945, 643152663287523

Offset: 0

N. J. A. Sloane, Nov 17 2002

Michael De Vlieger, Table of n, a(n) for n = 0..1675
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 19.
Index entries for linear recurrences with constant coefficients, signature (3,3,3).

Partial sums of S(n, x), for x=1...12, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-7.

Cf. A071675.

CoefficientList[Series[1/(1-3x-3x^2-3x^3),{x,0,30}],x] (* or *) LinearRecurrence[ {3,3,3},{1,3,12},30] (* Harvey P. Dale, Dec 25 2018 *)

Vec(1/(1-3*x-3*x^2-3*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n) = sum{k=0..n, T(n-k, k)3^(n-k)}, T(n, k) = trinomial coefficients (A027907). - Paul Barry, Feb 15 2005

a(n) = sum{k=0..n, sum{i=0..floor((n-k)/2), C(n-k-i, i)C(k, n-k-i)}*3^k}. - Paul Barry, Apr 26 2005

A077829 Expansion of 1/(1-3x-3x^2-2*x^3).

Original entry on oeis.org

1, 3, 12, 47, 183, 714, 2785, 10863, 42372, 165275, 644667, 2514570, 9808261, 38257827, 149227404, 582072215, 2270414511, 8855914986, 34543132921, 134737972743, 525555146964, 2049965624963, 7996038261267, 31189121952618, 121655411891581, 474525678055131

Offset: 0

N. J. A. Sloane, Nov 17 2002

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

Partial sums of S(n, x), for x=1...14, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139.

CoefficientList[Series[1/(1 - 3*x - 3*x^2 - 2*x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 20 2024 *)
LinearRecurrence[{3,3,2},{1,3,12},30] (* Harvey P. Dale, Dec 20 2024 *)

Vec(1/(1-3*x-3*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

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

a(n) = 3*a(n-1) + 3*a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, Jan 20 2024

A077831 Expansion of 1/(1-3x-2x^2-2*x^3).

Original entry on oeis.org

1, 3, 11, 41, 151, 557, 2055, 7581, 27967, 103173, 380615, 1404125, 5179951, 19109333, 70496151, 260067021, 959412031, 3539362437, 13057045415, 48168685181, 177698871247, 655548074933, 2418379337655, 8921631905325, 32912750541151, 121418274109413

Offset: 0

N. J. A. Sloane, Nov 17 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,2).

Partial sums of S(n, x), for x=1...16, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139, A097829, A097830.

CoefficientList[Series[1/(1-3x-2x^2-2x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,2,2},{1,3,11},30] (* Harvey P. Dale, Feb 28 2025 *)

Vec(1/(1-3*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

A214985 Array: T(m,n) = (F(n) + F(2n) + ... + F(nm))/F(n), by antidiagonals; transpose of A214984.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 5, 12, 7, 1, 8, 22, 33, 12, 1, 12, 56, 94, 88, 20, 1, 19, 134, 385, 399, 232, 33, 1, 30, 342, 1487, 2640, 1691, 609, 54, 1, 48, 872, 6138, 16492, 18096, 7164, 1596, 88, 1, 77, 2256, 25319, 110143, 182900, 124033, 30348, 4180, 143

Offset: 1

Clark Kimberling, Oct 28 2012

row 1: A001612 (except for initial term)
col 1: A000071
col 2: A027941
col 3: A049652
col 4: A092521
col 6: A049664
col 8: A156093 without minus signs

			Northwest corner:
1....1.....1......1.......1
2....4.....5......8.......12
4....12....22.....56......134
7....33....94.....385.....1487
12...88....399....2640....16492
20...232...1691...18096...182900

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A214978, A214984, A214986.

F[n_] := Fibonacci[n]; L[n_] := LucasL[n];
t[m_, n_] := (1/F[n])*Sum[F[k*n], {k, 1, m}]
TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]]
Flatten[Table[t[k, n + 1 - k], {n, 1, 12}, {k, 1, n}]]

For odd-numbered columns (m odd):
T(m,n) = (F(m*n+m) + F(m*n) - F(m))/(F(m)*L(m)).
For even-numbered columns (m even):
T(m,n) = (F(m*n+m) - F(m*n) - F(m))/(F(m)*(L(m)-1)).

A276472 Modified Pascal's triangle read by rows: T(n,k) = T(n-1,k) + T(n-1,k-1), 12. T(n,n) = T(n,n-1) + T(n-1,n-1), n>1. T(1,1) = 1, T(2,1) = 1. n>=1.

Original entry on oeis.org

1, 1, 2, 4, 3, 5, 11, 7, 8, 13, 29, 18, 15, 21, 34, 76, 47, 33, 36, 55, 89, 199, 123, 80, 69, 91, 144, 233, 521, 322, 203, 149, 160, 235, 377, 610, 1364, 843, 525, 352, 309, 395, 612, 987, 1597, 3571, 2207, 1368, 877, 661, 704, 1007, 1599, 2584, 4181

Offset: 1

Yuriy Sibirmovsky, Sep 12 2016

The recurrence relations for the border terms are the only way in which this differs from Pascal's triangle.
Column T(2n,n+1) appears to be divisible by 4 for n>=2; T(2n-1,n) divisible by 3 for n>=2; T(2n,n-2) divisible by 2 for n>=3.
The symmetry of T(n,k) can be observed in a hexagonal arrangement (see the links).
Consider T(n,k) mod 3 = q. Terms with q = 0 show reflection symmetry with respect to the central column T(2n-1,n), while q = 1 and q = 2 are mirror images of each other (see the link).

			Triangle T(n,k) begins:
n\k 1    2    3    4   5    6    7    8    9
1   1
2   1    2
3   4    3    5
4   11   7    8    13
5   29   18   15   21   34
6   76   47   33   36   55   89
7   199  123  80   69   91   144 233
8   521  322  203  149  160  235 377  610
9   1364 843  525  352  309  395 612  987  1597
...
In another format:
__________________1__________________
_______________1_____2_______________
____________4_____3_____5____________
________11_____7_____8_____13________
____29_____18_____15____21_____34____
_76_____47____33_____36____55_____89_

Yuriy Sibirmovsky, T(n,k), read by rows as a linear sequence a(j) for j = 1..5050
Yuriy Sibirmovsky, Symmetrical hexagonal arrangement for initial terms of T(n,k)
Yuriy Sibirmovsky, T(n,k) compared with Pascal's triangle
Yuriy Sibirmovsky, Illustration for T(n,k) mod 3

Cf. A000045, A000204, A001519, A001906, A002878, A005248, A054441, A088305, A122367, A258109, A026671, A026726, A026732, A004187, A049685, A033891, A206351, A092521, A081018, A049684, A058038, A081016, A003482, A049629, A133273, A049664, A049683, A119032, A023039, A007805, A033889.

Nm=12;
T=Table[0,{n,1,Nm},{k,1,n}];
T[[1,1]]=1;
T[[2,1]]=1;
T[[2,2]]=2;
Do[T[[n,1]]=T[[n-1,1]]+T[[n,2]];
T[[n,n]]=T[[n-1,n-1]]+T[[n,n-1]];
If[k!=1&&k!=n,T[[n,k]]=T[[n-1,k]]+T[[n-1,k-1]]],{n,3,Nm},{k,1,n}];
{Row[#,"\t"]}&/@T//Grid

T(n,k) = if (k==1, if (n==1, 1, if (n==2, 1, T(n-1,1) + T(n,2))), if (kMichel Marcus, Sep 14 2016

Conjectures:
Relations with other sequences:
T(n+1,1) = A002878(n-1), n>=1.
T(n,n) = A001519(n) = A122367(n-1), n>=1.
T(n+1,2) = A005248(n-1), n>=1.
T(n+1,n) = A001906(n) = A088305(n), n>=1.
T(2n-1,n) = 3*A054441(n-1), n>=2. [the central column].
Sum_{k=1..n} T(n,k) = 3*A105693(n-1), n>=2. [row sums].
Sum_{k=1..n} T(n,k)-T(n,1)-T(n,n) = 3*A258109(n), n>=2.
T(2n,n+1) - T(2n,n) = A026671(n), n>=1.
T(2n,n-1) - T(2n,n) = 2*A026726(n-1), n>=2.
T(n,ceiling(n/2)) - T(n-1,floor(n/2)) = 2*A026732(n-3), n>=3.
T(2n+1,2n) = 3*A004187(n), n>=1.
T(2n+1,2) = 3*A049685(n-1), n>=1.
T(2n+1,2n) + T(2n+1,2) = 3*A033891(n-1), n>=1.
T(2n+1,3) = 5*A206351(n), n>=1.
T(2n+1,2n)/3 - T(2n+1,3)/5 = 4*A092521(n-1), n>=2.
T(2n,1) = 1 + 5*A081018(n-1), n>=1.
T(2n,2) = 2 + 5*A049684(n-1), n>=1.
T(2n+1,2) = 3 + 5*A058038(n-1), n>=1.
T(2n,3) = 3 + 5*A081016(n-2), n>=2.
T(2n+1,1) = 4 + 5*A003482(n-1), n>=1.
T(3n,1) = 4*A049629(n-1), n>=1.
T(3n,1) = 4 + 8*A119032(n), n>=1.
T(3n+1,3) = 8*A133273(n), n>=1.
T(3n+2,3n+2) = 2 + 32*A049664(n), n>=1.
T(3n,3n-2) = 4 + 32*A049664(n-1), n>=1.
T(3n+2,2) = 2 + 16*A049683(n), n>=1.
T(3n+2,2) = 2*A023039(n), n>=1.
T(2n-1,2n-1) = A033889(n-1), n>=1.
T(3n-1,3n-1) = 2*A007805(n-1), n>=1.
T(5n-1,1) = 11*A097842(n-1), n>=1.
T(4n+5,3) - T(4n+1,3) = 15*A000045(8n+1), n>=1.
T(5n+4,3) - T(5n-1,3) = 11*A000204(10n-2), n>=1.
Relations between left and right sides:
T(n,1) = T(n,n) - T(n-2,n-2), n>=3.
T(n,2) = T(n,n-1) - T(n-2,n-3), n>=4.
T(n,1) + T(n,n) = 3*T(n,n-1), n>=2.

A354336 a(n) is the integer w such that (L(2n)^2, -L(2n-1)^2, -w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

Original entry on oeis.org

1, 11, 61, 401, 2731, 18701, 128161, 878411, 6020701, 41266481, 282844651, 1938646061, 13287677761, 91075098251, 624238009981, 4278590971601, 29325898791211, 201002700566861, 1377693005176801, 9442848335670731, 64722245344518301, 443612869075957361

Offset: 0

XU Pingya, Jun 20 2022

Subsequence of A017281.

			2*(L(4)^2)^3 + 2*(-L(3)^2)^3 + (-61)^3 = 2*(49)^3 + 2*(-1)^3 + (-61)^3 = 125, a(2) = 61.

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

Cf. A000032, A002878, A005248, A017281, A056914, A081015, A092521.

Cf. A337928, A354337.

LucasL[4*Range[22]-3] + 1 - LucasL[2*Range[22]-3]^2

a(n) = (-125 + 2*A005248(n)^6 - 2*A002878(n-1)^6)^(1/3).
a(n) = Lucas(4*n+1) - Lucas(4*n-2) + 3 = A056914(n) - 15*A092521(n-1), for n > 1.
a(n) = Lucas(4*n+1) + 1 - Lucas(2*n-1)^2.
a(n) = 2*A081015(n-1) + 1.
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: (1 + 3*x - 19*x^2)/((1 - x)*(1 - 7*x + x^2)). - Stefano Spezia, Jun 22 2022
a(n) = (F(2*n+1) + F(2*n-1))^2 + (F(2*n+1) + F(2*n-1)) * (F(2*n-1) + F(2*n-3)) - (F(2*n-1) + F(2*n-3))^2. - XU Pingya, Jul 17 2024

A354337 a(n) is the integer w such that (L(2n)^2, -L(2n + 1)^2, w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

Original entry on oeis.org

19, 149, 1039, 7139, 48949, 335519, 2299699, 15762389, 108037039, 740496899, 5075441269, 34787591999, 238437702739, 1634276327189, 11201496587599, 76776199786019, 526231901914549, 3606847113615839, 24721697893396339, 169445038140158549, 1161393569087713519

Offset: 1

XU Pingya, Jun 20 2022

Subsequence of A017377.

			2*(L(4)^2)^3 + 2*(-L(5)^2)^3 + (149)^3 = 2*(49)^3 + 2*(-121)^3 + (149)^3 = 125, a(2) = 149.

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

Cf. A000032, A002878, A005248, A017377, A081017, A089508, A092521, A288913.

Cf. A337929, A354336.

LinearRecurrence[{7,-1},{19,149},21]-1 + LucasL[2*Range[21]-3]^2

a(n) = (125 - 2*A005248(n)^6 + 2*A002878(n)^6)^(1/3).
a(n) = Lucas(4*n+2) + Lucas(4n-1) - 3 = 2*A056914(n)-3 = 15*A092521(n) + A288913(n-1).
a(n) = 2*A081017(n) - 1.
a(n) = 10*A089508(n) + 9.
a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).
G.f.: x*(19 - 3*x - x^2)/((1 - x)*(1 - 7*x + x^2)). - Stefano Spezia, Jun 22 2022

A077826 Expansion of (1-x)^(-1)/(1-2x-3x^2-2*x^3).

Original entry on oeis.org

1, 3, 10, 32, 101, 319, 1006, 3172, 10001, 31531, 99410, 313416, 988125, 3115319, 9821846, 30965900, 97627977, 307797347, 970410426, 3059468848, 9645763669, 30410754735, 95877738174, 302279267892, 953013259777, 3004619799579, 9472837914274, 29865561746840

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (3,1,-1,-2).

Partial sums of S(n, x), for x=1...10, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784.

Partial sums of A077833.

LinearRecurrence[{3,1,-1,-2},{1,3,10,32},30] (* Harvey P. Dale, May 12 2024 *)

Vec((1-x)^(-1)/(1-2*x-3*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

From Wesley Ivan Hurt, Jun 26 2022: (Start)
G.f.: (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).
a(n) = 3*a(n-1) + a(n-2) - a(n-3) - 2*a(n-4). (End)

A077827 Expansion of (1-x)^(-1)/(1-2x-2x^2-2*x^3).

Original entry on oeis.org

1, 3, 9, 27, 79, 231, 675, 1971, 5755, 16803, 49059, 143235, 418195, 1220979, 3564819, 10407987, 30387571, 88720755, 259032627, 756281907, 2208070579, 6446770227, 18822245427, 54954172467, 160446376243, 468445588275, 1367692273971, 3993168476979, 11658612678451

Offset: 0

N. J. A. Sloane, Nov 17 2002, Jun 05 2007

Index entries for linear recurrences with constant coefficients, signature (3,0,0,-2).

Partial sums of S(n, x), for x=1...11, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826.

CoefficientList[Series[(1-x)^(-1)/(1-2x-2x^2-2x^3),{x,0,40}],x]  (* Harvey P. Dale, Mar 27 2011 *)

Vec((1-x)^(-1)/(1-2*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

A077830 Expansion of 1/(1-3x-2x^2-3*x^3).

Original entry on oeis.org

1, 3, 11, 42, 157, 588, 2204, 8259, 30949, 115977, 434606, 1628619, 6103000, 22870056, 85702025, 321155187, 1203479779, 4509855786, 16899992477, 63330128340, 237319937332, 889320046107, 3332590398005, 12488371098225, 46798254229006, 175369276077483

Offset: 0

N. J. A. Sloane, Nov 17 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,3).

Partial sums of S(n, x), for x=1...15, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139, A097829.

CoefficientList[Series[1/(1-3x-2x^2-3x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,2,3},{1,3,11},40] (* Harvey P. Dale, Nov 05 2021 *)

cp's OEIS Frontend

A077828 Expansion of 1/(1-3*x-3*x^2-3*x^3).

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A077829 Expansion of 1/(1-3*x-3*x^2-2*x^3).

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A077831 Expansion of 1/(1-3*x-2*x^2-2*x^3).

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A214985 Array: T(m,n) = (F(n) + F(2*n) + ... + F(n*m))/F(n), by antidiagonals; transpose of A214984.

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A276472 Modified Pascal's triangle read by rows: T(n,k) = T(n-1,k) + T(n-1,k-1), 12. T(n,n) = T(n,n-1) + T(n-1,n-1), n>1. T(1,1) = 1, T(2,1) = 1. n>=1.

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A354336 a(n) is the integer w such that (L(2*n)^2, -L(2*n-1)^2, -w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

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A354337 a(n) is the integer w such that (L(2*n)^2, -L(2*n + 1)^2, w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

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A077826 Expansion of (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).

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A077828 Expansion of 1/(1-3x-3x^2-3*x^3).

A077829 Expansion of 1/(1-3x-3x^2-2*x^3).

A077831 Expansion of 1/(1-3x-2x^2-2*x^3).

A214985 Array: T(m,n) = (F(n) + F(2n) + ... + F(nm))/F(n), by antidiagonals; transpose of A214984.

A354336 a(n) is the integer w such that (L(2n)^2, -L(2n-1)^2, -w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

A354337 a(n) is the integer w such that (L(2n)^2, -L(2n + 1)^2, w) is a primitive solution to the Diophantine equation 2x^3 + 2y^3 + z^3 = 125, where L(n) is the n-th Lucas number (A000032).

A077826 Expansion of (1-x)^(-1)/(1-2x-3x^2-2*x^3).

A077827 Expansion of (1-x)^(-1)/(1-2x-2x^2-2*x^3).

A077830 Expansion of 1/(1-3x-2x^2-3*x^3).