A136175 -id:A136175 - cp's OEIS frontend

A214727 a(n) = a(n-1) + a(n-2) + a(n-3) with a(0) = 1, a(1) = a(2) = 2.

1, 2, 2, 5, 9, 16, 30, 55, 101, 186, 342, 629, 1157, 2128, 3914, 7199, 13241, 24354, 44794, 82389, 151537, 278720, 512646, 942903, 1734269, 3189818, 5866990, 10791077, 19847885, 36505952, 67144914, 123498751, 227149617, 417793282

Offset: 0

Abel Amene, Jul 27 2012

Part of a group of sequences defined by a(0), a(1)=a(2), a(n) = a(n-1) + a(n-2) + a(n-3) which is a subgroup of sequences with linear recurrences and constant coefficients listed in the index.
Note: A000073 (with offset=1), 1 followed by A000073, A000213, A141523, A214727, A214825 to A214831 completely define possible sequences with a(0)=0,1,2...9 and a(1)=a(2)=0,1,2...9 excluding any multiples of these sequences and the trivial case of a(0)=a(1)=a(2)=0.
Note: allowing a(0)=0 and a(1)=a(2)=1,2,3....9 leads to A000073 (with offset=1) and its multiples.
Note: allowing a(0)=1,2,3....9 a(1)=a(2)=0 leads to 1 followed by A000073 and its multiples.
With offset of 6 this sequence is the 8th row of tribonacci array A136175.

			G.f. = 1 + 2*x + 2*x^2 + 5 x^3 + 9*x^4 + 16*x^5 + 30*x^6 + 55*x^7 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,2,2];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

a214727 n = a214727_list !! n
a214727_list = 1 : 2 : 2 : zipWith3 (\x y z -> x + y + z)
   a214727_list (tail a214727_list) (drop 2 a214727_list)
-- Reinhard Zumkeller, Jul 31 2012

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{1,1,1},{1,2,2},40] (* Ray Chandler, Dec 08 2013 *)

a(n)=([0,1,0; 0,0,1; 1,1,1]^n*[1;2;2])[1,1] \\ Charles R Greathouse IV, Mar 22 2016

my(x='x+O('x^40)); Vec((1+x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

((1+x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

G.f.: (1+x-x^2)/(1-x-x^2-x^3).
a(n) = K(n) -2*T(n+1) + 3*T(n), where K(n) = A001644(n), T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019
a(n) = Sum_{r root of x^3-x^2-x-1} r^n/(-r^2+2*r+1). - Fabian Pereyra, Nov 20 2024

A214899 a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=2, a(1)=1, a(2)=2.

Original entry on oeis.org

2, 1, 2, 5, 8, 15, 28, 51, 94, 173, 318, 585, 1076, 1979, 3640, 6695, 12314, 22649, 41658, 76621, 140928, 259207, 476756, 876891, 1612854, 2966501, 5456246, 10035601, 18458348, 33950195, 62444144, 114852687, 211247026, 388543857

Offset: 0

Abel Amene, Jul 29 2012

With offset of 5 this sequence is the 4th row of the tribonacci array A136175.
For n>0, a(n) is the number of ways to tile a strip of length n with squares, dominoes, and trominoes, such that there must be exactly one "special" square (say, of a different color) in the first three cells. - Greg Dresden and Emma Li, Aug 17 2024
From Greg Dresden and Jiarui Zhou, Jun 30 2025: (Start)
For n >= 3, a(n) is the number of ways to tile this shape of length n-1 with squares, dominos, and trominos (of length 3):
  ._
  ||____________
  |||_|||_|||
    |_|
As an example, here is one of the a(9) = 173 ways to tile this shape of length 8:
  ._
  | |___________
  || |__|_|___|
    |_|. (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Robert Price)
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000073, A000213, A035513, A136175, A141036, A141523.

a:=[2,1,2];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{1,1,1},{2,1,2},34] (* Ray Chandler, Dec 08 2013 *)

a(n)=([0,1,0;0,0,1;1,1,1]^n*[2;1;2])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

my(x='x+O('x^40)); Vec((2-x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

((2-x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

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

a(n) = K(n) - T(n+1) + T(n), where K(n) = A001644(n), T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019

A214825 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 3.

Original entry on oeis.org

1, 3, 3, 7, 13, 23, 43, 79, 145, 267, 491, 903, 1661, 3055, 5619, 10335, 19009, 34963, 64307, 118279, 217549, 400135, 735963, 1353647, 2489745, 4579355, 8422747, 15491847, 28493949, 52408543, 96394339, 177296831, 326099713, 599790883, 1103187427

Offset: 0

Abel Amene, Jul 28 2012

Part of a group of sequences defined by a(0), a(1)=a(2), a(n) = a(n-1) + a(n-2) + a(n-3) which is a subgroup of sequences with linear recurrences and constant coefficients listed in the index. See Comments in A214727.

Indranil Ghosh, Table of n, a(n) for n = 0..3770
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000073, A000213, A000288, A000322, A000383, A001644, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,3,3];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+2*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{1,1,1},{1,3,3},40] (* Harvey P. Dale, Oct 05 2013 *)

a(n)=([0,1,0; 0,0,1; 1,1,1]^n*[1;3;3])[1,1] \\ Charles R Greathouse IV, Mar 22 2016

my(x='x+O('x^40)); Vec((1+2*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

((1+2*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

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

a(n) = K(n) - 2*T(n+1) + 4*T(n), where K(n) = A001644(n), and T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019

A136189 The 3rd-order Zeckendorf array, T(n,k), read by antidiagonals.

Original entry on oeis.org

1, 2, 5, 3, 8, 7, 4, 12, 11, 10, 6, 17, 16, 15, 14, 9, 25, 23, 22, 21, 18, 13, 37, 34, 32, 31, 27, 20, 19, 54, 50, 47, 45, 40, 30, 24, 28, 79, 73, 69, 66, 58, 44, 36, 26, 41, 116, 107, 101, 97, 85, 64, 53, 39, 29, 60, 170, 157, 148, 142, 125, 94, 77, 57, 43, 33, 88, 249, 230

Offset: 1

Clark Kimberling, Dec 20 2007

Rows satisfy this recurrence: T(n,k) = T(n,k-1) + T(n,k-3) for all k>=4.
Except for initial terms, (row 1) = A000930 (column 1) = A020942 (column 2) = A064105 (column 3) = A064106.
As a sequence, the array is a permutation of the natural numbers.
As an array, T is an interspersion (hence also a dispersion).

			Northwest corner:
  1  2  3  4  6  9  13  19 ...
  5  8 12 17 25 37  54  79 ...
  7 11 16 23 34 50  73 107 ...
 10 15 22 32 47 69 101 148 ...
 ...

Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Index entries for sequences that are permutations of the natural numbers

Cf. A000930, A035513, A134563, A136175, A136495, A202342.

Row 1 is the 3rd-order Zeckendorf basis, given by initial terms b(1)=1, b(2)=2, b(3)=3 and recurrence b(k) = b(k-1) + b(k-3) for k>=4. Every positive integer has a unique 3-Zeckendorf representation: n = b(i(1)) + b(i(2)) + ... + b(i(p)), where |i(h)-i(j)| >= 3. Rows of T are defined inductively: T(n,1) is the least positive integer not in an earlier row. T(n,2) is obtained from T(n,1) as follows: if T(n,1) = b(i(1)) + b(i(2)) + ... + b(i(p)), then T(n,k+1) = b(i(1+k)) + b(i(2+k)) + ... + b(i(p+k)) for k=1,2,3,... .

A(n, k) = A000930(k)*A202342(n) + A000930(k-2)*A136495(n) + A000930(k-1)*(n-1) for n > 1. - Alan Michael Gómez Calderón, Dec 23 2024

A214827 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 5.

Original entry on oeis.org

1, 5, 5, 11, 21, 37, 69, 127, 233, 429, 789, 1451, 2669, 4909, 9029, 16607, 30545, 56181, 103333, 190059, 349573, 642965, 1182597, 2175135, 4000697, 7358429, 13534261, 24893387, 45786077, 84213725, 154893189, 284892991, 523999905

Offset: 0

Abel Amene, Jul 29 2012

See comments in A214727.

Robert Price, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,5,5];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 24 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+4*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 24 2019

LinearRecurrence[{1,1,1},{1,5,5},40] (* Ray Chandler, Dec 08 2013 *)

my(x='x+O('x^40)); Vec((1+4*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 24 2019

((1+4*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 24 2019

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

a(n) = -A000073(n) + 4*A000073(n+1) + A000073(n+2). - R. J. Mathar, Jul 29 2012

A214831 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 9.

Original entry on oeis.org

1, 9, 9, 19, 37, 65, 121, 223, 409, 753, 1385, 2547, 4685, 8617, 15849, 29151, 53617, 98617, 181385, 333619, 613621, 1128625, 2075865, 3818111, 7022601, 12916577, 23757289, 43696467, 80370333, 147824089, 271890889, 500085311, 919800289, 1691776489

Offset: 0

Abel Amene, Aug 07 2012

Part of a group of sequences defined by a(0), a(1)=a(2), a(n)=a(n-1)+a(n-2)+a(n-3) which is a subgroup of sequences with linear recurrences and constant coefficients listed in the index. See comments in A214727.

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

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831, A244930, A244931.

a:=[1,9,9];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 24 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+8*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 24 2019

LinearRecurrence[{1,1,1},{1,9,9},40] (* Harvey P. Dale, Oct 11 2017 *)

Vec((x^2-8*x-1)/(x^3+x^2+x-1) + O(x^40)) \\ Michel Marcus, Jul 08 2014

((1+8*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 24 2019

G.f.: (1+8*x-x^2)/(1-x-x^2-x^3).

a(n) = -A000073(n) + 8*A000073(n+1) + A000073(n+2). - G. C. Greubel, Apr 24 2019

A214828 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 6.

Original entry on oeis.org

1, 6, 6, 13, 25, 44, 82, 151, 277, 510, 938, 1725, 3173, 5836, 10734, 19743, 36313, 66790, 122846, 225949, 415585, 764380, 1405914, 2585879, 4756173, 8747966, 16090018, 29594157, 54432141, 100116316, 184142614, 338691071, 622950001

Offset: 0

Abel Amene, Jul 30 2012

See comments in A214727.

Robert Price, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,6,6];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 24 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+5*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 24 2019

LinearRecurrence[{1,1,1},{1,6,6},33] (* Ray Chandler, Dec 08 2013 *)

my(x='x+O('x^40)); Vec((1+5*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 24 2019

((1+5*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 24 2019

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

a(n) = -A000073(n) + 5*A000073(n+1) + A000073(n+2). - G. C. Greubel, Apr 24 2019

A214829 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 7.

Original entry on oeis.org

1, 7, 7, 15, 29, 51, 95, 175, 321, 591, 1087, 1999, 3677, 6763, 12439, 22879, 42081, 77399, 142359, 261839, 481597, 885795, 1629231, 2996623, 5511649, 10137503, 18645775, 34294927, 63078205, 116018907, 213392039, 392489151, 721900097, 1327781287, 2442170535

Offset: 0

Abel Amene, Aug 07 2012

See comments in A214727.

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

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825, A214826, A214827, A214828, A214830, A214831.

a:=[1,7,7];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 24 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+6*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 24 2019

LinearRecurrence[{1,1,1}, {1,7,7}, 40] (* G. C. Greubel, Apr 24 2019 *)

Vec((x^2-6*x-1)/(x^3+x^2+x-1) + O(x^40)) \\ Michel Marcus, Jun 04 2017

((1+6*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 24 2019

G.f.: (1+6*x-x^2)/(1-x-x^2-x^3).

a(n) = -A000073(n) + 6*A000073(n+1) + A000073(n+2). - G. C. Greubel, Apr 24 2019

A353084 Column 0 of the extended Trithoff (tribonacci) array.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 80

Offset: 1

Tanya Khovanova and PRIMES STEP Senior Group, Apr 22 2022

nonn

This column is also called the wall of the Trithoff array.
These are the positions of letters a and b in the tribonacci word.
Complement of A003146: position of letter c in the tribonacci word.
Suppose number n_1 has tribonacci representation t that ends in 1 (such numbers are in column 1 of the Trithoff array). Then its tribonacci successor n_2 has tribonacci representation t0 (such numbers are in column 2 of the Trithoff array), and the successor of the successor n_3 has tribonacci representation t00 (such numbers are in column 3 of the Trithoff array). This sequence consists of numbers n_3-n_2-n_1.

			The first few tribonacci numbers are 1, 2, 4, 7, 13, 24, 44. The number 23 can be represented as 13+7+2+1. Thus, its tribonacci representation is 11011. The tribonacci successor of 23 is 24+13+4+2 = 43, and the next successor is 44+24+7+4 = 79. Thus, 79 - 43 - 23 = 13 is in this sequence.

Cf. A000073, A003144, A003145, A003146, A136175, A353083, A353086, A353090.

A214826 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 4.

Original entry on oeis.org

1, 4, 4, 9, 17, 30, 56, 103, 189, 348, 640, 1177, 2165, 3982, 7324, 13471, 24777, 45572, 83820, 154169, 283561, 521550, 959280, 1764391, 3245221, 5968892, 10978504, 20192617, 37140013, 68311134, 125643764, 231094911, 425049809

Offset: 0

Abel Amene, Jul 29 2012

See Comments in A214727.

Robert Price, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1).

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

a:=[1,4,4];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+3*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

LinearRecurrence[{1,1,1},{1,4,4},33] (* Ray Chandler, Dec 08 2013 *)

my(x='x+O('x^40)); Vec((1+3*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

((1+3*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

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

a(n) = K(n) - 2*T(n+1) + 5*T(n), where K(n) = A001644(n) and T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019

cp's OEIS Frontend

A214727 a(n) = a(n-1) + a(n-2) + a(n-3) with a(0) = 1, a(1) = a(2) = 2.

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A214899 a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=2, a(1)=1, a(2)=2.

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A214825 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 3.

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A136189 The 3rd-order Zeckendorf array, T(n,k), read by antidiagonals.

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A214827 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 5.

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A214831 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 9.

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