A242464 -id:A242464 - cp's OEIS frontend

A164001 Spiral of triangles around a hexagon.

1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081, 1432, 1897, 2513, 3329, 4410, 5842, 7739, 10252, 13581, 17991, 23833, 31572, 41824, 55405, 73396, 97229, 128801, 170625, 226030, 299426

Offset: 1

Omar E. Pol, Oct 27 2009

a(n) is the side length of the n-th triangle in a spiral around a hexagon with side length = 1.
Sequence very similar to A134816, but without repeated terms. Records in A134816. Also records in A000931, the Padovan sequence.
Column k=2 of A242464 (with offset 0). - Alois P. Heinz, May 19 2014
a(n) is the number of bitstrings of length n-1 without two consecutive 0's or three consecutive 1's. - Zachary Stier, Mar 16 2021

Alois P. Heinz, Table of n, a(n) for n = 1..1000
I. Amburg, K. Dasaratha, L. Flapan, T. Garrity, C. Lee, C. Mihailak, N. Neumann-Chun, S. Peluse, and M. Stoffregen, Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences, arXiv:1509.05239v1 [math.CO] 17 Sep 2015. See Conjecture 5.8.
Yuksel Soykan, Vedat Irge, and Erkan Tasdemir, A Comprehensive Study of K-Circulant Matrices Derived from Generalized Padovan Numbers, Asian Journal of Probability and Statistics 26 (12):152-70, (2024). See p. 154.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

The following are basically all variants of the same sequence: A000931, A078027, A096231, A124745, A133034, A134816, A164001, A182097, A228361 and probably A020720. However, each one has its own special features and deserves its own entry.

Cf. A060006.

LinearRecurrence[{0,1,1},{1,2,3,4},50] (* Harvey P. Dale, Jul 08 2017 *)

If n < 5 then a(n) = n, otherwise a(n) = a(n-2) + a(n-3).
G.f.: -x - 1 + (-x^2 - 2*x - 1)/(-1 + x^2 + x^3). a(n) = A000931(n+4) + A000931(n+5) = A000931(n+7), n > 1. - R. J. Mathar, Oct 29 2009
a(n) ~ 1.67873... * 1.32471...^(n-1) where 1.32471... is the real root of x^3 - x - 1 = 0 (see A060006), and 1.67873... is the real root of 23*x^3 - 46*x^2 + 13*x - 1 = 0. - Ricardo Bittencourt, May 14 2023

A242452 Number of length n words on {1,2,3} with no more than one consecutive 1 and no more than two consecutive 2's and no more than three consecutive 3's.

Original entry on oeis.org

1, 3, 8, 21, 54, 140, 362, 937, 2425, 6275, 16239, 42024, 108751, 281430, 728295, 1884709, 4877320, 12621710, 32662931, 84526348, 218740428, 566064618, 1464883079, 3790878933, 9810177543, 25387142435, 65697791726, 170015189725, 439971633412, 1138574962157

Offset: 0

Geoffrey Critzer and Alois P. Heinz, May 14 2014

			a(3) = 21 because there are 27 length 3 words on {1,2,3} but we don't count: 111, 112, 113, 211, 222, 311.

Fung Lam, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,4,3,2).

Cf. A000931 (binary words with at most one consecutive 1 and two consecutive 2's; offset=-8 for n>0).
Cf. A007283 (ternary words with no consecutive like letters).
Column k=3 of A242464.

nn=20;CoefficientList[Series[1/(1-Sum[v[i]/(1+v[i])/.v[i]->(z-z^(i+1))/(1-z),{i,1,3}]),{z,0,nn}],z]
(* replacing the 3 in this code with a positive integer k will return the number of words on {1,2,...,k} with no more than one consecutive 1 and no more than two consecutive 2's and ... no more than k consecutive k's *)

G.f.: (1 + x)*(1 + x^2)*(1 + x + x^2)/(1 - x - 2*x^2 - 4*x^3 - 3*x^4 - 2*x^5).

A242495 Number of length n words on {1,2,3,4} with at most one consecutive 1 and at most two consecutive 2's and at most three consecutive 3's and at most four consecutive 4's.

Original entry on oeis.org

1, 4, 15, 56, 208, 773, 2872, 10672, 39655, 147350, 547523, 2034486, 7559742, 28090486, 104378617, 387850022, 1441172953, 5355109869, 19898515060, 73938894118, 274742112508, 1020886629235, 3793410119173, 14095551768590

Offset: 0

Geoffrey Critzer, May 16 2014

Column k=4 of A242464.

			a(3) = 56 because there are 64 length 3 words on {1,2,3,4} but we don't count 111, 112, 113, 114, 211, 222, 311, or 411.

Fung Lam, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 5, 12, 18, 22, 20, 15, 8, 3).

Cf. A242452.

nn=23;CoefficientList[Series[1/(1-Sum[v[i]/(1+v[i])/.v[i]->(z-z^(i+1))/(1-z),{i,1,4}]),{z,0,nn}],z]

G.f.: (1 + x)*(1 + x^2)*(1 + x + x^2 )*(1 + x + x^2 + x^3 + x^4)/(1 - x - 5*x^2 - 12*x^3 - 18*x^4 - 22*x^5 - 20*x^6 - 15*x^7 - 8*x^8 - 3*x^9). (corrected by Fung Lam, May 18 2014)

A242509 Number of n-length words on {1,2,3,4,5} that contain at most one consecutive 1 and at most two consecutive 2's and at most three consecutive 3's and at most four consecutive 4's and at most five consecutive 5's.

Original entry on oeis.org

1, 5, 24, 115, 550, 2631, 12584, 60191, 287901, 1377066, 6586677, 31504891, 150691790, 720777469, 3447567781, 16490143094, 78874393932, 377265981421, 1804509849677, 8631193794141, 41284067429916, 197466800561799, 944508129929499, 4517699202928696

Offset: 0

Geoffrey Critzer and Alois P. Heinz, May 16 2014

Column k=5 of A242464.

Fung Lam, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, 5, 12, 17, 24, 24, 25, 19, 14, 7, 4).

Cf. A242452, A242495.

nn=23;CoefficientList[Series[1/(1-Sum[v[i]/(1+v[i])/.v[i]->(z-z^(i+1))/(1-z),{i,1,5}]),{z,0,nn}],z]
LinearRecurrence[{3,5,12,17,24,24,25,19,14,7,4},{1,5,24,115,550,2631,12584,60191,287901,1377066,6586677,31504891},30] (* Harvey P. Dale, Apr 13 2019 *)

G.f.: (1 + x)*(1 +x^2)*(1 - x + x^2)*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)/(1 - 3*x - 5*x^2 - 12*x^3 - 17*x^4 - 24*x^5 - 24*x^6 - 25*x^7 - 19*x^8 - 14*x^9 - 7*x^10 - 4*x^11).

A242629 Number of n-length words w over a 6-ary alphabet {a_1,...,a_6} such that w contains never more than j consecutive letters a_j (for 1<=j<=6).

Original entry on oeis.org

1, 6, 35, 204, 1188, 6919, 40295, 234672, 1366694, 7959418, 46354440, 269961210, 1572213035, 9156329637, 53325071447, 310557107219, 1808637367513, 10533228997581, 61343923944270, 357257684774972, 2080614429665859, 12117182049311250, 70568625653399251

Offset: 0

Geoffrey Critzer and Alois P. Heinz, May 19 2014

Geoffrey Critzer and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,10,26,49,81,114,148,170,181,174,155,123,90,57,32,14,5).

Column k=6 of A242464.

b:= proc(n, k, c, t) option remember;
      `if`(n=0, 1, add(`if`(c=t and j=c, 0,
       b(n-1, k, j, 1+`if`(j=c, t, 0))), j=1..k))
    end:
a:= n-> b(n, 6, 0$2):
seq(a(n), n=0..30);

G.f.: -(x^6+x^5+x^4+x^3+x^2+x+1) *(x+1)*(x^2-x+1) *(x^2+x+1) *(x^4+x^3+x^2+x+1) *(x^2+1) / (5*x^17 +14*x^16 +32*x^15 +57*x^14 +90*x^13 +123*x^12 +155*x^11 +174*x^10 +181*x^9 +170*x^8 +148*x^7 +114*x^6 +81*x^5 +49*x^4 +26*x^3 +10*x^2 +3*x-1).

A242630 Number of n-length words w over a 7-ary alphabet {a_1,...,a_7} such that w contains never more than j consecutive letters a_j (for 1<=j<=7).

Original entry on oeis.org

1, 7, 48, 329, 2254, 15443, 105804, 724892, 4966431, 34026362, 233123809, 1597194268, 10942809918, 74972150416, 513654479985, 3519185768909, 24110893526041, 165190252745398, 1131763100053353, 7754015102916294, 53124854674462893, 363972747889200054

Offset: 0

Geoffrey Critzer and Alois P. Heinz, May 19 2014

Geoffrey Critzer and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, 13, 33, 61, 106, 157, 220, 277, 331, 364, 382, 369, 340, 289, 233, 170, 117, 70, 39, 17, 6).

Column k=7 of A242464.

G.f.: -(x+1) *(x^2+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+x+1) *(x^2-x+1) *(x^4+x^3+x^2+x+1) / (6*x^21 +17*x^20 +39*x^19 +70*x^18 +117*x^17 +170*x^16 +233*x^15 +289*x^14 +340*x^13 +369*x^12 +382*x^11 +364*x^10 +331*x^9 +277*x^8 +220*x^7 +157*x^6 +106*x^5 +61*x^4 +33*x^3 +13*x^2 +4*x-1).

A242631 Number of n-length words w over an 8-ary alphabet {a_1,...,a_8} such that w contains never more than j consecutive letters a_j (for 1<=j<=8).

Original entry on oeis.org

1, 8, 63, 496, 3904, 30729, 241871, 1903792, 14984945, 117948062, 928381475, 7307387240, 57517205708, 452723914009, 3563437058402, 28048184061555, 220770176730345, 1737705044525640, 13677657310833723, 107658264618591797, 847389408675004032, 6669890253930098674

Offset: 0

Geoffrey Critzer and Alois P. Heinz, May 19 2014

Geoffrey Critzer and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, 16, 39, 79, 144, 229, 345, 480, 631, 782, 927, 1039, 1119, 1148, 1128, 1056, 950, 809, 659, 507, 369, 249, 159, 90, 46, 20, 7).

Column k=8 of A242464.

b:= proc(n, k, c, t) option remember;
      `if`(n=0, 1, add(`if`(c=t and j=c, 0,
       b(n-1, k, j, 1+`if`(j=c, t, 0))), j=1..k))
    end:
a:= n-> b(n, 8, 0$2):
seq(a(n), n=0..30);

G.f.: -(x^2+x+1) *(x^6+x^3+1) *(x+1) *(x^2+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2-x+1) *(x^4+x^3+x^2+x+1) / (7*x^27 +20*x^26 +46*x^25 +90*x^24 +159*x^23 +249*x^22 +369*x^21 +507*x^20 +659*x^19 +809*x^18 +950*x^17 +1056*x^16 +1128*x^15 +1148*x^14 +1119*x^13 +1039*x^12 +927*x^11 +782*x^10 +631*x^9 +480*x^8 +345*x^7 +229*x^6 +144*x^5 +79*x^4 +39*x^3 +16*x^2 +5*x-1).

A242632 Number of n-length words w over a 9-ary alphabet {a_1,...,a_9} such that w contains never more than j consecutive letters a_j (for 1<=j<=9).

Original entry on oeis.org

1, 9, 80, 711, 6318, 56143, 498896, 4433274, 39394819, 350068993, 3110771999, 27642843622, 245638961566, 2182789161071, 19396631915857, 172361736254288, 1531635402139359, 13610370004776711, 120944038906506659, 1074729088326395697, 9550223588843166996

Offset: 0

Geoffrey Critzer and Alois P. Heinz, May 19 2014

Geoffrey Critzer and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, 12, 34, 59, 109, 166, 258, 352, 483, 606, 754, 875, 1007, 1087, 1161, 1172, 1167, 1099, 1023, 895, 775, 628, 503, 371, 273, 179, 118, 66, 38, 15, 8).

Column k=9 of A242464.

b:= proc(n, k, c, t) option remember;
      `if`(n=0, 1, add(`if`(c=t and j=c, 0,
       b(n-1, k, j, 1+`if`(j=c, t, 0))), j=1..k))
    end:
a:= n-> b(n, 9, 0$2):
seq(a(n), n=0..30);

G.f.: -(x+1) *(x^4-x^3+x^2-x+1) *(x^4+x^3+x^2+x+1) *(x^2+x+1) *(x^6+x^3+1) *(x^2+1)*(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2-x+1) / (8*x^31 +15*x^30 +38*x^29 +66*x^28 +118*x^27 +179*x^26 +273*x^25 +371*x^24 +503*x^23 +628*x^22 +775*x^21 +895*x^20 +1023*x^19 +1099*x^18 +1167*x^17 +1172*x^16 +1161*x^15 +1087*x^14 +1007*x^13 +875*x^12 +754*x^11 +606*x^10 +483*x^9 +352*x^8 +258*x^7 +166*x^6 +109*x^5 +59*x^4 +34*x^3 +12*x^2 +7*x-1).

A242633 Number of n-length words w over a 10-ary alphabet {a_1,...,a_10} such that w contains never more than j consecutive letters a_j (for 1<=j<=10).

Original entry on oeis.org

1, 10, 99, 980, 9700, 96011, 950319, 9406280, 93103581, 921541438, 9121438862, 90284216730, 893635304019, 8845223290551, 87550228496839, 866574224082841, 8577372083864876, 84899030943287514, 840332608243515705, 8317631952113371291, 82328117000511661919

Offset: 0

Alois P. Heinz, May 19 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7, 21, 60, 128, 253, 444, 740, 1145, 1700, 2398, 3266, 4267, 5412, 6627, 7896, 9123, 10275, 11246, 12016, 12491, 12681, 12534, 12099, 11364, 10420, 9287, 8069, 6801, 5578, 4420, 3400, 2512, 1792, 1217, 793, 482, 278, 144, 69, 26, 9).

Column k=10 of A242464.

b:= proc(n, k, c, t) option remember;
      `if`(n=0, 1, add(`if`(c=t and j=c, 0,
       b(n-1, k, j, 1+`if`(j=c, t, 0))), j=1..k))
    end:
a:= n-> b(n, 10, 0$2):
seq(a(n), n=0..30);

G.f.: -(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1) *(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1) *(x^2+x+1) *(x^6+x^3+1) *(x^2+1) *(x^4+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2-x+1) / (9*x^41 +26*x^40 +69*x^39 +144*x^38 +278*x^37 +482*x^36 +793*x^35 +1217*x^34 +1792*x^33 +2512*x^32 +3400*x^31 +4420*x^30 +5578*x^29 +6801*x^28 +8069*x^27 +9287*x^26 +10420*x^25 +11364*x^24 +12099*x^23 +12534*x^22 +12681*x^21 +12491*x^20 +12016*x^19 +11246*x^18 +10275*x^17 +9123*x^16 +7896*x^15 +6627*x^14 +5412*x^13 +4267*x^12 +3266*x^11 +2398*x^10 +1700*x^9 +1145*x^8 +740*x^7 +444*x^6 +253*x^5 +128*x^4 +60*x^3 +21*x^2 +7*x-1).

A242635 Number of n-length words w over an n-ary alphabet {a_1,...,a_n} such that w contains never more than j consecutive letters a_j for 1<=j<=n.

Original entry on oeis.org

1, 1, 3, 21, 208, 2631, 40295, 724892, 14984945, 350068993, 9121438862, 262285777567, 8250643190038, 281849526767134, 10390959086757005, 411219228179234026, 17387847546353549435, 782342249208357483984, 37321230268969840324231, 1881590248383756833279387

Offset: 0

Geoffrey Critzer and Alois P. Heinz, May 19 2014

nonn

Geoffrey Critzer and Alois P. Heinz, Table of n, a(n) for n = 0..386

Main diagonal of A242464.

a:= proc(n) option remember; local v;
      v:= i-> (x-x^(i+1))/(1-x);
      coeff(series(1/(1-add(v(i)/(1+v(i)), i=1..n)), x, n+1), x, n)
    end:
seq(a(n), n=0..25);

b[n_, k_, c_, t_] := b[n, k, c, t] = If[n == 0, 1, Sum[If[c == t && j == c, 0, b[n - 1, k, j, 1 + If[j == c, t, 0]]], {j, 1, k}]];
a[n_] := b[n, n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 28 2020, from Maple code of A242464 *)

a(n) = [x^n] 1/(1-Sum_{i=1..n} v(i)/(1+v(i))) with v(i) = (x-x^(i+1))/(1-x).

a(n) ~ n^n. - Vaclav Kotesovec, Aug 27 2014

cp's OEIS Frontend

A164001 Spiral of triangles around a hexagon.

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A242452 Number of length n words on {1,2,3} with no more than one consecutive 1 and no more than two consecutive 2's and no more than three consecutive 3's.

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A242495 Number of length n words on {1,2,3,4} with at most one consecutive 1 and at most two consecutive 2's and at most three consecutive 3's and at most four consecutive 4's.

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A242509 Number of n-length words on {1,2,3,4,5} that contain at most one consecutive 1 and at most two consecutive 2's and at most three consecutive 3's and at most four consecutive 4's and at most five consecutive 5's.

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A242629 Number of n-length words w over a 6-ary alphabet {a_1,...,a_6} such that w contains never more than j consecutive letters a_j (for 1<=j<=6).

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A242630 Number of n-length words w over a 7-ary alphabet {a_1,...,a_7} such that w contains never more than j consecutive letters a_j (for 1<=j<=7).

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A242631 Number of n-length words w over an 8-ary alphabet {a_1,...,a_8} such that w contains never more than j consecutive letters a_j (for 1<=j<=8).

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A242632 Number of n-length words w over a 9-ary alphabet {a_1,...,a_9} such that w contains never more than j consecutive letters a_j (for 1<=j<=9).

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A242633 Number of n-length words w over a 10-ary alphabet {a_1,...,a_10} such that w contains never more than j consecutive letters a_j (for 1<=j<=10).

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