A097333 -id:A097333 - cp's OEIS frontend

A179070 a(1)=a(2)=a(3)=1, a(4)=3; thereafter a(n) = a(n-1) + a(n-3).

1, 1, 1, 3, 4, 5, 8, 12, 17, 25, 37, 54, 79, 116, 170, 249, 365, 535, 784, 1149, 1684, 2468, 3617, 5301, 7769, 11386, 16687, 24456, 35842, 52529, 76985, 112827, 165356, 242341, 355168, 520524, 762865, 1118033, 1638557, 2401422, 3519455, 5158012, 7559434

Offset: 1

Mark Dols, Jun 27 2010

Also (essentially), coordination sequence for (2,4,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
Column sums of shifted (1,2) Pascal array:
1 1 1 1 1 1 1 1 1
......2 3 4 5 6 7
............2 5 9
.................
----------------- +
1 1 1 3 4 5 8 ...
a(n+1) is the number of multus bitstrings of length n with no runs of 2 0's. - Steven Finch, Mar 25 2020
From Areebah Mahdia and Greg Dresden, Jun 13 2020: (Start)
For n >= 5, a(n) gives the number of ways to tile the following board of length n-3 with squares and trominos:
. 
|||
|||_   _ _
|||_|||_|_| ... . (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
J. W. Cannon, P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000930, A029635, A097333, A214626.

Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

a179070 n = a179070_list !! (n-1)
a179070_list = 1 : zs where zs = 1 : 1 : 3 : zipWith (+) zs (drop 2 zs)
-- Reinhard Zumkeller, Jul 23 2012

Join[{1}, LinearRecurrence[{1, 0, 1}, {1, 1, 3}, 80]] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2012 *)

a(n)=([0,1,0; 0,0,1; 1,0,1]^(n-1)*[1;1;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

a(n) = A000930(n-1) + A000930(n-4).
G.f.: x - x^2*(1+2*x^2) / ( -1+x+x^3 ). - R. J. Mathar, Oct 30 2011
a(n) = A000930(n-2)+2*A000930(n-4) for n>3. - R. J. Mathar, May 19 2024

Simpler definition from N. J. A. Sloane, Aug 29 2013

A171861 Expansion of x(1+x+x^2) / ( (x-1)(x^3+x^2-1) ).

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 18, 25, 34, 46, 62, 83, 111, 148, 197, 262, 348, 462, 613, 813, 1078, 1429, 1894, 2510, 3326, 4407, 5839, 7736, 10249, 13578, 17988, 23830, 31569, 41821, 55402, 73393, 97226, 128798, 170622, 226027, 299423, 396652, 525453, 696078, 922108

Offset: 1

Ed Pegg Jr, Oct 16 2010

Number of wins in Penney's game if the two players start HHT and TTT and HHT beats TTT.

HHT beats TTT 70% of the time. - Geoffrey Critzer, Mar 01 2014

			a(n) enumerates length n+2 sequences on {H,T} that end in HHT but do not contain the contiguous subsequence TTT.
a(3)=4 because we have: TTHHT, THHHT, HTHHT, HHHHT.
a(4)=6 because we have: TTHHHT, THTHHT, THHHHT, HTTHHT, HTHHHT, HHHHHT. - _Geoffrey Critzer_, Mar 01 2014

G. C. Greubel, Table of n, a(n) for n = 1..1000
Wikipedia, Penney's game
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

Related sequences are A000045 (HHH beats HHT, HTT beats TTH), A006498 (HHH beats HTH), A023434 (HHH beats HTT), A000930 (HHH beats THT, HTH beats HHT), A000931 (HHH beats TTH), A077868 (HHT beats HTH), A002620 (HHT beats HTT), A000012 (HHT beats THH), A004277 (HHT beats THT), A070550 (HTH beats HHH), A000027 (HTH beats HTT), A097333 (HTH beats THH), A040000 (HTH beats TTH), A068921 (HTH beats TTT), A054405 (HTT beats HHH), A008619 (HTT beats HHT), A038718 (HTT beats THT), A128588 (HTT beats TTT).

Cf. A164315 (essentially the same sequence).

A171861 := proc(n) option remember; if n <=4 then op(n,[1,2,4,6]); else procname(n-1)+procname(n-2)-procname(n-4) ; end if; end proc:

nn=44;CoefficientList[Series[x(1+x+x^2)/(1-x-x^2+x^4),{x,0,nn}],x] (* Geoffrey Critzer, Mar 01 2014 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,1,1]^(n-1)*[1;2;4;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = a(n-1) +a(n-2) -a(n-4) = A000931(n+10)-3 = A134816(n+6)-3 = A078027(n+12)-3.

a(n) = A164315(n-1). - Alois P. Heinz, Oct 12 2017

A038718 Number of permutations P of {1,2,...,n} such that P(1)=1 and |P^-1(i+1)-P^-1(i)| equals 1 or 2 for i=1,2,...,n-1.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 14, 21, 31, 46, 68, 100, 147, 216, 317, 465, 682, 1000, 1466, 2149, 3150, 4617, 6767, 9918, 14536, 21304, 31223, 45760, 67065, 98289, 144050, 211116, 309406, 453457, 664574, 973981, 1427439, 2092014, 3065996, 4493436, 6585451

Offset: 1

John W. Layman, May 02 2000

This sequence is the number of digits of each term of A061583. - Dmitry Kamenetsky, Jan 17 2009

Michael De Vlieger, Table of n, a(n) for n = 1..6023
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1).

Cf. A003274, A003410.

LinearRecurrence[{2,-1,1,-1},{1,1,2,4},50] (* or *) CoefficientList[ Series[(x^2-x+1)/(x^4-x^3+x^2-2x+1),{x,0,50}],x] (* Harvey P. Dale, Apr 24 2011 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,1,-1,2]^(n-1)*[1;1;2;4])[1,1] \\ Charles R Greathouse IV, Apr 07 2016

From Joseph Myers, Feb 03 2004: (Start)
G.f.: (1 -x +x^2)/(1-2*x+x^2-x^3+x^4).
a(n) = a(n-1) + a(n-3) + 1. (End)
a(n) = Sum_{i=1..n} A058278(i) = A097333(n) - 1. - R. J. Mathar, Oct 16 2010

More terms from Joseph Myers, Feb 03 2004

A003410 Expansion of (1+x)(1+x^2)/(1-x-x^3).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982, 1427440, 2092015, 3065997, 4493437, 6585452, 9651449

Offset: 0

N. J. A. Sloane

From Emeric Deutsch, Feb 15 2010: (Start)
a(n) is the number of binary words of length n that have no pair of adjacent 1's and have no 0000 subwords. Example: a(4)=7 because we have 0101, 1010, 0010, 1001, 0100, 0001, and 1000.
a(n) = A171855(n,0). (End)
a(n) is the number of solus bitstrings of length n with no runs of 4 zeros. - Steven Finch, Mar 25 2020

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Essentially the same as A058278 and A097333, partial sums and first differences of A058278, first and second differences of itself and A038718. Equals A038718(n+1) + 1, n>0.

Cf. A171855. - Emeric Deutsch, Feb 15 2010

G:=series((1+x)*(1+x^2)/(1-x-x^3),x=0,42): 1,seq(coeff(G,x^n),n=1..38);
A003410:=-(1+z)*(1+z**2)/(-1+z+z**3); # Simon Plouffe in his 1992 dissertation

Join[{1}, LinearRecurrence[{1, 0, 1}, {2, 3, 5}, 80]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
CoefficientList[Series[((1+x)(1+x^2))/(1-x-x^3),{x,0,50}],x] (* Harvey P. Dale, Jul 07 2025 *)

a(n)=([0,1,0; 0,0,1; 1,0,1]^n*[1;2;3])[1,1] \\ Charles R Greathouse IV, Mar 25 2020

a(n) = a(n-1) + a(n-3) for n>3, see also A000930. - Reinhard Zumkeller, Oct 26 2005
For n>1, a(n) = 2*A000930(n) + A000930(n-2). - Gerald McGarvey, Sep 10 2008
a(n) = A058278(n+4) = A097333(n+1) for n >= 1. - Jianing Song, Aug 11 2023

More terms from Emeric Deutsch, Dec 11 2004

A013979 Expansion of 1/(1 - x^2 - x^3 - x^4) = 1/((1 + x)*(1 - x - x^3)).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 5, 8, 11, 17, 24, 36, 52, 77, 112, 165, 241, 354, 518, 760, 1113, 1632, 2391, 3505, 5136, 7528, 11032, 16169, 23696, 34729, 50897, 74594, 109322, 160220, 234813, 344136, 504355, 739169, 1083304, 1587660, 2326828, 3410133, 4997792, 7324621

Offset: 0

N. J. A. Sloane

For n>0, number of compositions (ordered partitions) of n into 2's, 3's and 4's. - Len Smiley, May 08 2001
Diagonal sums of trinomial triangle A071675 (Riordan array (1, x*(1+x+x^2))). - Paul Barry, Feb 15 2005
For n>1, a(n) is number of compositions of n-2 into parts 1 and 2 with no 3 consecutive 1's. For example: a(7) = 5 because we have: 2+2+1, 2+1+2, 1+2+2, 1+2+1+1, 1+1+2+1. - Geoffrey Critzer, Mar 15 2014
In the same way [per 2nd comment for A006498, by Sreyas Srinivasan] that the sum of any two alternating terms (terms separated by one term) of A006498 produces a term from A000045 (the Fibonacci sequence), so it could therefore be thought of as a "metaFibonacci," the sum of any two (nonalternating) terms of this sequence produces a term from A000930 (Narayana’s cows), so this sequence could analogously be called "meta-Narayana’s cows" (e.g. 4+5=9, 5+8=13, 8+11=19, 11+17=28). - Michael Cohen and Yasuyuki Kachi, Jun 13 2024

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 5*x^7 + 8*x^8 + 11*x^9 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Michael Cohen and Yasuyuki Kachi, Recurrence Relations Rhythm. In: Noll, T., Montiel, M., Gómez, F., Hamido, O.C., Besada, J.L., Martins, J.O. (eds) Mathematics and Computation in Music. MCM 2024. Lecture Notes in Computer Science, vol. 14639. Springer, Cham.
C. K. Fan, A Hecke algebra quotient and some combinatorial applications, J. Algebraic Combin. 5 (1996), no. 3, 175-189.
C. K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f^0_n.]
R. Mullen, On Determining Paint by Numbers Puzzles with Nonunique Solutions, JIS 12 (2009) 09.6.5.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1).

Cf. A060945 (Ordered partitions into 1's, 2's and 4's).
First differences of A023435.
Cf. A000045, A000930, A001634, A006498, A071675, A077889, A078012, A097333, A107458.

a013979 n = a013979_list !! n
a013979_list = 1 : 0 : 1 : 1 : zipWith (+) a013979_list
   (zipWith (+) (tail a013979_list) (drop 2 a013979_list))
-- Reinhard Zumkeller, Mar 23 2012

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( 1/((1+x)*(1-x-x^3)) )); // G. C. Greubel, Jul 17 2023

a[n_]:= If[n<0, SeriesCoefficient[x^4/(1 +x +x^2 -x^4), {x, 0, -n}], SeriesCoefficient[1/(1 -x^2 -x^3 -x^4), {x,0,n}]]; (* Michael Somos, Jun 20 2015 *)
LinearRecurrence[{0,1,1,1}, {1,0,1,1}, 50] (* G. C. Greubel, Jul 17 2023 *)

@CachedFunction
def b(n): return 1 if (n<3) else b(n-1) + b(n-3) # b = A000930
def A013979(n): return ((-1)^n +2*b(n) -b(n-1) +b(n-2) -int(n==1))/3
[A013979(n) for n in (0..50)] # G. C. Greubel, Jul 17 2023

a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..floor(n/2)} C(k, 2i+3k-n)*C(2i+3k-n, i). - Paul Barry, Feb 15 2005
a(n) = a(n-4) + a(n-3) + a(n-2). - Jon E. Schoenfield, Aug 07 2006
a(n) + a(n+1) = A000930(n+1). - R. J. Mathar, Mar 14 2011
a(n) = (1/3)*(A000930(n) + A097333(n-2) + (-1)^n), n>1. - Ralf Stephan, Aug 15 2013
a(n) = (-1)^n * A077889(-4-n) = A107458(n+4) for all n in Z. - Michael Somos, Jun 20 2015
a(n) = Sum_{i=0..floor(n/2)} A078012(n-2*i). - Paul Curtz, Aug 18 2021
a(n) = (1/3)*((-1)^n + 2*b(n) - b(n-1) + b(n-2) - [n=1]), where b(n) = A000930(n). - G. C. Greubel, Jul 17 2023

A058278 Expansion of (1 - x^2)/(1 - x - x^3).

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 3, 5, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982, 1427440, 2092015, 3065997, 4493437

Offset: 0

Robert G. Wilson v, Dec 06 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
Engin Özkan, Bahar Kuloǧlu, and James Peters, k-Narayana sequence self-similarity, hal-03242990 [math.CO], 2021. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Essentially the same as A003410 and A097333.

Cf. A000930.

CoefficientList[Series[(1 - x^2)/(1 - x - x^3), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Dec 28 2010 *)
LinearRecurrence[{1,0,1},{1,1,0},50] (* Harvey P. Dale, Apr 03 2015 *)

Vec((1-x^2)/(1-x-x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

G.f.: (1 - x^2)/(1 - x - x^3).
a(n+3) = Sum_{k=0..n} binomial(n-k, floor(k/2)). - Paul Barry, Jul 06 2004
a(n) = a(n-3) + a(n-1). - Graeme McRae, Apr 26 2010
From Wolfdieter Lang, Apr 21 2015 : (Start)
a(n) = A097333(n-3), n >= 3.
a(n) = A000930(n) - A000930(n-2), n >= 2. (End)
a(n) = A003410(n-4) for n >= 5. - Jianing Song, Aug 11 2023

Edited: Offset corrected to 0. The formula by P. Barry corrected. Old formulas adapted to new offset. - Wolfdieter Lang, Apr 21 2015

A213274 Irregular array T(n,k) of numbers/2 of non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 2, n >= 2.

Original entry on oeis.org

4, 4, 4, 2, 4, 4, 6, 6, 4, 4, 6, 10, 10, 2, 4, 4, 6, 10, 14, 16, 8, 4, 4, 6, 10, 14, 20, 26, 18, 2, 4, 4, 6, 10, 14, 20, 30, 40, 34, 10, 4, 4, 6, 10, 14, 20, 30, 44, 60, 60, 28, 2, 4, 4, 6, 10, 14, 20, 30, 44, 64, 90, 100, 62, 12, 4, 4, 6, 10, 14, 20, 30, 44, 64, 94, 134, 160, 122, 40, 2

Offset: 2

Christopher Hunt Gribble, Jun 08 2012

The irregular array of numbers is:
....k..3...4...5...6...7...8...9..10..11..12..13..14..15..16..17
..n
..2....4
..3....4...4...2
..4....4...4...6...6
..5....4...4...6..10..10...2
..6....4...4...6..10..14..16...8
..7....4...4...6..10..14..20..26..18...2
..8....4...4...6..10..14..20..30..40..34..10
..9....4...4...6..10..14..20..30..44..60..60..28...2
.10....4...4...6..10..14..20..30..44..64..90.100..62..12
.11....4...4...6..10..14..20..30..44..64..94.134.160.122..40...2
where k is the path length in nodes.
In an attempt to define the irregularity of the array, it appears that the maximum value of k is (3n + n mod 2)/2 for n >= 2.
Reading this array by rows gives the sequence.
One half of the numbers of paths constitute the sequence to remove the effect of the bilateral symmetry of a rectangle.

			T(2,3) = One half of the number of complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 2 node rectangle.

Cf. A213106, A213249.

The asymptotic sequence for the number of paths of each nodal length k for n >> 0 appears to be 2*A097333(1:), that is, 2*(Sum(j=0..k-2, C(k-2-j, floor(j/2)))), for k >= 3.

A226247 Let S be the set of numbers defined by these rules: 0 is in S; if x is in S, then x+1 is in S, and if nonzero x is in S, then -1/x are in S. (See Comments.)

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 1, 5, 4, 3, 2, 2, 3, 1, 6, 5, 4, 3, 3, 5, 2, 5, 3, 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1, 1, 8, 7, 6, 5, 5, 9, 4, 11, 7, 3, 11, 8, 5, 2, 2, 9, 7, 5, 3, 3, 4, 1, 9, 8, 7, 6, 6, 11, 5, 14, 9, 4, 15, 11, 7, 3, 3

Offset: 1

Clark Kimberling, Jun 01 2013

Let S be the set of numbers defined by these rules:  0 is in S; if x is in S, then x+1 is in S, and if nonzero x is in S, then -1/x are in S.  Then S is the set of all rational numbers, produced in generations as follows:
g(1) = (0), g(2) = (1), g(3) = (2, -1), g(4) = (3, -1/2), g(5) = (4, -1/3, 1/2), ... For n > 2, once g(n-1) = (c(1), ..., c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2), ..., c(z)+1, -1/c(z)) by deleting previously generated elements.
Let S'' denote the sequence formed by concatenating the generations.
  A226247:  Denominators of terms of S''
  A226248:  Numerators of terms of S''
  A226249:  Positions of nonnegative numbers in S''
  A226250:  Positions of positive numbers in S''
A closely related sequence S' (for which the rules of generation are shorter but the resulting sequence is slightly less natural) is discussed at A226130.  For both S' and S'', the number of numbers in g(n) is given by A097333.

			The denominators and numerators are read from S'':
  0/1, 1/1, 2/1, -1/1, 3, -1/2, 4/1, -1/3, 1/2, 5, -1/4, 2/3, 3/2, -2, ...
Table begins:
  n |
  --+-----------------------------------------------
  1 | 1;
  2 | 1;
  3 | 1, 1;
  4 | 1, 2;
  5 | 1, 3, 2;
  6 | 1, 4, 3, 2, 1;
  7 | 1, 5, 4, 3, 2, 2, 3;
  8 | 1, 6, 5, 4, 3, 3, 5, 2, 5, 3;
  9 | 1, 7, 6, 5, 4, 4, 7, 3, 8, 5, 2, 7, 5, 3, 1;

Clark Kimberling, Table of n, a(n) for n = 1..1000
Peter Kagey, Illustration of first seven generations.

Cf. A226080 (rabbit ordering of positive rationals), A226130.

Cf. A226248, A226249, A226250.

z = 12; g[1] := {0}; g[2] := {1}; g[n_] :=  g[n] = DeleteCases[Flatten[Transpose[{# + 1, -1/#}]] &[g[n - 1]], Apply[Alternatives, Flatten[Map[g, Range[n - 1]]]]]; f = Flatten[Map[g, Range[z]]]; Take[Denominator[f], 100]  (*A226247*)
t = Take[Numerator[f], 100]  (*A226248*)
s[n_] := If[t[[n]] > 0, 1, 0]; u = Table[s[n], {n, 1, Length[t]}]
Flatten[Position[u, 1]] (*A226249*)
p = Flatten[Position[u, 0]] (*A226250*) (* Peter J. C. Moses, May 30 2013 *)

from fractions import Fraction
from itertools import count, islice
def agen():
    rats = [Fraction(0, 1)]
    seen = {Fraction(0, 1)}
    for n in count(1):
        yield from [r.denominator for r in rats]
        newrats = []
        for r in rats:
            f = 1+r
            if f not in seen:
                newrats.append(1+r)
                seen.add(f)
            if r != 0:
                g = -1/r
                if g not in seen:
                    newrats.append(-1/r)
                    seen.add(g)
        rats = newrats
print(list(islice(agen(), 84))) # Michael S. Branicky, Jan 17 2022

A213431 Irregular array T(n,k) of the numbers of distinct shapes under rotation of the non-extendable (complete) non-self-adjacent simple paths of each length within a square lattice bounded by rectangles with nodal dimensions n and 2, n >= 2.

Original entry on oeis.org

2, 2, 4, 2, 2, 4, 6, 6, 2, 4, 6, 10, 10, 2, 2, 4, 6, 10, 14, 16, 8, 2, 4, 6, 10, 14, 20, 26, 18, 2, 2, 4, 6, 10, 14, 20, 30, 40, 34, 10, 2, 4, 6, 10, 14, 20, 30, 44, 60, 60, 28, 2, 2, 4, 6, 10, 14, 20, 30, 44, 64, 90, 100, 62, 12

Offset: 2

Christopher Hunt Gribble, Jun 11 2012

The irregular array of numbers is:
....k..3...4...5...6...7...8...9..10..11..12..13..14..15
..n
..2....2
..3....2...4...2
..4....2...4...6...6
..5....2...4...6..10..10...2
..6....2...4...6..10..14..16...8
..7....2...4...6..10..14..20..26..18...2
..8....2...4...6..10..14..20..30..40..34..10
..9....2...4...6..10..14..20..30..44..60..60..28...2
.10....2...4...6..10..14..20..30..44..64..90.100..62..12
where k is the path length in nodes. In an attempt to define the irregularity of the array, it appears that the maximum value of k is n + floor((n+1)/2) for n >= 2. Reading this array by rows gives the sequence.

			T(2,3) = The number of distinct shapes under rotation of the complete non-self-adjacent simple paths of length 3 nodes within a square lattice bounded by a 2 X 2 node rectangle.

Cf. A213106, A213249.

The asymptotic sequence for the number of distinct shapes under rotation of the complete non-self-adjacent simple paths of each nodal length k for n >> 0 appears to be 2*A097333(2:), that is, 2*(Sum(j=0..k-2, C(k-2-j, floor(j/2)))), for k >= 4.

A097334 a(n) = Sum_{k=0..n} C(n-k, floor(k/2))*2^k.

Original entry on oeis.org

1, 3, 3, 7, 19, 31, 59, 135, 259, 495, 1035, 2071, 4051, 8191, 16475, 32679, 65443, 131343, 262059, 523831, 1049203, 2097439, 4192763, 8389575, 16779331, 33550383, 67108683, 134226007, 268427539, 536862271, 1073766299, 2147476455

Offset: 0

Paul Barry, Aug 05 2004

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,4).

Cf. A089977, A097333, A097335.

Table[Sum[Binomial[n-k,Floor[k/2]]2^k,{k,0,n}],{n,0,40}] (* or *) LinearRecurrence[{1,0,4},{1,3,3},40] (* Harvey P. Dale, May 17 2021 *)

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

cp's OEIS Frontend

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