A056453 -id:A056453 - cp's OEIS frontend

A056450 a(n) = (3*2^n - (-2)^n)/2.

1, 4, 4, 16, 16, 64, 64, 256, 256, 1024, 1024, 4096, 4096, 16384, 16384, 65536, 65536, 262144, 262144, 1048576, 1048576, 4194304, 4194304, 16777216, 16777216, 67108864, 67108864, 268435456, 268435456, 1073741824, 1073741824, 4294967296

Offset: 0

Marks R. Nester

Number of palindromes of length n using a maximum of four different symbols.
Number of achiral rows of n colors using up to four colors. - Robert A. Russell, Nov 09 2018
Interleaving of A000302 and 4*A000302.
Unsigned version of A141125.
Binomial transform is A164907. Second binomial transform is A164908. Third binomial transform is A057651. Fourth binomial transform is A016129.

			At length n=1 there are a(1)=4 palindromes, A, B, C, D.
At length n=2, there are a(2)=4 palindromes, AA, BB, CC, DD.
At length n=3, there are a(3)=16 palindromes, AAA, BBB, CCC, DDD, ABA, BAB, ... , CDC, DCD.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Sean A. Irvine, Walks on Graphs.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,4).

Column k=4 of A321391.
Cf. A000302 (powers of 4), A056450, A141125, A164907, A164908, A057651, A016129.
Cf. A016116.
Essentially the same as A213173.
Cf. A000302 (oriented), A032121 (unoriented), A032087(n>1) (chiral).

Magma
```
[ (3*2^n-(-2)^n)/2: n in [0..31] ];
```

[4^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

Table[4^Ceiling[n/2], {n,0,40}] (* or *)
CoefficientList[Series[(1 + 4 x)/((1 + 2 x) (1 - 2 x)), {x, 0, 31}], x] (* or *)
LinearRecurrence[{0, 4}, {1, 4}, 40] (* Robert A. Russell, Nov 07 2018 *)

a(n)=4^((n+1)\2) \\ Charles R Greathouse IV, Apr 08 2012

a(n)=(3*2^n-(-2)^n)/2 \\ Charles R Greathouse IV, Oct 03 2016

a(n) = 4^floor((n+1)/2).
a(n) = 4*a(n-2) for n > 1; a(0) = 1, a(1) = 4.
G.f.: (1+4*x) / (1-4*x^2). - R. J. Mathar, Jan 19 2011 [Adapted to offset 0 by Robert A. Russell, Nov 07 2018]
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = 4*abs(A164111(n-1)). - R. J. Mathar, Jan 19 2011
a(n) = C(4,0)*A000007(n) + C(4,1)*A057427(n) + C(4,2)*A056453(n) + C(4,3)*A056454(n) + C(4,4)*A056455(n). - Robert A. Russell, Nov 08 2018

a(0)=1 prepended by Robert A. Russell, Nov 07 2018

Edited by N. J. A. Sloane, Sep 29 2019

A122746 G.f.: 1/((1-2x)(1-2*x^2)).

Original entry on oeis.org

1, 2, 6, 12, 28, 56, 120, 240, 496, 992, 2016, 4032, 8128, 16256, 32640, 65280, 130816, 261632, 523776, 1047552, 2096128, 4192256, 8386560, 16773120, 33550336, 67100672, 134209536, 268419072, 536854528, 1073709056, 2147450880, 4294901760, 8589869056

Offset: 0

N. J. A. Sloane, Sep 24 2006

Equals row sums of triangle A156665. - Gary W. Adamson, Feb 12 2009
a(n) is the number of subsets of {1,2,...,n+1} that contain at least one odd integer. - Geoffrey Critzer, Mar 03 2009
a(n-3) is the number of chiral pairs of color patterns of length n using two colors. Two color patterns are equivalent if the colors are permuted. For example, a string of five colors using exactly two different colors has six chiral pairs: AAAAB-ABBBB, AAABA-ABAAA, AAABB-AABBB, AABAB-ABABB, AABBA-ABBAA, and ABAAB-ABBAB. The number of color patterns of length n using exactly k colors when chiral pairs are counted twice is the Stirling subset number S2(n,k). The number of achiral color patterns of length n using exactly 2 colors is S2(floor(n/2)+1,2). The value of a(n-3) is half the difference of these two. - Robert A. Russell, Feb 01 2018
a(n-2) is the number of chiral pairs for a row of n colors with exactly 2 different colors. If the reverse of a sequence is different, the combination of the two is a chiral pair. For a row of 4 colors using exactly 2 different colors, the chiral pairs are AAAB-BAAA, AABA-ABAA, AABB-BBAA, ABAB-BABA, ABBB-BBBA, and BABB-BBAB. Thus a(4-2) = a(2) = 6. - Robert A. Russell, Jun 10 2018

			G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 28*x^4 + 56*x^5 + 120*x^6 + 240*x^7 + 496*x^8 + ... - _Michael Somos_, Jul 01 2018

Harvey P. Dale, Table of n, a(n) for n = 0..1000
S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
E. Filiol and C. Fontaine, Highly nonlinear balanced Boolean functions with a good correlation-immunity, Lect. Not. Comp. Sci 1403 (1998), 475-488, NL(F_n).
Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, arXiv:2112.03338 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Essentially the same as A032085.
Cf. A000079, A060546.
Cf. A006516, A020522, A156665.
Cf. A000918, A056309, A056453.

List([0..35],n->2^(n+1)-2^(QuoInt(n+1,2))); # Muniru A Asiru, Sep 27 2018

seq(coeff(series(((1-2*x)*(1-2*x^2))^(-1),x,n+1), x, n), n = 0..35); # Muniru A Asiru, Sep 27 2018

RecurrenceTable[{a[n] == 2 (BitOr[a[n - 1], a[n - 2]]), a[0] == 1, a[1] == 2}, a, {n, 0, 32}] (* Geoffrey Critzer, Jan 09 2011 *)
CoefficientList[Series[1/((1-2x)(1-2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,2,-4},{1,2,6},40] (* Harvey P. Dale, Jun 25 2013 *)
Table[(StirlingS2[n,2] - StirlingS2[Floor[n/2]+1,2])/2, {n,3,30}] (* Robert A. Russell, Jan 29 2018 *)
a[ n_] := 2^(n + 1) - 2^Quotient[n + 1, 2]; (* Michael Somos, Jul 01 2018 *)

{a(n) = 2^(n+1) - 2^((n+1)\2)}; /* Michael Somos, Jul 01 2018 */

From Alexander Adamchuk, Sep 25 2006: (Start)
a(2k) = A006516(k+1) = 2^k*(2^(k+1) - 1) = A020522(k+1) /2.
a(2k+1) = 2*A006516(k+1) = 2^(k+1)*(2^(k+1) - 1) = A020522(k+1). (End)
a(n) = 2^(n+1) - 2^(floor((n+1)/2)). - Geoffrey Critzer, Mar 03 2009
a(n) = 2*(a(n-1) bitwiseOR a(n-2)), a(0)=1, a(1)=2. - Pierre Charland, Dec 12 2010
G.f.: (1+x*Q(0))/(1-x)^2, where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 23 2013
a(0)=1, a(1)=2, a(2)=6, a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3). - Harvey P. Dale, Jun 25 2013
a(n) = (A000079(n+2) - A060546(n+2))/ 2. - Robert A. Russell, Jun 19 2018
a(n) = -a(-3-n) * 2^(n+2 + floor((n+1)/2)) for all n in Z. - Michael Somos, Jul 01 2018
a(n) = (A000918(n+2) - A056453(n+2)) / 2 = A000918(n+2) - A056309(n+2) = A056309(n+2) - A056453(n+2). - Robert A. Russell, Sep 26 2018

A056449 a(n) = 3^floor((n+1)/2).

Original entry on oeis.org

1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467

Offset: 0

Marks R. Nester

One followed by powers of 3 with positive exponent, repeated. - Omar E. Pol, Jul 27 2009

Number of achiral rows of n colors using up to three colors. E.g., for a(3) = 9, the rows are AAA, ABA, ACA, BAB, BBB, BCB, CAC, CBC, and CCC. - Robert A. Russell, Nov 07 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0,3).

Column k=3 of A321391.
Cf. A000007, A000244, A016116, A056391, A056453, A056454, A057427.
Essentially the same as A108411 and A162436.
Cf. A000244 (oriented), A032120 (unoriented), A032086(n>1) (chiral).

[3^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

Riffle[3^Range[0, 20], 3^Range[20]] (* Harvey P. Dale, Jan 21 2015 *)
Table[3^Ceiling[n/2], {n,0,40}] (* or *)
LinearRecurrence[{0, 3}, {1, 3}, 40] (* Robert A. Russell, Nov 07 2018 *)

a(n)=3^floor((n+1)/2); \\ Joerg Arndt, Apr 23 2013

def A056449(n): return 3**(n+1>>1) # Chai Wah Wu, Oct 28 2024

G.f.: (1 + 3*x) / (1 - 3*x^2). - R. J. Mathar, Jul 06 2011 [Adapted to offset 0 by Robert A. Russell, Nov 07 2018]
a(n) = k^ceiling(n/2), where k = 3 is the number of possible colors. - Robert A. Russell, Nov 07 2018
a(n) = C(3,0)*A000007(n) + C(3,1)*A057427(n) + C(3,2)*A056453(n) + C(3,3)*A056454(n). - Robert A. Russell, Nov 08 2018
E.g.f.: cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x). - Stefano Spezia, Dec 31 2022

Edited by N. J. A. Sloane at the suggestion of Klaus Brockhaus, Jul 03 2009

a(0)=1 prepended by Robert A. Russell, Nov 07 2018

A056451 Number of palindromes using a maximum of five different symbols.

Original entry on oeis.org

1, 5, 5, 25, 25, 125, 125, 625, 625, 3125, 3125, 15625, 15625, 78125, 78125, 390625, 390625, 1953125, 1953125, 9765625, 9765625, 48828125, 48828125, 244140625, 244140625, 1220703125, 1220703125, 6103515625, 6103515625, 30517578125, 30517578125, 152587890625, 152587890625

Offset: 0

Marks R. Nester

Number of achiral rows of n colors using up to five colors. For a(3) = 25, the rows are AAA, ABA, ACA, ADA, AEA, BAB, BBB, BCB, BDB, BEB, CAC, CBC, CCC, CDC, CEC, DAD, DBD, DCD, DDD, DED, EAE, EBE, ECE, EDE, and EEE. - Robert A. Russell, Nov 09 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0,5).

Column k=5 of A321391.
Cf. A000007, A016116, A056391, A056453, A056454, A056455, A056456, A057427,
Cf. A000351 (oriented), A032122 (unoriented), A032088(n>1) (chiral).

[5^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

LinearRecurrence[{0,5},{1,5},30] (* or *) Riffle[5^Range[0, 20], 5^Range[20]] (* Harvey P. Dale, Jul 28 2018 *)
Table[5^Ceiling[n/2], {n,0,40}] (* Robert A. Russell, Nov 07 2018 *)

vector(40, n, n--; 5^floor((n+1)/2)) \\ G. C. Greubel, Nov 07 2018

a(n) = 5^floor((n+1)/2).
a(n) = 5*a(n-2). - Colin Barker, May 06 2012
G.f.: (1+5*x) / (1-5*x^2). - Colin Barker, May 06 2012 [Adapted to offset 0 by Robert A. Russell, Nov 07 2018]
a(n) = C(5,0)*A000007(n) + C(5,1)*A057427(n) + C(5,2)*A056453(n) + C(5,3)*A056454(n) + C(5,4)*A056455(n) + C(5,5)*A056456(n). - Robert A. Russell, Nov 08 2018
E.g.f.: cosh(sqrt(5)*x) + sqrt(5)*sinh(sqrt(5)*x). - Stefano Spezia, Jun 06 2023

a(0)=1 prepended by Robert A. Russell, Nov 07 2018

A056452 a(n) = 6^floor((n+1)/2).

Original entry on oeis.org

1, 6, 6, 36, 36, 216, 216, 1296, 1296, 7776, 7776, 46656, 46656, 279936, 279936, 1679616, 1679616, 10077696, 10077696, 60466176, 60466176, 362797056, 362797056, 2176782336, 2176782336, 13060694016, 13060694016, 78364164096

Offset: 0

Marks R. Nester

Number of achiral rows of length n using up to six different colors. For a(3) = 36, the rows are AAA, ABA, ACA, ADA, AEA, AFA, BAB, BBB, BCB, BDB, BEB, BFB, CAC, CBC, CCC, CDC, CEC, CFC, DAD, DBD, DCD, DDD, DED, DFD, EAE, EBE, ECE, EDE, EEE, EFE, FAF, FBF, FCF, FDF, FEF, and FFF. - Robert A. Russell, Nov 08 2018

Also: a(n) is the number of palindromes with n digits using a maximum of six different symbols. - David A. Corneth, Nov 09 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0,6).

Column k=6 of A321391.
Cf. A016116.
Cf. A000400 (oriented), A056308 (unoriented), A320524 (chiral).

[6^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

Maple
```
A056452:=n->6^floor((n+1)/2);
```

Riffle[6^Range[0, 20], 6^Range[20]] (* Harvey P. Dale, Jun 18 2017 *)
Table[6^Ceiling[n/2], {n,0,40}] (* or *)
LinearRecurrence[{0, 6}, {1, 6}, 40] (* Robert A. Russell, Nov 08 2018 *)

a(n) = 6^floor((n+1)/2).
a(n) = 6*a(n-2). - Colin Barker, May 06 2012
G.f.: (1+6*x) / (1-6*x^2). - Colin Barker, May 06 2012 [Adapted to offset 0 by Robert A. Russell, Nov 08 2018]
a(n) = C(6,0)*A000007(n) + C(6,1)*A057427(n) + C(6,2)*A056453(n) + C(6,3)*A056454(n) + C(6,4)*A056455(n) + C(6,5)*A056456(n) + C(6,6)*A056457(n). - Robert A. Russell, Nov 08 2018

a(0)=1 prepended by Robert A. Russell, Nov 08 2018

Name corrected by David A. Corneth, Nov 08 2018

A056309 Number of reversible strings with n beads using exactly two different colors.

Original entry on oeis.org

0, 1, 4, 8, 18, 34, 70, 134, 270, 526, 1054, 2078, 4158, 8254, 16510, 32894, 65790, 131326, 262654, 524798, 1049598, 2098174, 4196350, 8390654, 16781310, 33558526, 67117054, 134225918, 268451838, 536887294, 1073774590, 2147516414, 4295032830, 8590000126

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent.

			For n=3, the four rows are ABA, BAB, AAB, and ABB, the last two being respectively equivalent to BAA and BBA, with which they form chiral pairs. - _Robert A. Russell_, Sep 25 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-6,4).

Cf. A000079, A005418.
Equals (A000918 + A056453) / 2.
a(n) = A000918(n) - A122746(n-2) = A122746(n-2) + A056453(n).

[2^(n-1)+2^((n-1) div 2)-2: n in [1..40]]; // Vincenzo Librandi, Sep 29 2018

seq(2^(n-1) + 2^floor((n-1)/2) - 2, n=1..34); # Peter Luschny, Nov 25 2017

Rest[CoefficientList[Series[x^2(1+x-4x^2)/(1-3x+6x^3-4x^4),{x,0,30}],x]] (* or *) LinearRecurrence[{3,0,-6,4},{0,1,4,8},30] (* Harvey P. Dale, Feb 18 2012 *)

Vec(x^2*(1+x-4*x^2)/(1-3*x+6*x^3-4*x^4) + O(x^40)) \\ Colin Barker, Nov 24 2017

a(n) = 2^(n-1)+2^((n-1)\2)-2; \\ Altug Alkan, Sep 25 2018

a(n) = A005418(n+1) - 2.
G.f.: x^2*(1 + x - 4*x^2)/(1 - 3*x + 6*x^3 - 4*x^4). - Colin Barker, Feb 03 2012
a(1)=0, a(2)=1, a(3)=4, a(4)=8, a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4). - Harvey P. Dale, Feb 18 2012
From Colin Barker, Nov 24 2017: (Start)
a(n) = 2^(n/2-1) + 2^(n-1) - 2 for n even.
a(n) = 2^((n-1)/2) + 2^(n-1) - 2 for n odd. (End)
a(n) = A000079(n-1) + A056453(n-2). - Peter Luschny, Nov 25 2017
a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=2 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A208900 Number of bitstrings of length n which (if having two or more runs) the last two runs have different lengths.

Original entry on oeis.org

2, 2, 6, 10, 26, 50, 114, 226, 482, 962, 1986, 3970, 8066, 16130, 32514, 65026, 130562, 261122, 523266, 1046530, 2095106, 4190210, 8384514, 16769026, 33546242, 67092482, 134201346, 268402690, 536838146, 1073676290, 2147418114, 4294836226, 8589803522

Offset: 1

David Nacin, Mar 03 2012

A run is a maximal subsequence of (possibly just one) identical bits.

			If n=3 the bitstrings where the last runs have different lengths are 111,000,100,011,110,001 so a(3) = 6.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Aruna Gabhe, Problem 11623, Am. Math. Monthly 119 (2012) 161.
Index entries for linear recurrences with constant coefficients, signature (3,0,-6,4).

Cf. A056453, A208901, A208902, A208903.

Table[2 + 2^n - 2^(Floor[n/2] + 1) ,  {n, 1, 40}]
LinearRecurrence[{3, 0, -6, 4}, {2, 2, 6, 10}, 40]

a(n) = 2^n + 2 - 2^(floor(n/2)+1).
a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4), a(0) = 2, a(1) = 2, a(2) = 6, a(3) = 10.
G.f.: x*(2 - 4*x +  4*x^3)/((1-x)*(1-2*x^2)*(1-2*x)).
E.g.f.: - 2*cosh(sqrt(2)*x) + 2*exp(x)*(1 + sinh(x)) - sqrt(2)*sinh(sqrt(2)*x). - Stefano Spezia, Jun 06 2023

A208901 Number of bitstrings of length n (with at least two runs) where the last two runs have different lengths.

Original entry on oeis.org

0, 0, 4, 8, 24, 48, 112, 224, 480, 960, 1984, 3968, 8064, 16128, 32512, 65024, 130560, 261120, 523264, 1046528, 2095104, 4190208, 8384512, 16769024, 33546240, 67092480, 134201344, 268402688, 536838144, 1073676288, 2147418112, 4294836224, 8589803520

Offset: 1

David Nacin, Mar 03 2012

A run is a maximal subsequence of (possibly just one) identical bits.

			If n=3 the bitstrings (with at least two runs) where the last runs have different lengths are 100,011,110,001 so a(3) = 4.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Aruna Gabhe, Problem 11623, Am. Math. Monthly 119 (2012) 161.
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A056453, A208900, A208902, A208903.

Table[2^n - 2^(Floor[ n/2] + 1) ,  {n, 1, 40}]
LinearRecurrence[{2, 2, -4}, {0, 0, 4}, 40]

a(n) = 2^n - 2^(floor(n/2)+1).
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3), a(0) = 0, a(1) = 0, a(2) = 4.
G.f.: 4*x^2/((1 - 2*x)*(1 - 2*x^2)).
E.g.f.: 2*(cosh(2*x) - cosh(sqrt(2)*x) + sinh(2*x) - sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Jun 06 2023

A208902 The sum over all bitstrings b of length n of the number of runs in b not immediately followed by a longer run.

Original entry on oeis.org

2, 6, 14, 34, 78, 182, 414, 942, 2110, 4702, 10366, 22718, 49406, 106878, 229886, 492286, 1049598, 2229758, 4720638, 9964542, 20975614, 44046334, 92282878, 192950270, 402669566, 838885374, 1744863230, 3623927806, 7516258302, 15569354750, 32212385790

Offset: 1

David Nacin, Mar 03 2012

A run is a maximal subsequence of (possibly just one) identical bits.

			When n=3, 000,111 each have 1 such run, 101,010 each have 3, 100,011 each have 1, 001, 110 each have 2, summing these gives 2+6+2+4=14 so a(3) = 14.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Aruna Gabhe, Problem 11623, Am. Math. Monthly 119 (2012) 161.
Index entries for linear recurrences with constant coefficients, signature (5,-6,-6,16,-8).

Cf. A056453, A208900, A208901, A208903.

Table[2^n*(2 + (n - 1)/2 - (1/2)^(n - 1) - 2*(1 - (1/2)^Floor[n/2]) + (1/2)^(Floor[n/2] + 1) (1 + (-1)^n)), {n, 1, 40}]
LinearRecurrence[{5, -6, -6, 16, -8}, {2, 6, 14, 34, 78}, 40]

a(n) = 2^n * (2 + (n - 1)/2 - (1/2)^(n - 1) - 2 (1 - (1/2)^floor(n/2)) + (1/2)^(floor(n/2) + 1) (1 + (-1)^n)).
a(n) = A208903(n) + 2.
a(n) = 5*a(n-1) - 6*a(n-2) - 6*a(n-3) + 16*a(n-4) - 8*a(n-5), a(1) = 2, a(2) = 6, a(3) = 14, a(4) = 34, a(5) = 78.
G.f.: (2 - 4*x - 4*x^2 + 12*x^3 - 4*x^4)/(1 - 5*x + 6*x^2 + 6*x^3 - 16*x^4 + 8*x^5).

A208903 The sum over all bitstrings b of length n with at least two runs of the number of runs in b not immediately followed by a longer run.

Original entry on oeis.org

0, 4, 12, 32, 76, 180, 412, 940, 2108, 4700, 10364, 22716, 49404, 106876, 229884, 492284, 1049596, 2229756, 4720636, 9964540, 20975612, 44046332, 92282876, 192950268, 402669564, 838885372, 1744863228, 3623927804, 7516258300, 15569354748, 32212385788

Offset: 1

David Nacin, Mar 03 2012

A run is a maximal subsequence of (possibly just one) identical bits.

			n=3: 101, 010 each have 3; 100, 011 each have 1; 001, 110 each have 2. (000, 111 do not have at least two runs so they do not contribute.) Summing these gives 6+2+4 = 12 so a(3) = 12.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Aruna Gabhe, Problem 11623, Am. Math. Monthly 119 (2012) 161.
Index entries for linear recurrences with constant coefficients, signature (5,-6,-6,16,-8).

Cf. A056453, A208900, A208901, A208902.

Table[2^n*(2 + (n-1)/2 - (1/2)^(n-1) - 2*(1 - (1/2)^Floor[n/2]) + (1/2)^(Floor[n/2] + 1) (1 + (-1)^n)) - 2, {n, 1, 40}]
LinearRecurrence[{5, -6, -6, 16, -8}, {0, 4, 12, 32, 76}, 40]

a(n) = 2^n * (2 + (n - 1)/2 - (1/2)^(n - 1) - 2 (1 - (1/2)^floor(n/2)) + (1/2)^(floor(n/2) + 1) (1 + (-1)^n)) - 2.
a(n) = A208902(n) - 2.
a(n) = 5*a(n-1) - 6*a(n-2) - 6*a(n-3) + 16*a(n-4) - 8*a(n-5), a(1) = 0, a(2) = 4, a(3) = 12, a(4) = 32, a(5) = 76.
G.f.: (4*x - 8*x^2 - 4*x^3 + 12*x^4)/(1 - 5*x + 6*x^2 + 6*x^3 - 16*x^4 +
   8*x^5).

cp's OEIS Frontend

A056450 a(n) = (3*2^n - (-2)^n)/2.

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A122746 G.f.: 1/((1-2*x)*(1-2*x^2)).

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A056449 a(n) = 3^floor((n+1)/2).

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A056451 Number of palindromes using a maximum of five different symbols.

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A056452 a(n) = 6^floor((n+1)/2).

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A056309 Number of reversible strings with n beads using exactly two different colors.

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A122746 G.f.: 1/((1-2x)(1-2*x^2)).