A082412 -id:A082412 - cp's OEIS frontend

A103333 Number of closed walks on the graph of the (7,4) Hamming code.

1, 3, 24, 192, 1536, 12288, 98304, 786432, 6291456, 50331648, 402653184, 3221225472, 25769803776, 206158430208, 1649267441664, 13194139533312, 105553116266496, 844424930131968, 6755399441055744, 54043195528445952, 432345564227567616

Offset: 0

Paul Barry, Jan 31 2005

Counts closed walks of length 2n at the degree 3 node of the graph of the (7,4) Hamming code. With interpolated zeros, counts paths of length n at this node.
a(n+1) = A157176(A016945(n)). - Reinhard Zumkeller, Feb 24 2009
For n>0: a(n) = A083713(n) - A083713(n-1). - Reinhard Zumkeller, Feb 22 2010

David J.C. Mackay, Information Theory, Inference and Learning Algorithms, CUP, 2003, p. 19

Nathaniel Johnston, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A082412, A103334.

Cf. A000302, A004171. - Vincenzo Librandi, Jan 22 2009

seq((3*8^n+5*`if`(n=0,1,0))/8,n=0..20); # Nathaniel Johnston, Jun 26 2011

Join[{1},NestList[8#&,3,20]] (* Harvey P. Dale, Mar 02 2012 *)

G.f.: (1-5*x)/(1-8*x);
a(n) = (3*8^n + 5*0^n)/8.
a(n) = 8*a(n-1) for n > 0. - Harvey P. Dale, Mar 02 2012

A083076 Third row of number array A083075.

Original entry on oeis.org

1, 5, 33, 229, 1601, 11205, 78433, 549029, 3843201, 26902405, 188316833, 1318217829, 9227524801, 64592673605, 452148715233, 3165041006629, 22155287046401, 155087009324805, 1085609065273633, 7599263456915429, 53194844198408001

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A067411. Inverse binomial transform of A082412.

Trinomial transform of Jacobsthal numbers A001045. - Paul Barry, Sep 10 2007

Nathaniel Johnston, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A034478, A066443, A082761, A083077.

[(2*7^n+1)/3: n in [0..30]]; // Vincenzo Librandi, Nov 06 2011

seq((2*7^n+1)/3,n=0..20); # Nathaniel Johnston, Jun 26 2011

a(n) = (2*7^n + 1)/3.
G.f.: (1-3*x)/((1-x)*(1-7*x)).
E.g.f.: (2*exp(7*x) + exp(x))/3.
a(n) = Sum_{k=0..2*n} trinomial(n,k)*Fibonacci(k+1), where trinomial(n,k) are the trinomial coefficients (A027907). - Paul Barry, Sep 10 2007
a(n) = 7*a(n-1) - 2, a(n) = 8*a(n-1) - 7*a(n-2). - Vincenzo Librandi, Nov 06 2011

A082413 a(n) = (2*9^n + 3^n)/3.

Original entry on oeis.org

1, 7, 57, 495, 4401, 39447, 354537, 3189375, 28700001, 258286887, 2324542617, 20920765455, 188286534801, 1694577750327, 15251196564297, 137260759512735, 1235346806916801, 11118121176157767, 100063090327139577, 900567812169415215

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A082412.

Nathaniel Johnston, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (12,-27).

Cf. A082412, A082414.

seq((2*9^n+3^n)/3,n=0..19); # Nathaniel Johnston, Jun 26 2011

G.f.: (1-5*x)/((1-3*x)*(1-9*x));
E.g.f.: (2*exp(9*x) + exp(3*x))/3.
a(n) = (2*9^n + 3^n)/3.

A103334 Number of closed walks of length 2n at any of the nodes of degree 1 on the graph of the (7,4) Hamming code.

Original entry on oeis.org

1, 1, 4, 24, 176, 1376, 10944, 87424, 699136, 5592576, 44739584, 357914624, 2863312896, 22906494976, 183251943424, 1466015514624, 11728124051456, 93824992280576, 750599937982464, 6004799503335424, 48038396025634816

Offset: 0

Paul Barry, Jan 31 2005

a(n+1)=8^n/3+2^(n+1)/3 with g.f. (1-6x)/(1-10x+16x^2) counts walks of length 2n+1 between adjacent nodes of degrees 1 and 4 on the graph of the (7,4) Hamming code.

David J.C. Mackay, Information Theory, Inference and Learning Algorithms, CUP, 2003, p. 19.

Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. A082412, A103333.

G.f.: (1-9x+10x^2)/(1-10x+16x^2); a(n)=8^(n-1)/3+2^(n)/3+5*0^n/8.

A082414 a(n) = (2*10^n + 4^n)/3.

Original entry on oeis.org

1, 8, 72, 688, 6752, 67008, 668032, 6672128, 66688512, 666754048, 6667016192, 66668064768, 666672259072, 6666689036288, 66666756145152, 666667024580608, 6666668098322432, 66666672393289728, 666666689573158912, 6666666758292635648

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A082413.

Nathaniel Johnston, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (14,-40).

Cf. A082412, A082413.

seq((2*10^n+4^n)/3,n=0..19); # Nathaniel Johnston, Jun 26 2011

a(n) = (2*10^n+4^n)/3; \\ Altug Alkan, Sep 08 2018

G.f.: (1-6*x)/((1-4*x)*(1-10*x)).
E.g.f.: (2*exp(10*x) + exp(4*x))/3.
a(n) = (2*10^n + 4^n)/3.

A083217 a(n) = (2*5^n + (-1)^n)/3.

Original entry on oeis.org

1, 3, 17, 83, 417, 2083, 10417, 52083, 260417, 1302083, 6510417, 32552083, 162760417, 813802083, 4069010417, 20345052083, 101725260417, 508626302083, 2543131510417, 12715657552083, 63578287760417, 317891438802083

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A003683 (without leading zero). Inverse binomial transform of A067411.

a(n) is the number of compositions of n when there are 3 types of 1 and 8 types of other natural numbers. - Milan Janjic, Aug 13 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,5).

Cf. A003683, A067411, A082412.

[(2*5^n +(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Feb 17 2023

LinearRecurrence[{4,5},{1,3},30] (* Harvey P. Dale, Sep 18 2018 *)

from sage.combinat.sloane_functions import recur_gen2b
it = recur_gen2b(1,3,4,5, lambda n: 0)
[next(it) for i in range(1,24)] # Zerinvary Lajos, Jul 03 2008

a(n) = (2*5^n + (-1)^n)/3.
G.f.: (1-x)/((1-5*x)*(1+x)).
E.g.f.: (2*exp(5*x) + exp(-x))/3
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(n,j)*C(n-j,k)*J(n-j+1) where J(n) = A001045(n). - Paul Barry, May 19 2006
a(0)=1, a(n) = 5*a(n-1) - 2 if n is odd, and a(n) = 5*a(n) + 2 if n is even. - Vincenzo Librandi, Nov 18 2010

A083333 a(n) = 10a(n-2) - 16a(n-4) for n>=4, with a(0)=a(1)=1, a(2)=6, a(3)=10.

Original entry on oeis.org

1, 1, 6, 10, 44, 84, 344, 680, 2736, 5456, 21856, 43680, 174784, 349504, 1398144, 2796160, 11184896, 22369536, 89478656, 178956800, 715828224, 1431655424, 5726623744, 11453245440, 45812985856, 91625967616, 366503878656

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 24 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-16).

Cf. A016131, A082412 (bisections).

Cf. A001045, A016116, A083332.

I:=[1,1,6,10]; [n le 4 select I[n] else 10*Self(n-2) -16*Self(n-4): n in [1..41]]; // G. C. Greubel, Dec 27 2024

CoefficientList[Series[(1+x-4x^2)/(1-10x^2+16x^4), {x, 0, 30}], x]
LinearRecurrence[{0,10,0,-16},{1,1,6,10},30] (* Harvey P. Dale, Aug 04 2024 *)

def A083333(n): return 2^((n-1)/2)*( (n%2)*(2^(n+1) -1) + ((n+1)%2)*sqrt(2)*(2^(n+1) +1))/3
print([A083333(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

G.f.: (1+x-4*x^2)/(1-10*x^2+16*x^4).
Limit_{n -> oo} A083332(n)/a(n) = 3.
a(n) = A001045(n+1)*A016116(n). - R. J. Mathar, Jul 08 2009
From G. C. Greubel, Dec 27 2024: (Start)
a(n) = (1/3)*2^((n-3)/2)*( (1-(-1)^n)*(2^(n+1) - 1) + (1+(-1)^n)*sqrt(2)*(2^(n+1) + 1) ).
E.g.f.: (1/3)*(2*cosh(2*sqrt(2)*x) + cosh(sqrt(2)*x)) + (1/(3*sqrt(2)))*(2*sinh(2*sqrt(2)*x) - sinh(sqrt(2)*x)). (End)

cp's OEIS Frontend

A103333 Number of closed walks on the graph of the (7,4) Hamming code.

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A083076 Third row of number array A083075.

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A082413 a(n) = (2*9^n + 3^n)/3.

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A103334 Number of closed walks of length 2n at any of the nodes of degree 1 on the graph of the (7,4) Hamming code.

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A082414 a(n) = (2*10^n + 4^n)/3.

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A083217 a(n) = (2*5^n + (-1)^n)/3.

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A083333 a(n) = 10*a(n-2) - 16*a(n-4) for n>=4, with a(0)=a(1)=1, a(2)=6, a(3)=10.

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Magma

Mathematica

SageMath

Formula

A083333 a(n) = 10a(n-2) - 16a(n-4) for n>=4, with a(0)=a(1)=1, a(2)=6, a(3)=10.