A062112 -id:A062112 - cp's OEIS frontend

A007068 a(n) = a(n-1) + (3+(-1)^n)*a(n-2)/2.

1, 3, 4, 10, 14, 34, 48, 116, 164, 396, 560, 1352, 1912, 4616, 6528, 15760, 22288, 53808, 76096, 183712, 259808, 627232, 887040, 2141504, 3028544, 7311552, 10340096, 24963200, 35303296, 85229696, 120532992, 290992384, 411525376, 993510144, 1405035520, 3392055808

Offset: 1

N. J. A. Sloane, Mira Bernstein

First row of spectral array W(sqrt 2).

Row sums of the square of the matrix with general term binomial(floor(n/2),n-k). - Paul Barry, Feb 14 2005

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Aviezri S. Fraenkel and Clark Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discr. Math. 126 (1-3) (1994) 137-149.
Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

Cf. A062112, A001882, A007070, A007052, A000034.

a007068 n = a007068_list !! (n-1)
a007068_list = 1 : 3 : zipWith (+)
   (tail a007068_list) (zipWith (*) a000034_list a007068_list)
-- Reinhard Zumkeller, Jan 21 2012

RecurrenceTable[{a[1]==1,a[2]==3,a[n]==a[n-1]+(3+(-1)^n) a[n-2]/2},a[n],{n,40}] (* Harvey P. Dale, Nov 12 2012 *)

a(2n+1) = a(2n)+a(2n-1); a(2n) = a(2n-1)+2*a(2n-2); same recurrence (mod parity) as A001882. - Len Smiley, Feb 05 2001
a(n) = Sum_{k=0..n} Sum_{j=0..n} C(floor(n/2), n-j)*C(floor(j/2), j-k). - Paul Barry, Feb 14 2005
a(n) = 4*a(n-2)-2*a(n-4). G.f.: -x*(1+x)*(2*x^2-2*x-1)/(1-4*x^2+2*x^4). a(2n+1)=A007070(n). a(2n)=A007052(n). [R. J. Mathar, Aug 17 2009]
a(n) = a(n-1) + a(n-2) * A000034(n-1). [Reinhard Zumkeller, Jan 21 2012]

Better description and more terms from Olivier Gérard, Jun 05 2001

A062113 a(0)=1; a(1)=2; a(n) = a(n-1) + a(n-2)*(3 - (-1)^n)/2.

Original entry on oeis.org

1, 2, 3, 7, 10, 24, 34, 82, 116, 280, 396, 956, 1352, 3264, 4616, 11144, 15760, 38048, 53808, 129904, 183712, 443520, 627232, 1514272, 2141504, 5170048, 7311552, 17651648, 24963200, 60266496, 85229696, 205762688, 290992384, 702517760

Offset: 0

Olivier Gérard, Jun 05 2001

A bistable recurrence.

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

Cf. A007068, A062112.

a062113 n = a062113_list !! n
a062113_list = 1 : 2 : zipWith (+)
   (tail a062113_list) (zipWith (*) a000034_list a062113_list)
-- Reinhard Zumkeller, Jan 21 2012

I:=[1,2,3,7]; [n le 4 select I[n] else 4*Self(n-2) - 2*Self(n-4): n in [1..40]]; // G. C. Greubel, Oct 16 2018

LinearRecurrence[{0,4,0,-2}, {1,2,3,7}, 40] (* G. C. Greubel, Oct 16 2018 *)

{ for (n=0, 200, if (n>1, a=a1 + a2*(3 - (-1)^n)/2; a2=a1; a1=a, if (n==0, a=a2=1, a=a1=2)); write("b062113.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 01 2009

x='x+O('x^40); Vec((1+2*x-x^2-x^3)/(1-4*x^2+2*x^4)) \\ G. C. Greubel, Oct 16 2018

a(n) = a(n-1) + a(n-2) * A000034(n). - Reinhard Zumkeller, Jan 21 2012
From Colin Barker, Apr 20 2012: (Start)
a(n) = 4*a(n-2) - 2*a(n-4).
G.f.: (1+2*x-x^2-x^3)/(1-4*x^2+2*x^4). (End)

A384598 Expansion of (1-3x^2) / (1-x-4x^2+2x^3+2x^4).

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 38, 86, 186, 418, 914, 2042, 4490, 9994, 22042, 48954, 108154, 239898, 530522, 1175898, 2601882, 5764634, 12759322, 28262298, 62566554, 138567834, 306790810, 679404442, 1504298906, 3331199386, 7376004506, 16333395354, 36166416794

Offset: 0

Sean A. Irvine, Jun 04 2025

Number of walks of length n on the following graph starting at vertex 0:
           3
          /|
     0-1-2 |
          \|
           4.
Also, for n>=1, the number of walks of length n-1 starting from vertex 1 in the same graph.

			a(3)=4 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-2-4.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-2).

Cf. A384599 (vertex 2), A384600 (vertex 3), A062112 (missing edge {3,4}), A382683 (missing edge {0,1}).

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-2|-2|4|1>>^n. <<1,1,2,4>>)[1,1]:
seq(a(n), n=0..32);  # Alois P. Heinz, Jun 04 2025

CoefficientList[Series[(1 - 3*x^2)/(1 - x - 4*x^2 + 2*x^3 + 2*x^4), {x, 0, 32}], x] (* Michael De Vlieger, Jun 04 2025 *)

A384599 Expansion of (1+3x+2x^2) / (1-4x^2-2x^3).

Original entry on oeis.org

1, 3, 6, 14, 30, 68, 148, 332, 728, 1624, 3576, 7952, 17552, 38960, 86112, 190944, 422368, 936000, 2071360, 4588736, 10157440, 22497664, 49807232, 110305536, 244224256, 540836608, 1197508096, 2651794944, 5871705600, 13002195968, 28790412288, 63752195072

Offset: 0

Sean A. Irvine, Jun 04 2025

Number of walks of length n on the following graph starting at vertex 2:
           3
          /|
     0-1-2 |
          \|
           4.

			a(2)=6 because we have the walks 2-1-0, 2-1-2, 2-3-2, 2-3-4, 2-4-2, 2-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (0,4,2).

Cf. A384598 (vertices 0 and 1), A384600 (vertex 3), A062112 (missing edge {3,4}), A382683 (missing edge {0,1}).

a:= n-> (<<0|1|0>, <0|0|1>, <2|4|0>>^n. <<1,3,6>>)[1,1]:
seq(a(n), n=0..31);  # Alois P. Heinz, Jun 04 2025

CoefficientList[Series[(1 + 3*x + 2*x^2)/(1 - 4*x^2 - 2*x^3), {x, 0, 31}], x] (* Michael De Vlieger, Jun 04 2025 *)

A384600 Expansion of (1+x-x^2) / (1-x-4x^2+2x^3+2*x^4).

Original entry on oeis.org

1, 2, 5, 11, 25, 55, 123, 271, 603, 1331, 2955, 6531, 14483, 32035, 70995, 157107, 348051, 770419, 1706419, 3777779, 8366515, 18523955, 41021619, 90828851, 201134387, 445358643, 986195251, 2183703347, 4835498291, 10707203891, 23709399859, 52499812147

Offset: 0

Sean A. Irvine, Jun 04 2025

Number of walks of length n on the following graph starting at vertex 3:
           3
          /|
     0-1-2 |
          \|
           4.

			a(2)=5 because we have the walk 3-2-1, 3-2-3, 3-2-4, 3-4-2, 3-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-2).

Cf. A384598 (vertices 0 and 1), A384599 (vertex 2), A062112 (missing edge {3,4}), A382683 (missing edge {0,1}).

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-2|-2|4|1>>^n. <<1,2,5,11>>)[1,1]:
seq(a(n), n=0..31);  # Alois P. Heinz, Jun 04 2025

CoefficientList[Series[(1 + x - x^2)/(1 - x - 4*x^2 + 2*x^3 + 2*x^4), {x, 0, 31}], x] (* Michael De Vlieger, Jun 04 2025 *)

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2677, 73327, 3578339, 329652351

Offset: 1

Thomas Scheuerle, Aug 19 2022

This sequence can be calculated by a recursive algorithm:
Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
There exist some transformation pairs of infinite sequences in the database:
  A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
  A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
  A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
  A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
  A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
  A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
  A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
  A279312 <--> A------; A031923 <--> A------.
These transformation pairs are conjectured:
  A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
  A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
  A349079 <--> A------.
("A------" means not yet in the database.)
Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.

			a(4) = 17. The 17 transformation pairs of length 4 are:
  {1, 2, 3, 4}  = S({1, 2, 6, 24}).
  {1, 2, 3, 5}  = S({1, 2, 6, 15}).
  {1, 2, 3, 6}  = S({1, 2, 6, 12}).
  {1, 2, 3, 9}  = S({1, 2, 6, 9}).
  {1, 2, 3, 12} = S({1, 2, 6, 8}).
  {1, 2, 3, 21} = S({1, 2, 6, 7}).
  {1, 2, 4, 5}  = S({1, 2, 4, 20}).
  {1, 2, 4, 6}  = S({1, 2, 4, 12}).
  {1, 2, 4, 8}  = S({1, 2, 4, 8}).
  {1, 2, 4, 12} = S({1, 2, 4, 6}).
  {1, 2, 4, 20} = S({1, 2, 4, 5}).
  {1, 2, 6, 7}  = S({1, 2, 3, 21}).
  {1, 2, 6, 8}  = S({1, 2, 3, 12}).
  {1, 2, 6, 9}  = S({1, 2, 3, 9}).
  {1, 2, 6, 12} = S({1, 2, 3, 6}).
  {1, 2, 6, 15} = S({1, 2, 3, 5}).
  {1, 2, 6, 24} = S({1, 2, 3, 4}).
b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
a(3) = A000005(b(2)*b(2)) = 3.
The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.

Cf. A000005.
Cf. A000027, A000045, A000079, A000142, A001405.
Cf. A001654, A004277, A006501, A008233, A019442.
Cf. A010551, A019464, A026549, A031923, A038754.
Cf. A057895, A058295, A062112, A066332, A100071.
Cf. A079352, A082458, A087811, A093968, A098011.
Cf. A098558, A100538, A111286, A166447, A137326.
Cf. A138278, A171647, A205825, A208147, A264557.
Cf. A264635, A308546, A329227, A336496, A349079.
Cf. A349080, A359039.

cp's OEIS Frontend

A007068 a(n) = a(n-1) + (3+(-1)^n)*a(n-2)/2.

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A062113 a(0)=1; a(1)=2; a(n) = a(n-1) + a(n-2)*(3 - (-1)^n)/2.

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A384598 Expansion of (1-3*x^2) / (1-x-4*x^2+2*x^3+2*x^4).

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A384599 Expansion of (1+3*x+2*x^2) / (1-4*x^2-2*x^3).

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A384600 Expansion of (1+x-x^2) / (1-x-4*x^2+2*x^3+2*x^4).

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A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

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A384598 Expansion of (1-3x^2) / (1-x-4x^2+2x^3+2x^4).

A384599 Expansion of (1+3x+2x^2) / (1-4x^2-2x^3).

A384600 Expansion of (1+x-x^2) / (1-x-4x^2+2x^3+2*x^4).