A122950 -id:A122950 - cp's OEIS frontend

A026597 Expansion of (1+x)/(1-x-4*x^2).

1, 2, 6, 14, 38, 94, 246, 622, 1606, 4094, 10518, 26894, 68966, 176542, 452406, 1158574, 2968198, 7602494, 19475286, 49885262, 127786406, 327327454, 838473078, 2147782894, 5501675206, 14092806782, 36099507606, 92470734734

Offset: 0

Clark Kimberling

This sequence can generated by the following formula: a(n) = a(n-1) + 4*a(n-2) when n > 2; a[1] = 1, a[2] = 2. - Alex Vinokur (alexvn(AT)barak-online.net), Oct 21 2004
An elephant sequence, see A175654 and A175655. For the corner squares just one A[5] vector, with decimal value 325, leads to the sequence given above. For the central square this vector leads to a companion sequence that is 4 times this very same sequence with n >= -1. - Johannes W. Meijer, Aug 15 2010
Equals INVERTi transform of A180168. - Gary W. Adamson, Aug 14 2010
Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 2 X 2 cells and remove the cells that have one '1' in their modulo 3 coordinates. a(n) is the number of cells after n iterations. Cell configuration converges to a fractal with approximate dimension 1.357. - Peter Karpov, Apr 20 2017
Also, the number of walks of length n starting at vertex 1 in the graph with 4 vertices and edges {{0,1}, {0,2}, {0,3}, {1,2}, {2,3}}. - Sean A. Irvine, Jun 02 2025

Nathaniel Johnston, Table of n, a(n) for n = 0..500
Sean A. Irvine, Walks on Graphs.
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..8)
Index entries for linear recurrences with constant coefficients, signature (1,4).

Cf. A006131, A006138, A007482, A026581, A026584, A122950, A175654, A175655, A180168.

[n le 2 select n else Self(n-1) + 4*Self(n-2): n in [1..41]]; // G. C. Greubel, Dec 08 2021

LinearRecurrence[{1,4},{1,2},40] (* Harvey P. Dale, Nov 28 2011 *)

[(2*i)^n*( chebyshev_U(n, -i/4) - (i/2)*chebyshev_U(n-1, -i/4) ) for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1+x)/(1-x-4*x^2).
a(n) = T(n,0) + T(n,1) + ... + T(n,2*n), T given by A026584.
a(n) = Sum_{k=0..n} binomial(floor((2*n-k-1)/2), n-k)*2^k. - Paul Barry, Feb 11 2005
a(n) = A006131(n) + A006131(n-1), n >= 1. - R. J. Mathar, Oct 20 2006
a(n) = Sum_{k=0..n} binomial(floor((2*n-k)/2),n-k)*4^floor(k/2). - Paul Barry, Feb 02 2007
Inverse binomial transform of A007482: (1, 3, 11, 39, 139, 495, ...). - Gary W. Adamson, Dec 04 2007
a(n) = Sum_{k=0..n+1} A122950(n+1,k)*3^(n+1-k). - Philippe Deléham, Jan 04 2008
a(n) = (1/2 + 3*sqrt(17)/34)*(1/2 + sqrt(17)/2)^n + (1/2 - 3*sqrt(17)/34)*(1/2 - sqrt(17)/2)^n. - Antonio Alberto Olivares, Jun 07 2011
a(n) = (2*i)^n*( chebyshevU(n, -i/4) - (i/2)*chebyshevU(n-1, -i/4) ). - G. C. Greubel, Dec 08 2021
E.g.f.: exp(x/2)*(17*cosh(sqrt(17)*x/2) + 3*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, Jan 31 2023

Better name from Ralf Stephan, Jul 14 2013

A006138 a(n) = a(n-1) + 3*a(n-2).

Original entry on oeis.org

1, 2, 5, 11, 26, 59, 137, 314, 725, 1667, 3842, 8843, 20369, 46898, 108005, 248699, 572714, 1318811, 3036953, 6993386, 16104245, 37084403, 85397138, 196650347, 452841761, 1042792802, 2401318085, 5529696491, 12733650746, 29322740219, 67523692457, 155491913114

Offset: 0

N. J. A. Sloane

The binomial transform of a(n) is b(n) = A006190(n+1), which satisfies b(n) = 3*b(n-1) + b(n-2). - Paul Barry, May 21 2006
Partial sums of A105476. - Paul Barry, Feb 02 2007
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 69, 261, 321 and 324, lead to this sequence. For the central square these vectors lead to the companion sequence A105476 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
Equals the INVERTi transform of A063782: (1, 3, 10, 32, 104, ...). - Gary W. Adamson, Aug 14 2010
Pisano period lengths: 1, 3, 1, 6, 24, 3, 24, 6, 3, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24, ... - R. J. Mathar, Aug 10 2012
The sequence is the INVERT transform of A016116: (1, 1, 2, 2, 4, 4, 8, 8, ...). - Gary W. Adamson, Jul 17 2015

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..200
N. T. Gridgeman, A new look at Fibonacci generalization, Fib,. Quart., 11 (1973), 40-55.
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 (1,3).

Cf. A063782. - Gary W. Adamson, Aug 14 2010

Cf. A006130, A006190, A016116, A105476, A122950, A175654, A274977.

a:=[1,2];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; a; # G. C. Greubel, Nov 19 2019

[n le 2 select n else Self(n-1)+3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Sep 15 2016

A006138:=-(1+z)/(-1+z+3*z**2); # Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(1+z)/(1-z-3*z^2), {z,0,40}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
Table[(I*Sqrt[3])^(n-1)*(I*Sqrt[3]*ChebyshevU[n, 1/(2*I*Sqrt[3])] + ChebyshevU[n-1, 1/(2*I*Sqrt[3])]), {n, 0, 40}]//Simplify (* G. C. Greubel, Nov 19 2019 *)
LinearRecurrence[{1,3},{1,2},40] (* Harvey P. Dale, May 29 2025 *)

main(size)={my(v=vector(size),i);v[1]=1;v[2]=2;for(i=3,size,v[i]=v[i-1]+3*v[i-2]);return(v);} /* Anders Hellström, Jul 17 2015 */

def A006138_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x)/(1-x-3*x^2)).list()
A006138_list(40) # G. C. Greubel, Nov 19 2019

a(n) = Sum_{k=0..n+1} A122950(n+1,k)*2^(n+1-k). - Philippe Deléham, Jan 04 2008
G.f.: (1+x)/(1-x-3*x^2). - Paul Barry, May 21 2006
a(n) = Sum_{k=0..n} C(floor((2n-k)/2),n-k)*3^floor(k/2). - Paul Barry, Feb 02 2007
a(n) = A006130(n) + A006130(n-1). - R. J. Mathar, Jun 22 2011
a(n) = (i*sqrt(3))^(n-1)*(i*sqrt(3)*ChebyshevU(n, 1/(2*i*sqrt(3))) + ChebyshevU(n-1, 1/(2*i*sqrt(3)))), where i=sqrt(-1). - G. C. Greubel, Nov 19 2019

Typo in formula corrected by Johannes W. Meijer, Aug 15 2010

A133558 a(n) = a(n-1) + 9*a(n-2) for n >= 2, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 11, 29, 128, 389, 1541, 5042, 18911, 64289, 234488, 813089, 2923481, 10241282, 36552611, 128724149, 457697648, 1616214989, 5735493821, 20281428722, 71900873111, 254433731609, 901541589608, 3191445174089, 11305319480561

Offset: 0

Philippe Deléham, Jan 03 2008

Muniru A Asiru, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (1, 9).

a:=[1,2]: for n in [3..510] do a[n]:=a[n-1]+9*a[n-2]; od; a; # Muniru A Asiru, Aug 04 2018

a:=n->(<<0|1>,<9|1>>^n. <<1,2>>)[1,1]: seq(a(n),n=0..25); # Muniru A Asiru, Aug 04 2018

LinearRecurrence[{1,9},{1,2},30] (* or *) CoefficientList[Series[ (1+x)/(1-x-9x^2),{x,0,30}],x]  (* Harvey P. Dale, Apr 21 2011 *)

G.f.: (1+x)/(1-x-9*x^2).

a(n) = Sum_{k=0..n+1} A122950(n+1,k)*8^(n+1-k). - Philippe Deléham, Jan 08 2008

A133577 a(n) = a(n-1) + 10*a(n-2) for n >= 2, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 12, 32, 152, 472, 1992, 6712, 26632, 93752, 360072, 1297592, 4898312, 17874232, 66857352, 245599672, 914173192, 3370169912, 12511901832, 46213600952, 171332619272, 633468628792, 2346794821512, 8681481109432

Offset: 0

Philippe Deléham, Jan 03 2008

Index entries for linear recurrences with constant coefficients, signature (1,10).

LinearRecurrence[{1, 10}, {1, 2}, 24] (* or *)
CoefficientList[Series[(1 + x)/(1 - x - 10 x^2), {x, 0, 23}], x] (* Michael De Vlieger, Jul 20 2017 *)

G.f.: (1+x)/(1-x-10*x^2).

a(n) = Sum_{k=0..n+1} A122950(n+1,k)*9^(n+1-k). - Philippe Deléham, Jan 08 2008

A133592 a(n) = 2a(n-1)+6a(n-2) for n>=3, a(0)=1, a(1)=2, a(2)=8.

Original entry on oeis.org

1, 2, 8, 28, 104, 376, 1376, 5008, 18272, 66592, 242816, 885184, 3227264, 11765632, 42894848, 156383488, 570136064, 2078573056, 7577962496, 27627363328, 100722501632, 367209183232, 1338753376256, 4880761851904, 17794043961344

Offset: 0

Philippe Deléham, Dec 31 2007

Index entries for linear recurrences with constant coefficients, signature (2,6).

Cf. A083099, A122950.

A133592 := proc(n)
        option remember;
        if n <=1 then
                n+1;
        elif n = 2 then
                8;
        else
                2*procname(n-1)+6*procname(n-2) ;
        fi ;
end proc: # R. J. Mathar, Jul 15 2017

Join[{1}, LinearRecurrence[{2, 6}, {2, 8}, 24]] (* Jean-François Alcover, Jul 01 2023 *)

G.f.: (1-2*x^2)/(1-2*x-6*x^2).
a(n) = Sum_{k=0..n} A122950(n,k)*2^k .
a(n) = ((7+2*sqrt(7))/21)*(1+sqrt(7))^n+((7-2*sqrt(7))/21)*(1-sqrt(7))^n for n=>1. [Richard Choulet, Nov 19 2008]
a(n) = A083099(n+1) - 2*A083099(n-1). - R. J. Mathar, Jun 20 2015

a(16) corrected by R. J. Mathar, Jun 20 2015

A133407 a(n) = a(n-1) + 5*a(n-2) for n >= 2, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 7, 17, 52, 137, 397, 1082, 3067, 8477, 23812, 66197, 185257, 516242, 1442527, 4023737, 11236372, 31355057, 87536917, 244312202, 681996787, 1903557797, 5313541732, 14831330717, 41399039377, 115555692962, 322550889847, 900329354657, 2513083803892

Offset: 0

Philippe Deléham, Jan 03 2008

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 100 terms from Matt C. Anderson)
Index entries for linear recurrences with constant coefficients, signature (1,5).

Cf. A030195 (shifted binomial transform).

a:= n-> (<<0|1>, <5|1>>^n. <<1, 2>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 20 2025

LinearRecurrence[{1,5},{1,2},30] (* Harvey P. Dale, Jul 23 2013 *)

x='x+O('x^99); Vec((1+x)/(1-x-5*x^2)) \\ Altug Alkan, Aug 28 2017

G.f.: (1+x)/(1-x-5*x^2).
a(n) = Sum_{k=0..n+1} A122950(n+1,k)*4^(n+1-k). - Philippe Deléham, Jan 08 2008
a(n) = ((21 - 3*sqrt(21))/42)*(1/2 - (1/2)*sqrt(21))^n + ((21 + 3*sqrt(21))/42)*(1/2 + (1/2)*sqrt(21))^n. - Richard Choulet, Nov 20 2008

A133467 a(n) = a(n-1) + 6*a(n-2) for n >= 2, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 8, 20, 68, 188, 596, 1724, 5300, 15644, 47444, 141308, 425972, 1273820, 3829652, 11472572, 34450484, 103285916, 309988820, 929704316, 2789637236, 8367863132, 25105686548, 75312865340, 225946984628, 677824176668, 2033506084436, 6100451144444, 18301487651060

Offset: 0

Philippe Deléham, Jan 03 2008

From Sean A. Irvine, Jun 07 2025: (Start)
The number of walks of length n starting at vertex 0 (or, by symmetry, vertex 2 or 3) in the graph K_{1,1,3}:
    1---2
   /|\ /
  0 | X
   \|/ \
    4---3. (End)

			a(2)=8 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-4-0, 0-4-1, 0-4-2, 0-4-3. - _Sean A. Irvine_, Jun 07 2025

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (1,6).

Digits := 50:
for n from 0 to 40 do round(.8*3^n+.2*(-2)^n) end do;
# Matt C. Anderson, Jul 18 2017

LinearRecurrence[{1,6},{1,2},30] (* Harvey P. Dale, Apr 05 2014 *)

from sage.combinat.sloane_functions import recur_gen2b; it = recur_gen2b(1,2,1,6, lambda n: 0); [next(it) for i in range(0,29)] # Zerinvary Lajos, Jul 03 2008

G.f.: (1+x)/((1+2*x)*(1-3*x)).
a(n) = Sum_{k=0..n+1} A122950(n+1,k)*5^(n+1-k). - Philippe Deléham, Jan 08 2008
a(n) = (4 * 3^n + (-2)^n) / 5. - Werner Schulte, Dec 18 2024

A133469 a(n) = a(n-1) + 7*a(n-2) for n >= 2, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 9, 23, 86, 247, 849, 2578, 8521, 26567, 86214, 272183, 875681, 2780962, 8910729, 28377463, 90752566, 289394807, 924662769, 2950426418, 9423065801, 30076050727, 96037511334, 306569866423, 978832445761, 3124821510722

Offset: 0

Philippe Deléham, Jan 03 2008

Robert Israel, Table of n, a(n) for n = 0..1981
Index entries for linear recurrences with constant coefficients, signature (1, 7).

f:= gfun:-rectoproc({a(n) = a(n-1) + 7*a(n-2), a(0)=1, a(1)=2}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Jul 23 2017

LinearRecurrence[{1,7},{1,2},30] (* Harvey P. Dale, Dec 09 2013 *)

x='x+O('x^99); Vec((1+x)/(1-x-7*x^2)) \\ Altug Alkan, Jul 23 2017

G.f.: (1+x)/(1-x-7*x^2) .
a(n) = Sum_{k=0..n+1} A122950(n+1,k)*6^(n+1-k). - Philippe Deléham, Jan 08 2008
a(n) = ((29 + 3*sqrt(29))/58)*(0.5 + 0.5*sqrt(29))^n + ((29 - 3*sqrt(29))/58)*(0.5 - 0.5*sqrt(29))^n. - Richard Choulet, Nov 20 2008

A133479 a(n) = a(n-1) + 8*a(n-2) for n >= 2, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 10, 26, 106, 314, 1162, 3674, 12970, 42362, 146122, 485018, 1653994, 5534138, 18766090, 63039194, 213167914, 717481466, 2422824778, 8162676506, 27545274730, 92846686778, 313208884618, 1055982378842, 3561653455786

Offset: 0

Philippe Deléham, Jan 03 2008

Index entries for linear recurrences with constant coefficients, signature (1,8).

G.f.: (1+x)/(1-x-8*x^2).
a(n) = Sum_{k=0..n+1} A122950(n+1,k)*7^(n+1-k). - Philippe Deléham, Jan 08 2008
a(n) = ((11+sqrt(33))/22)*(1/2 + (1/2)*sqrt(33))^n + ((11-sqrt(33))/22)*(1/2 - (1/2)*sqrt(33))^n. - Richard Choulet, Nov 20 2008

A133646 a(n)=5a(n-1)+30a(n-2) for n>=3, a(0)=1, a(1)=5, a(2)=50 .

Original entry on oeis.org

1, 5, 50, 400, 3500, 29500, 252500, 2147500, 18312500, 155987500, 1329312500, 11326187500, 96510312500, 822337187500, 7006995312500, 59705092187500, 508735320312500, 4334829367187500, 36936206445312500, 314725913242187500

Offset: 0

Philippe Deléham, Dec 31 2007

Index entries for linear recurrences with constant coefficients, signature (5, 30).

G.f.: (1-5*x^2)/(1-5*x-30*x^2) . a(n) = Sum_{k, 0<=k<=n}A122950(n,k)*5^k .

cp's OEIS Frontend

A026597 Expansion of (1+x)/(1-x-4*x^2).

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A006138 a(n) = a(n-1) + 3*a(n-2).

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A133558 a(n) = a(n-1) + 9*a(n-2) for n >= 2, a(0)=1, a(1)=2.

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A133577 a(n) = a(n-1) + 10*a(n-2) for n >= 2, a(0)=1, a(1)=2.

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A133592 a(n) = 2*a(n-1)+6*a(n-2) for n>=3, a(0)=1, a(1)=2, a(2)=8.

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A133407 a(n) = a(n-1) + 5*a(n-2) for n >= 2, a(0)=1, a(1)=2.

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A133467 a(n) = a(n-1) + 6*a(n-2) for n >= 2, a(0)=1, a(1)=2.

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A133592 a(n) = 2a(n-1)+6a(n-2) for n>=3, a(0)=1, a(1)=2, a(2)=8.

A133646 a(n)=5a(n-1)+30a(n-2) for n>=3, a(0)=1, a(1)=5, a(2)=50 .