A161731 -id:A161731 - cp's OEIS frontend

A048776 First partial sums of A048739; second partial sums of A000129.

1, 4, 12, 32, 81, 200, 488, 1184, 2865, 6924, 16724, 40384, 97505, 235408, 568336, 1372096, 3312545, 7997204, 19306972, 46611168, 112529329, 271669848, 655869048, 1583407968, 3822685009, 9228778012, 22280241060, 53789260160, 129858761409, 313506783008

Offset: 0

Barry E. Williams

Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Cf. A001333, A000129, A048739.

with(combinat):seq((fibonacci(n+3, 2)-n-3)/2, n=0..25); # Zerinvary Lajos, Jun 02 2008

a=b=0;Table[c=2*b+a+n;a=b;b=c,{n,1,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011*)
LinearRecurrence[{4,-4,0,1},{1,4,12,32},30] (* Harvey P. Dale, Aug 27 2014 *)

a(n) = 2*a(n-1) + a(n-2) + n + 1; a(0)=1, a(1)=4.
a(n) = (((7/2 + (5/2)*sqrt(2))*(1+sqrt(2))^n - (7/2 - (5/2)*sqrt(2))*(1-sqrt(2))^n)/2*sqrt(2)) - (n+3)/2.
a(n) = (A000129(n+3) - (n+3))/2 = Sum_{j} A047662(n-j+1, j+1). - Henry Bottomley, Jul 09 2001
From R. J. Mathar, Feb 06 2010: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).
G.f.: -1/((x^2+2*x-1) * (x-1)^2). (End)
Define an array with m(n,1)=1 and m(1,k) = k*(k+1)/2 for n=1,2,3,...  The interior terms are m(n,k) = m(n,k-1) + m(n-1,k-1) + m(n-1,k). The sum of the terms in each antidiagonal=a(n). - J. M. Bergot, Dec 01 2012 [This is A154948 without the first column. The diagonal is m(n,n) = A161731(n-1). R. J. Mathar, Dec 06 2012]
E.g.f.: exp(x)*(10*cosh(sqrt(2)*x) + 7*sqrt(2)*sinh(sqrt(2)*x) - 2*(3 + x))/4. - Stefano Spezia, May 13 2023

More terms from Harvey P. Dale, Aug 27 2014

A081568 Third binomial transform of Fibonacci(n+1).

Original entry on oeis.org

1, 4, 17, 75, 338, 1541, 7069, 32532, 149965, 691903, 3193706, 14745009, 68084297, 314394980, 1451837593, 6704518371, 30961415074, 142980203437, 660285858245, 3049218769908, 14081386948661, 65028302171639, 300302858766202, 1386808687475385, 6404329365899473

Offset: 0

Paul Barry, Mar 22 2003

Binomial transform of A081567.
Case k=3 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2) for n >= 2, with a(0) = 1 and a(1) = k + 1.
a(n) = 4^n*a(n;1/4) = Sum_{k=0..n} binomial(n,k) * (-1)^k * F(k-1) * 4^(n-k), which also implies Deléham's formula given below and where a(n;d), n = 0, 1, ..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al. papers). - Roman Witula, Jul 12 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Edyta Hetmaniok, Bożena Piątek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Mathematics, 15(1) (2017), 477-485.
Roman Witula and Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009), 310-329, MR2555042.
Index entries for linear recurrences with constant coefficients, signature (7,-11).

Cf. A000045, A161731 (INVERT transform), A007582 (INVERTi transform), A028387, A081567, A081569 (binomial transform), A094441, A099453.

a:=[1,4];; for n in [3..30] do a[n]:=7*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, Aug 12 2019

I:=[1, 4]; [n le 2 select I[n] else 7*Self(n-1)-11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013

seq(coeff(series((1-3*x)/(1-7*x+11*x^2), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Aug 12 2019

CoefficientList[Series[(1-3x)/(1 -7x +11x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)
LinearRecurrence[{7,-11},{1,4},30] (* Harvey P. Dale, Feb 01 2015 *)

Vec((1-3*x)/(1-7*x+11*x^2) + O(x^30)) \\ Altug Alkan, Dec 10 2015

def A081568_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)/(1-7*x+11*x^2)).list()
A081568_list(30) # G. C. Greubel, Aug 12 2019

a(n) = 7*a(n-1) - 11*a(n-2) for n >= 2, with a(0) = 1 and a(1) = 4.
a(n) = (1/2 - sqrt(5)/10)*(7/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)*(sqrt(5)/2 + 7/2)^n = A099453(n) - 3*A099453(n-1).
G.f.: (1 - 3*x)/(1 - 7*x + 11*x^2).
a(n) = Sum_{k=0..n} A094441(n,k)*3^k. - Philippe Deléham, Dec 14 2009
G.f.: Q(0,u)/x - 1/x, where u = x/(1 - 3*x), Q(k,u) = 1 + u^2 + (k+2)*u - u*(k + 1 + u)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
E.g.f.: exp(7*x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Jun 03 2024

A164095 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 6.

Original entry on oeis.org

5, 6, 10, 12, 20, 24, 40, 48, 80, 96, 160, 192, 320, 384, 640, 768, 1280, 1536, 2560, 3072, 5120, 6144, 10240, 12288, 20480, 24576, 40960, 49152, 81920, 98304, 163840, 196608, 327680, 393216, 655360, 786432, 1310720, 1572864, 2621440, 3145728

Offset: 1

Klaus Brockhaus, Aug 10 2009

nonn

Interleaving of A020714 and A007283 without initial term 3.
Partial sums are in A164096.
Binomial transform is A048655 without initial 1, second binomial transform is A161941 without initial 2, third binomial transform is A164037, fourth binomial transform is A161731 without initial 1, fifth binomial transform is A164038, sixth binomial transform is A164110.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0, 2).

Cf. A020714 (5*2^n), A007283 (3*2^n), A164096, A048655, A161941, A164037, A161731, A164038, A164110, A070876.

[ n le 2 select n+4 else 2*Self(n-2): n in [1..40] ];

LinearRecurrence[{0,2},{5,6},50] (* or *) With[{nn=20},Riffle[NestList[ 2#&,5,nn],NestList[2#&,6,nn]]] (* Harvey P. Dale, Aug 15 2020 *)

a(n) = A070876(n)/3.
a(n) = (4-(-1)^n)*2^(1/4*(2*n-1+(-1)^n)).
G.f.: x*(5+6*x)/(1-2*x^2).

A164038 Expansion of (5-19x)/(1-10x+23*x^2).

Original entry on oeis.org

5, 31, 195, 1237, 7885, 50399, 322635, 2067173, 13251125, 84966271, 544886835, 3494644117, 22414043965, 143763624959, 922113238395, 5914569009893, 37937085615845, 243335768930911, 1560804720144675, 10011324516035797

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

Binomial transform of A161731 without initial 1. Fifth binomial transform of A164095. Inverse binomial transform of A164110.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (10,-23).

Cf. A161731, A164095, A164110.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(5+r)^n+(5-3*r)*(5-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 10 2009

LinearRecurrence[{10,-23},{5,31},50] (* or *) CoefficientList[Series[(5 - 19*x)/(1 - 10*x + 23*x^2), {x,0,50}], x] (* G. C. Greubel, Sep 08 2017 *)

Vec((5-19*x)/(1-10*x+23*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 10*a(n-1) - 23*a(n-2) for n > 1; a(0) = 5, a(1) = 31.
G.f.: (5-19*x)/(1-10*x+23*x^2).
a(n) = ((5+3*sqrt(2))*(5+sqrt(2))^n + (5-3*sqrt(2))*(5-sqrt(2))^n)/2.
E.g.f: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(5*x). - G. C. Greubel, Sep 08 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 10 2009

A164037 Expansion of (5-9x)/(1-6x+7*x^2).

Original entry on oeis.org

5, 21, 91, 399, 1757, 7749, 34195, 150927, 666197, 2940693, 12980779, 57299823, 252933485, 1116502149, 4928478499, 21755355951, 96032786213, 423909225621, 1871225850235, 8259990522063, 36461362180733, 160948239429957

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

Binomial transform of A161941 without initial 2. Third binomial transform of A164095. Inverse binomial transform of A161731 without initial 1.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (6,-7).

Cf. A161941, A164095 (5, 6, 10, 12, 20, 24, ...), A161731.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(3+r)^n+(5-3*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 10 2009

CoefficientList[Series[(5-9x)/(1-6x+7x^2),{x,0,30}],x] (* or *) LinearRecurrence[{6,-7},{5,21},30] (* Harvey P. Dale, Apr 27 2017 *)

Vec((5-9*x)/(1-6*x+7*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 6*a(n-1)-7*a(n-2) for n > 1; a(0) = 5, a(1) = 21.
G.f.: (5-9*x)/(1-6*x+7*x^2).
a(n) = ((5+3*sqrt(2))*(3+sqrt(2))^n+(5-3*sqrt(2))*(3-sqrt(2))^n)/2.
E.g.f.: (5*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))*exp(3*x). - G. C. Greubel, Sep 08 2017

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 10 2009

cp's OEIS Frontend

A048776 First partial sums of A048739; second partial sums of A000129.

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A081568 Third binomial transform of Fibonacci(n+1).

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A164095 a(n) = 2*a(n-2) for n > 2; a(1) = 5, a(2) = 6.

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A164038 Expansion of (5-19*x)/(1-10*x+23*x^2).

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A164037 Expansion of (5-9*x)/(1-6*x+7*x^2).

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A164038 Expansion of (5-19x)/(1-10x+23*x^2).

A164037 Expansion of (5-9x)/(1-6x+7*x^2).