A097162 -id:A097162 - cp's OEIS frontend

A057651 a(n) = (3*5^n - 1)/2.

1, 7, 37, 187, 937, 4687, 23437, 117187, 585937, 2929687, 14648437, 73242187, 366210937, 1831054687, 9155273437, 45776367187, 228881835937, 1144409179687, 5722045898437, 28610229492187, 143051147460937, 715255737304687, 3576278686523437, 17881393432617187, 89406967163085937

Offset: 0

N. J. A. Sloane, Oct 13 2000

Sum of n-th row of triangle of powers of 5: 1; 1 5 1; 1 5 25 5 1 ; 1 5 25 125 25 5 1; ... - Philippe Deléham, Feb 23 2014

			a(0) = 1;
a(1) = 1 + 5 + 1 = 7;
a(2) = 1 + 5 + 25 + 5 + 1 = 37;
a(3) = 1 + 5 + 25 + 125 + 25 + 5 + 1 = 187; etc. - _Philippe Deléham_, Feb 23 2014
G.f. = 1 + 7*x + 37*x^2 + 187*x^3 + 937*x^4 + 4687*x^5 + 23437*x^6 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A024049, A081655, A097162, A198762.

Cf. A020989, A061801, A112468, A112739.

[(3*5^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Oct 30 2011

G.f=(1+x)/(1-5*x)/(1-x): gser:=series(g, x=0, 43): seq(coeff(gser, x, n), n=0..30); # Zerinvary Lajos, Jan 11 2009

Table[(3*5^n-1)/2,{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)

a(n)=3*5^n\2 \\ Charles R Greathouse IV, Dec 22 2011

G.f.: (1+x)/(1 - 6*x + 5*x^2).
a(0)=1, a(n) = 5*a(n-1) + 2; a(n) = a(n-1) + 6*(5^(n-1)). - Amarnath Murthy, May 27 2001
a(n) = 6*a(n-1) - 5*a(n-2), n > 1. - Vincenzo Librandi, Oct 30 2011
a(n) = Sum_{k=0..n} A112468(n,k)*6^k. - Philippe Deléham, Feb 23 2014
From Elmo R. Oliveira, Mar 29 2025: (Start)
E.g.f.: exp(x)*(3*exp(4*x) - 1)/2.
a(n) = A097162(2*n) = A198762(n)/2. (End)

A097165 Expansion of (1-3x)/((1-x)(1-4x)(1-5x)).

Original entry on oeis.org

1, 7, 41, 227, 1221, 6447, 33601, 173467, 889181, 4533287, 23015961, 116477907, 587981941, 2962279327, 14900875121, 74862289547, 375743103501, 1884442140567, 9445117195081, 47317211944387, 236952563597861

Offset: 0

Paul Barry, Jul 30 2004

Partial sums of A085351. Convolution of A034478 and 4^n. Convolution of A047849 and 5^n. a(n)=A097162(2n+1)/3. Third binomial transform of A097164.

Index entries for linear recurrences with constant coefficients, signature (10,-29,20).

CoefficientList[Series[(1-3x)/((1-x)(1-4x)(1-5x)),{x,0,30}],x] (* or *) LinearRecurrence[{10,-29,20},{1,7,41},30] (* Harvey P. Dale, Jan 24 2012 *)

a(n)=5*5^n/2-4*4^n/3-1/6; a(n)=sum{k=0..n, (5^k+1)4^(n-k)/2}; a(n)=sum{k=0..n, (4^k+2)5^(n-k)/3}; a(n)=10a(n-1)-29a(n-2)+20a(n-3).

A097169 a(n) = Sum_{k=0..n} C(floor((n+1)/2),floor((k+1)/2)) * 3^k.

Original entry on oeis.org

1, 4, 13, 52, 133, 604, 1333, 6772, 13333, 74284, 133333, 801892, 1333333, 8550364, 13333333, 90286612, 133333333, 945912844, 1333333333, 9846548932, 13333333333, 101952273724, 133333333333, 1050903796852, 1333333333333

Offset: 0

Paul Barry, Jul 30 2004

a(n) = (4/3){1,10,10,100,100,1000...} -9{0,1,0,9,0,81...} -(1/3){1,1,1,1,1,1...} .
a(2n) = A097166(n).
a(2n+1)/4 = A097168(n).

Index entries for linear recurrences with constant coefficients, signature (1,19,-19,-90,90)

Cf. A097162, A075427.

LinearRecurrence[{1,19,-19,-90,90},{1,4,13,52,133},30] (* Harvey P. Dale, Dec 15 2017 *)

G.f.: (1+3x-10x^2-18x^3)/((1-x)*(1-9x^2)*(1-10x^2)).
a(n) = 2((1-sqrt(10))(-sqrt(10))^n+(1+sqrt(10))(sqrt(10))^n)/3+3((-3)^n-3^n)/2-1/3.
a(n) = a(n-1) +19a(n-2) -19a(n-3) -90a(n-4) +90a(n-5).

cp's OEIS Frontend

A057651 a(n) = (3*5^n - 1)/2.

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A097165 Expansion of (1-3x)/((1-x)(1-4x)(1-5x)).

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A097169 a(n) = Sum_{k=0..n} C(floor((n+1)/2),floor((k+1)/2)) * 3^k.

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