A108350 -id:A108350 - cp's OEIS frontend

A108351 Diagonal sums of symmetric triangle A108350.

1, 1, 2, 3, 5, 8, 14, 24, 43, 77, 140, 255, 467, 856, 1572, 2888, 5309, 9761, 17950, 33011, 60713, 111664, 205378, 377744, 694775, 1277885, 2350392, 4323039, 7951303, 14624720, 26899048, 49475056, 90998809, 167372897, 307846746

Offset: 0

Paul Barry, May 31 2005

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1,0,1).

LinearRecurrence[{2,1,-2,-1,0,1},{1,1,2,3,5,8},40] (* Harvey P. Dale, Nov 24 2017 *)

G.f.: (1-x-x^2)/((1+x)(1-x)^2(1-x-x^2-x^3)); a(n)=2a(n-1)+a(n-2)-2a(n-3)-a(n-4)+a(n-6); a(n)=sum{k-0..floor(n/2), sum{j=0..n-2k, C(k, j)C(n-k-j, k)*(1+(-1)^j)/2}}.

A108363 Triangle read by rows, generated from A108350.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 16, 6, 1, 1, 7, 21, 35, 38, 24, 7, 1

Offset: 0

Gary W. Adamson, Jun 01 2005

			Binomial transform of row 4 in the form [1, 4, 7, 4, 1, 0, 0, 0...] of A108350 given first row (a "1") has 0 offset = column 4 of A108363: [1, 5, 16, 38, 76...].

Cf. A108350.

Columns of A108363 are binomial transforms of A108350 rows.

A100131 a(n) = Sum_{k=0..floor(n/4)} binomial(n-2k, 2k)*2^(n-4k).

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 88, 208, 497, 1194, 2876, 6936, 16737, 40398, 97520, 235424, 568353, 1372114, 3312564, 7997224, 19306993, 46611190, 112529352, 271669872, 655869073, 1583407994, 3822685036, 9228778040, 22280241089, 53789260190, 129858761440, 313506783040

Offset: 0

Paul Barry, Nov 06 2004

Binomial transform of 1,1,1,1,2,2,4,4,8,8,... (g.f.: (1-x)(1+x)^2/(1-2x^2)).

Row sums of number triangle A108350. - Paul Barry, May 31 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Cf. A098576, A100132, A100133.

I:=[1, 2, 4, 8]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2)+Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 25 2012

with(combinat): seq((n+1+fibonacci(n+1, 2))/2, n=0..30); # Zerinvary Lajos, Jun 02 2008

CoefficientList[Series[(1-2x)/((1-2x)^2-x^4),{x,0,40}],x]  (* Harvey P. Dale, Mar 22 2011 *)
LinearRecurrence[{4,-4,0,1},{1,2,4,8},40] (* Vincenzo Librandi, Jun 25 2012 *)

G.f.: (1-2x)/((1-2x)^2-x^4) = (1-2x)/((1-x)^2(1-2x-x^2));
a(n) = 4a(n-1) - 4a(n-2) + a(n-4);
a(n) = ((sqrt(2)+1)^(n+1) + (sqrt(2)-1)^(n+1)(-1)^n)/(4*sqrt(2)) + (n+1)/2;
a(n) = Sum_{k=0..n} (1-k)*A000129(n-k+1).
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} binomial(k, j)*binomial(n-j, k)*((j+1) mod 2). - Paul Barry, May 31 2005
a(n) = (1/2)*(Pell(n+1) + n + 1), where Pell(n) = A000129(n). - Ralf Stephan, May 15 2007 [corrected by Jon E. Schoenfield, Feb 19 2019]

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A108351 Diagonal sums of symmetric triangle A108350.

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A108363 Triangle read by rows, generated from A108350.

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A100131 a(n) = Sum_{k=0..floor(n/4)} binomial(n-2k, 2k)*2^(n-4k).

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