A131326 -id:A131326 - cp's OEIS frontend

A131325 Triangle |3*|A049310(n,k)| - 2| read by rows, 0 <= k <= n.

1, 2, 1, 1, 2, 1, 2, 4, 2, 1, 1, 2, 7, 2, 1, 2, 7, 2, 10, 2, 1, 1, 2, 16, 2, 13, 2, 1, 2, 10, 2, 28, 2, 16, 2, 1, 1, 2, 28, 2, 43, 2, 19, 2, 1, 2, 13, 2, 58, 2, 61, 2, 22, 2, 1, 1, 2, 43, 2, 103, 2, 82, 2, 25, 2, 1, 2, 16, 2, 103, 2, 166, 2, 106, 2, 28, 2, 1, 1, 2, 61, 2, 208, 2, 250, 2

Offset: 0

Gary W. Adamson, Jun 28 2007

			First few rows of the triangle:
  1;
  2,  1;
  1,  2,  1;
  2,  4,  2,  1;
  1,  2,  7,  2,  1;
  2,  7,  2, 10,  2,  1;
  1,  2, 16,  2, 13,  2,  1;
  ...

Cf. A049310, A131324, A131326 (row sums), A131327.

A131325 := proc(n,k)
    abs(3*abs(A049310(n,k))-2) ;
end proc:
seq(seq(A131325(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Aug 13 2012

Definition corrected by David Scambler, Aug 12 2012

A131327 Triangle |4*|A049310(n,k)| - 3| read by rows, 0<=k<=n.

Original entry on oeis.org

1, 3, 1, 1, 3, 1, 3, 5, 3, 1, 1, 3, 9, 3, 1, 3, 9, 3, 13, 3, 1, 1, 3, 21, 3, 17, 3, 1, 3, 13, 3, 37, 3, 21, 3, 1, 1, 3, 37, 3, 57, 3, 25, 3, 1, 3, 17, 3, 77, 3, 81, 3, 29, 3, 1, 1, 3, 57, 3, 137, 3, 109, 3, 33, 3, 1, 3, 21, 3, 137, 3, 221, 3, 141, 3, 37, 3, 1, 1, 3, 81, 3, 277, 3, 333, 3, 177

Offset: 0

Gary W. Adamson, Jun 28 2007

			First few rows of the triangle are:
1;
3, 1;
1, 3, 1;
3, 5, 3, 1;
1, 3, 9, 3, 1;
3, 9, 3, 13, 3, 1;
1, 3, 21, 3, 17, 3, 1;
...

Cf. A049310, A131328 (row sums), A131324, A131325, A131326.

A131327 := proc(n,k)
    abs(4*abs(A049310(n,k))-3) ;
end proc:
seq(seq(A131327(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Aug 13 2012

Definition corrected. - R. J. Mathar, Aug 13 2012

A131324 2*A049310 - A000012(signed).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 11, 1, 9, 1, 1, 1, 7, 1, 19, 1, 11, 1, 1, 1, 1, 19, 1, 29, 1, 13, 1, 1, 1, 9, 1, 39, 1, 41, 1, 15, 1, 1

Offset: 0

Gary W. Adamson, Jun 28 2007

Row sums = A062114: (1, 2, 3, 6, 9, 16, 25, ...).

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  1,  5,  1,  1;
  1,  5,  1,  7,  1,  1,
  1,  1, 11,  1,  9,  1,  1;
  1,  7,  1, 19,  1, 11,  1,  1;
  ...

Cf. A049310, A062114, A131325, A131326, A131327.

2*A049310 - A000012(signed + - + - ... by columns).

A131328 Row sums of triangle A131327.

Original entry on oeis.org

1, 4, 5, 12, 17, 32, 49, 84, 133, 220, 353, 576, 929, 1508, 2437, 3948, 6385, 10336, 16721, 27060, 43781, 70844, 114625, 185472, 300097, 485572, 785669, 1271244, 2056913, 3328160, 5385073, 8713236, 14098309, 22811548, 36909857, 59721408, 96631265, 156352676

Offset: 0

Gary W. Adamson, Jun 28 2007

a(n)/a(n-1) tends to phi. (Cf. A062114).

			a(3) = 12 = sum of row 3 terms of A131327: (3 + 5 + 3 + 1).
a(3) = (9 + 3) since we add terms of A131326: (1, 3, 4, 9, 13,...) to A052952: (0, 1, 1, 3, 4,...), getting (9 + 3 ) = 12.

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

Cf. A062114, A052952, A131324, A131325, A131326, A131327.

Vec((1 + 3*x - x^2) / ((1 - x)*(1 + x)*(1 - x - x^2)) + O(x^50)) \\ Colin Barker, Jul 12 2017

a(n+1) = A131326(n) + A052952(n+1).
a(n) = -3*(1+(-1)^n)/2 +4*A000045(n+1). - R. J. Mathar, Aug 13 2012
G.f.: ( 1+3*x-x^2 ) / ( (x-1)*(1+x)*(x^2+x-1) ). - R. J. Mathar, Aug 13 2012
From Colin Barker, Jul 12 2017: (Start)
a(n) = (2^(1-n)*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1))) / sqrt(5) - 3 for n even.
a(n) = (2^(1-n)*((1+sqrt(5))^(n+1) - (1-sqrt(5))^(n+1))) / sqrt(5) for n odd.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>3.
(End)

More terms from Colin Barker, Jul 12 2017

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A131325 Triangle |3*|A049310(n,k)| - 2| read by rows, 0 <= k <= n.

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A131327 Triangle |4*|A049310(n,k)| - 3| read by rows, 0<=k<=n.

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A131324 2*A049310 - A000012(signed).

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A131328 Row sums of triangle A131327.

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