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A135222 Triangle A049310 + A000012 - I, read by rows. 2
1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 4, 1, 1, 1, 4, 1, 5, 1, 1, 2, 1, 7, 1, 6, 1, 1, 1, 5, 1, 11, 1, 7, 1, 1, 2, 1, 11, 1, 16, 1, 8, 1, 1, 1, 6, 1, 21, 1, 22, 1, 9, 1, 1, 2, 1, 16, 1, 36, 1, 29, 1, 10, 1, 1, 1, 7, 1, 36, 1, 57, 1, 37, 1, 11, 1, 1, 2, 1, 22, 1, 71, 1, 85, 1, 46, 1, 12, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums = A081659: (1, 2 4, 6, 9, 13, 19, 28,...).

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

FORMULA

T(n,k) = A049310(n,k) + A000012(n,k) - Identity matrix, as infinite lower triangular matrices.

T(n,k) = 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) ), with T(n,n) = 1. - G. C. Greubel, Nov 20 2019

EXAMPLE

First few rows of the triangle are:

  1;

  1, 1;

  2, 1,  1;

  1, 3,  1,  1;

  2, 1,  4,  1,  1;

  1, 4,  1,  5,  1, 1;

  2, 1,  7,  1,  6, 1, 1;

  1, 5,  1, 11,  1, 7, 1, 1;

  2, 1, 11,  1, 16, 1, 8, 1, 1;

...

MAPLE

T:= proc(n, k) option remember;

      if k=n then 1

    else 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) )

      fi; end:

seq(seq(T(n, k), k=0..n), n=0..15); # G. C. Greubel, Nov 20 2019

MATHEMATICA

T[n_, k_]:= T[n, k]= If[k==n, 1, 1 + Abs[Simplify[((1+(-1)^(n-k))/2)* Binomial[(n+k)/2, (n-k)/2]*Cos[(n-k)*Pi/2]]] ]; Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

PROG

(Sage)

@CachedFunction

def T(n, k):

    if (k==n): return 1

    else: return 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*pi/2) )

[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Nov 20 2019

CROSSREFS

Cf. A049310, A081659.

Sequence in context: A324247 A138904 A196660 * A285706 A333381 A124094

Adjacent sequences:  A135219 A135220 A135221 * A135223 A135224 A135225

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Nov 23 2007

EXTENSIONS

More terms added and offset changed by G. C. Greubel, Nov 20 2019

STATUS

approved

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Last modified July 9 14:04 EDT 2020. Contains 335543 sequences. (Running on oeis4.)