A079977 -id:A079977 - cp's OEIS frontend

A289845 p-INVERT of A079977, where p(S) = 1 - S - S^2.

1, 2, 4, 9, 19, 43, 91, 202, 433, 952, 2055, 4494, 9737, 21236, 46099, 100403, 218164, 474833, 1032256, 2245929, 4883690, 10623848, 23103985, 50255443, 109298635, 237734446, 517055409, 1124617945, 2446001258, 5320100761, 11571106298, 25167245524, 54738437517

Offset: 0

Clark Kimberling, Aug 14 2017

Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A289780 for a guide to related sequences.

Clark Kimberling, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (1, 3, -1, 1, -1, -2, 0, -1)

Cf. A079977, A289780.

z = 60; s = -x/(x^4 + x^2 - 1); p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A079977 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (*A289845*)
LinearRecurrence[{1,3,-1,1,-1,-2,0,-1},{1,2,4,9,19,43,91,202},40] (* Harvey P. Dale, Jan 16 2019 *)

G.f.: (1 + x - x^2 - x^4)/(1 - x - 3 x^2 + x^3 - x^4 + x^5 + 2 x^6 + x^8).

a(n) = a(n-1) + 3*a(n-2) - a(n-3) + a(n-4) - a(n-5) - 2*a(n-6) - a(n-8).

A289780 p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.

Original entry on oeis.org

1, 4, 14, 47, 156, 517, 1714, 5684, 18851, 62520, 207349, 687676, 2280686, 7563923, 25085844, 83197513, 275925586, 915110636, 3034975799, 10065534960, 33382471801, 110713382644, 367182309614, 1217764693607, 4038731742156, 13394504020957, 44423039068114

Offset: 0

Clark Kimberling, Aug 10 2017

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).
Taking p(S) = 1 - S gives the INVERT transform of s, so that p-INVERT is a generalization of the INVERT transform (e.g., A033453).
Guide to p-INVERT sequences using p(S) = 1 - S - S^2:
t(A000012) = t(1,1,1,1,1,1,1,...)    = A001906
t(A000290) = t(1,4,9,16,25,36,...)   = A289779
t(A000027) = t(1,2,3,4,5,6,7,8,...)  = A289780
t(A000045) = t(1,2,3,5,8,13,21,...)  = A289781
t(A000032) = t(2,1,3,4,7,11,14,...)  = A289782
t(A000244) = t(1,3,9,27,81,243,...)  = A289783
t(A000302) = t(1,4,16,64,256,...)    = A289784
t(A000351) = t(1,5,25,125,625,...)   = A289785
t(A005408) = t(1,3,5,7,9,11,13,...)  = A289786
t(A005843) = t(2,4,6,8,10,12,14,...) = A289787
t(A016777) = t(1,4,7,10,13,16,...)   = A289789
t(A016789) = t(2,5,8,11,14,17,...)   = A289790
t(A008585) = t(3,6,9,12,15,18,...)   = A289795
t(A000217) = t(1,3,6,10,15,21,...)   = A289797
t(A000225) = t(1,3,7,15,31,63,...)   = A289798
t(A000578) = t(1,8,27,64,625,...)    = A289799
t(A000984) = t(1,2,6,20,70,252,...)  = A289800
t(A000292) = t(1,4,10,20,35,56,...)  = A289801
t(A002620) = t(1,2,4,6,9,12,16,...)  = A289802
t(A001906) = t(1,3,8,21,55,144,...)  = A289803
t(A001519) = t(1,1,2,5,13,34,...)    = A289804
t(A103889) = t(2,1,4,3,6,5,8,7,,...) = A289805
t(A008619) = t(1,1,2,2,3,3,4,4,...)  = A289806
t(A080513) = t(1,2,2,3,3,4,4,5,...)  = A289807
t(A133622) = t(1,2,1,3,1,4,1,5,...)  = A289809
t(A000108) = t(1,1,2,5,14,42,...)    = A081696
t(A081696) = t(1,1,3,9,29,97,...)    = A289810
t(A027656) = t(1,0,2,0,3,0,4,0,5...) = A289843
t(A175676) = t(1,0,0,2,0,0,3,0,...)  = A289844
t(A079977) = t(1,0,1,0,2,0,3,...)    = A289845
t(A059841) = t(1,0,1,0,1,0,1,...)    = A289846
t(A000040) = t(2,3,5,7,11,13,...)    = A289847
t(A008578) = t(1,2,3,5,7,11,13,...)  = A289828
t(A000142) = t(1!, 2!, 3!, 4!, ...)  = A289924
t(A000201) = t(1,3,4,6,8,9,11,...)   = A289925
t(A001950) = t(2,5,7,10,13,15,...)   = A289926
t(A014217) = t(1,2,4,6,11,17,29,...) = A289927
t(A000045*) = t(0,1,1,2,3,5,...)     = A289975 (* indicates prepended 0's)
t(A000045*) = t(0,0,1,1,2,3,5,...)   = A289976
t(A000045*) = t(0,0,0,1,1,2,3,5,...) = A289977
t(A290990*) = t(0,1,2,3,4,5,...)     = A290990
t(A290990*) = t(0,0,1,2,3,4,5,...)   = A290991
t(A290990*) = t(0,0,01,2,3,4,5,...)  = A290992

			Example 1:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S.
S(x) = x + 2x^2 + 3x^3 + 4x^4 + ...
p(S(x)) = 1 - (x + 2x^2 + 3x^3 + 4x^4 + ... )
- p(0) + 1/p(S(x)) = -1 + 1 + x + 3x^2 + 8x^3 + 21x^4 + ...
T(x) = 1 + 3x + 8x^2 + 21x^3 + ...
t(s) = (1,3,8,21,...) = A001906.
***
Example 2:  s = (1,2,3,4,5,6,...) = A000027 and p(S) = 1 - S - S^2.
S(x) =  x + 2x^2 + 3x^3 + 4x^4 + ...
p(S(x)) = 1 - ( x + 2x^2 + 3x^3 + 4x^4 + ...) - ( x + 2x^2 + 3x^3 + 4x^4 + ...)^2
- p(0) + 1/p(S(x)) = -1 + 1 + x + 4x^2 + 14x^3 + 47x^4 + ...
T(x) = 1 + 4x + 14x^2 + 47x^3 + ...
t(s) = (1,4,14,47,...) = A289780.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -7, 5, -1)

Cf. A000027.

P:=[1,4,14,47];; for n in [5..10^2] do P[n]:=5*P[n-1]-7*P[n-2]+5*P[n-3]-P[n-4]; od; P; # Muniru A Asiru, Sep 03 2017

z = 60; s = x/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289780 *)

x='x+O('x^99); Vec((1-x+x^2)/(1-5*x+7*x^2-5*x^3+x^4)) \\ Altug Alkan, Aug 13 2017

G.f.: (1 - x + x^2)/(1 - 5 x + 7 x^2 - 5 x^3 + x^4).

a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) - a(n-4).

A053602 a(n) = a(n-1) - (-1)^n*a(n-2), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 5, 3, 8, 5, 13, 8, 21, 13, 34, 21, 55, 34, 89, 55, 144, 89, 233, 144, 377, 233, 610, 377, 987, 610, 1597, 987, 2584, 1597, 4181, 2584, 6765, 4181, 10946, 6765, 17711, 10946, 28657, 17711, 46368, 28657, 75025, 46368, 121393, 75025

Offset: 0

Michael Somos, Jan 17 2000

If b(0)=0, b(1)=1 and b(n) = b(n-1) + (-1)^n*b(n-2), then a(n) = b(n+3). - Jaume Oliver Lafont, Oct 03 2009
a(n) is the number of palindromic compositions of n-1 into parts of 1 and 2. a(7) = 5 because we have 2+2+2, 2+1+1+2, 1+2+2+1, 1+1+2+1+1, 1+1+1+1+1+1. - Geoffrey Critzer, Mar 17 2014
a(n) is the number of palindromic compositions of n into odd parts (the corresponding generating function follows easily from Theorem 1.2 of the Hoggatt et al. reference). Example: a(7) = 5 because we have 7, 1+5+1, 3+1+3, 1+1+3+1+1, 1+1+1+1+1+1+1. - Emeric Deutsch, Aug 16 2016
The ratio of a(n)/a(n-1) oscillates between phi-1 and phi+1 as n tends to infinity, where phi is golden ratio (A001622). - Waldemar Puszkarz, Oct 10 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239. - _Ron Knott_, Oct 29 2010
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
M. A. Nyblom, Counting Palindromic Binary Strings Without r-Runs of Ones, J. Int. Seq. 16 (2013) #13.8.7, P_2(n)
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1)

Cf. A000045, A001622, A051792, A079977.

I:=[0,1,1,2]; [n le 4 select I[n] else Self(n-2)+Self(n-4): n in [1..50]]; // Vincenzo Librandi Oct 10 2017

a[0] := 0: a[1] := 1: for n from 2 to 60 do a[n] := a[n-1]-(-1)^n*a[n-2] end do: seq(a[n], n = 0 .. 50); # Emeric Deutsch, Oct 09 2017

nn=50;CoefficientList[Series[x (1+x+x^2)/(1-x^2-x^4),{x,0,nn}],x] (* Geoffrey Critzer, Mar 17 2014 *)
LinearRecurrence[{0,1,0,1},{0,1,1,2},60] (* Harvey P. Dale, Nov 07 2016 *)
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-1]-(-1)^n a[n-2]}, a, {n, 50}] (* Vincenzo Librandi, Oct 10 2017 *)

PARI
```
a(n)=fibonacci(n\2+n%2*2)
```

[fibonacci(n//2 + 2*(n%2)) for n in range(61)] # G. C. Greubel, Dec 06 2022

G.f.: x*(1 + x + x^2)/(1 - x^2 - x^4).
a(n) = a(n-2) + a(n-4).
a(2n) = F(n), a(2n-1) = F(n+1) where F() is Fibonacci sequence.
a(3-n) = A051792(n).
a(3)=1, a(4)=2, a(n+2) = a(n+1) + sign(a(n) - a(n+1))*a(n), n > 4. - Benoit Cloitre, Apr 08 2002
a(n) = A079977(n-1) + A079977(n-2) + A079977(n-3), n > 2. - Ralf Stephan, Apr 26 2003
a(0) = 0, a(1) = 1; a(2n) = a(2n-1) - a(2n-2); a(2n+1) = a(2n) + a(2n-1). - Amarnath Murthy, Jul 21 2005

A226206 Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles of area > 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 3, 1, 3, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 5, 0, 7, 0, 5, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 7, 7, 0, 0, 0, 0, 0, 1

Offset: 0

Alois P. Heinz, May 31 2013

			A(6,4) = A(4,6) = 3:
  ._._._._._._.   ._._._._._._.   ._._._._._._.
  |   |   |   |   |       |   |   |   |       |
  |___|___|___|   |       |___|   |___|       |
  |   |   |   |   |       |   |   |   |       |
  |___|___|___|   |_______|___|   |___|_______|  .
Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1,  1, 1,   1,  1,   1, ...
  1, 0, 0, 0, 0, 0,  0, 0,   0,  0,   0, ...
  1, 0, 1, 0, 1, 0,  1, 0,   1,  0,   1, ...
  1, 0, 0, 1, 0, 0,  1, 0,   0,  1,   0, ...
  1, 0, 1, 0, 2, 0,  3, 0,   5,  0,   8, ...
  1, 0, 0, 0, 0, 1,  2, 0,   0,  0,   1, ...
  1, 0, 1, 1, 3, 2,  7, 7,  16, 19,  40, ...
  1, 0, 0, 0, 0, 0,  7, 1,   0,  0,   2, ...
  1, 0, 1, 0, 5, 0, 16, 0,  48,  0, 160, ...
  1, 0, 0, 1, 0, 0, 19, 0,   0, 50,  17, ...
  1, 0, 1, 0, 8, 1, 40, 2, 160, 17, 796, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..34, flattened

Columns (or rows) k=0-12 give: A000012, A000007, A059841, A079978, A079977, A226369, A226370, A226371, A226372, A226373, A226374, A226375, A226376.
Main diagonal gives A347800.
Cf. A219924.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:=0;
         for i from k+1 to nops(l) while l[i]=0 do s:=s+
           b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
         od; s
      fi
    end:
A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which [Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = 0; For[i = k+1, i <= Length[l] && l[[i]] == 0, i++, s = s + b[n, Join [l[[1 ;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1 ;; -1]] ]]]; s]]; a [n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

A051792 a(n) = (-1)^(n-1)*(a(n-1) - a(n-2)), a(1)=1, a(2)=1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, -1, 1, 2, -1, -3, 2, 5, -3, -8, 5, 13, -8, -21, 13, 34, -21, -55, 34, 89, -55, -144, 89, 233, -144, -377, 233, 610, -377, -987, 610, 1597, -987, -2584, 1597, 4181, -2584, -6765, 4181, 10946, -6765, -17711, 10946, 28657, -17711, -46368

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 10 1999

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,1).

Cf. A000045, A053602.

[Fibonacci(1 -Floor((n-4)/2) -2*((n-4) mod 2)): n in [1..60]]; // G. C. Greubel, Dec 06 2022

LinearRecurrence[{0,-1,0,1},{1,1,0,1},60] (* Harvey P. Dale, May 08 2017 *)

PARI
```
a(n)=fibonacci((3-n)\2+(3-n)%2*2)
```

def A051792():
    x, y, b = 1, 1, true
    while True:
        yield x
        x, y = y, x - y
        y = -y if b else y
        b = not b
a = A051792()
print([next(a) for  in range(51)]) # _Peter Luschny, Mar 19 2020

a(3-n) = A053602(n).
From Michael Somos: (Start)
G.f.: x*(1 + x + x^2 + 2*x^3)/(1 + x^2 - x^4).
a(n) = -a(n-2) + a(n-4). (End)
a(n) = b(n-1) + b(n-2) + b(n-3) + 2*b(n-4), where b(n) = i^n * A079977(n). - G. C. Greubel, Dec 06 2022

A096748 Expansion of (1+x)^2/(1-x^2-x^4).

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 26, 34, 42, 55, 68, 89, 110, 144, 178, 233, 288, 377, 466, 610, 754, 987, 1220, 1597, 1974, 2584, 3194, 4181, 5168, 6765, 8362, 10946, 13530, 17711, 21892, 28657, 35422, 46368, 57314, 75025, 92736, 121393, 150050

Offset: 0

Paul Barry, Jul 07 2004

The ratio a(n+1) / a(n) increasingly approximates two constants connected to the golden ratio phi = (1 + sqrt(5))/2: (phi+1)/2 = 1.30901699... = A239798 and (phi-1)*2 = 1.23606797... = A134972, according to whether n is odd or even. - Davide Rotondo, Jul 31 2020

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Davide Rotondo, Perfino I Capelli Sono Tutti Contati (in Italian), see p. 11.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Cf. A000045, A079977.

Cf. A134972 and A239798 (limiting ratios for a(n+1)/a(n)).

CoefficientList[Series[(1+x)^2/(1-x^2-x^4),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,0,1},{1,2,2,2},50] (* Harvey P. Dale, Jan 29 2012 *)

a(n) = a(n-2) + a(n-4).
a(n) = 2*F((n+1)/2)*(1-(-1)^n)/2 + F((n+4)/2)*(1+(-1)^n)/2.
a(2*n) = A000045(n+2); a(2*n+1) = 2*A000045(n+1).
a(n) = Sum_{k=0..n} binomial(floor((n-k)/2), floor(k/2)). - Paul Barry, Jul 24 2004
a(n) = A079977(n) + A079977(n-2) + 2*A079977(n-1). - R. J. Mathar, Jul 15 2013

A099574 Diagonal sums of triangle A099573.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 9, 11, 18, 23, 37, 48, 74, 97, 147, 195, 290, 387, 568, 763, 1108, 1495, 2152, 2915, 4167, 5662, 8047, 10962, 15506, 21168, 29825, 40787, 57280, 78448, 109870, 150657, 210521, 288969, 403020, 553677, 770963, 1059932, 1473898

Offset: 0

Paul Barry, Oct 23 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A003269, A079977, A099573, A099577, A103609.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x^4)/((1-x^2-x^4)*(1-x-x^4)) )); // G. C. Greubel, Jul 25 2022

a[n_]:= a[n]= Sum[Binomial[n-k-j, j], {k,0,Floor[n/2]}, {j,0,Floor[k/2]}];
Table[a[n], {n, 0, 40}] (* G. C. Greubel, Jul 25 2022 *)

@CachedFunction
def A099574(n): return sum(sum(binomial(n-k-j, j) for j in (0..(k//2))) for k in (0..(n//2)))
[A099574(n) for n in (0..40)] # G. C. Greubel, Jul 25 2022

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..floor(k/2)} binomial(n-k-j, j).
G.f.: (1-x)*(1+x)*(1+x^2) / ( (1-x-x^4)*(1-x^2-x^4) ). - R. J. Mathar, Nov 11 2014
From G. C. Greubel, Jul 25 2022: (Start)
a(n) = A003269(n+5) - A079977(n+3) - A079977(n+2).
a(n) = A003269(n+5) - A103609(n+5). (End)

A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 5, 0, 1, 0, 1, 1, 0, 2, 0, 8, 1, 1, 0, 1, 0, 1, 0, 3, 0, 13, 0, 1, 0, 1, 0, 1, 1, 1, 4, 1, 21, 1, 1, 0, 1, 1, 0, 1, 2, 0, 6, 0, 34, 0, 1, 0, 1, 1, 2, 0, 1, 3, 0, 9, 0, 55, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 01 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1,  1, 1,  1, 1,  1, 1, 1,   1, 1, 1, 1,   1, ...
  0, 1, 0,  1, 0,  1, 0,  1, 0, 0,   1, 1, 0, 0,   1, ...
  0, 1, 1,  2, 0,  1, 0,  1, 1, 0,   2, 1, 1, 0,   2, ...
  0, 1, 0,  3, 1,  2, 0,  1, 1, 0,   4, 1, 0, 0,   3, ...
  0, 1, 1,  5, 0,  3, 1,  2, 1, 0,   7, 1, 2, 0,   6, ...
  0, 1, 0,  8, 0,  4, 0,  3, 2, 1,  13, 2, 0, 0,  10, ...
  0, 1, 1, 13, 1,  6, 0,  4, 2, 0,  24, 3, 3, 1,  18, ...
  0, 1, 0, 21, 0,  9, 0,  5, 3, 0,  44, 4, 0, 0,  31, ...
  0, 1, 1, 34, 0, 13, 1,  7, 4, 0,  81, 5, 5, 0,  55, ...
  0, 1, 0, 55, 1, 19, 0, 10, 5, 0, 149, 6, 0, 0,  96, ...
  0, 1, 1, 89, 0, 28, 0, 14, 7, 1, 274, 8, 8, 0, 169, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns k=0-21, 23, 25-28 give: A000007, A000012, A059841, A000045(n+1), A079978, A000930, A121262, A003269(n+1), A182097, A079998, A000073(n+2), A003520, A079977, A079979, A060945, A005708, A001687(n+1), A017817, A082784, A079971, A006498, A005709, A052920, A120400, A060961, A005710, A013979.
Main diagonal gives A246691.
Cf. A246688, A246720 (the same for partitions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(i>n, 0, g(n-i, l)), i=l))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n==0, 1, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A124304 Riordan array (1, x*(1-x^2)).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, 0, -2, 0, 1, 0, 0, 0, -3, 0, 1, 0, 0, 1, 0, -4, 0, 1, 0, 0, 0, 3, 0, -5, 0, 1, 0, 0, 0, 0, 6, 0, -6, 0, 1, 0, 0, 0, -1, 0, 10, 0, -7, 0, 1, 0, 0, 0, 0, -4, 0, 15, 0, -8, 0, 1, 0, 0, 0, 0, 0, -10, 0, 21, 0, -9, 0, 1, 0, 0, 0, 0, 1, 0, -20, 0, 28, 0, -10, 0, 1

Offset: 0

Paul Barry, Oct 25 2006

T(2n,n) is a signed aerated version of C(2n,n).

Inverse is A124305.

			Triangle begins
  1;
  0,  1;
  0,  0,  1;
  0, -1,  0,  1;
  0,  0, -2,  0,  1;
  0,  0,  0, -3,  0,  1;
  0,  0,  1,  0, -4,  0,  1;
  0,  0,  0,  3,  0, -5,  0,  1;
  0,  0,  0,  0,  6,  0, -6,  0,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.

Cf. A014021 (diagonal sums), A050935 (row sums), A124305 (inverse).

Cf. A002054, A079977, A126869, A138364, A176971.

A124304:= func< n,k | (&+[(-1)^j*Binomial(k,k-j)*Binomial(k,n-k-j) : j in [0..n]]) >;
[A124304(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Aug 18 2023

A124304[n_, k_]:= Binomial[k, (n-k)/2]*(-1)^((n-k)/2)*(1+(-1)^(n-k))/2;
Table[A124304[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 18 2023 *)

def A124304(n, k): return binomial(k, (n-k)//2)*(-1)^((n-k)//2)*(1+(-1)^(n-k))/2
flatten([[A124304(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Aug 18 2023

T(n, k) = Sum_{j=0..n} C(k,k-j)*C(k,n-k-j)*(-1)^j.
T(n, k) = C(k,(n-k)/2)*(-1)^((n-k)/2)*(1 + (-1)^(n-k))/2.
Sum_{k=0..n} T(n, k) = A050935(n+2).
Sum_{k=0..floor(n/2)} T(n-k, k) = A014021(n).
T(2*n, n) = (1 - 2*0^(n+2 mod 4))*A126869(n).
From G. C. Greubel, Aug 18 2023: (Start)
T(2*n-1, n-1) = (1 - 2*0^(n+1 mod 4))*A138364(n-1).
T(2*n-1, n+1) = (1 - 2*0^(n mod 4))*((1+(-1)^n)/2)*A002054(floor(n/2)).
Sum_{k=0..n} (-1)^k*T(n, k) = A176971(n+3).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (1 - 2*0^(n+2 mod 4))*A079977(n).
G.f.: 1/(1 - x*y*(1-x^2)). (End)

A174618 For n odd a(n) = a(n-2) + a(n-3), for n even a(n) = a(n-2) + a(n-5); with a(1) = 0, a(2) = 1.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 4, 4, 6, 7, 10, 11, 17, 17, 28, 27, 45, 44, 72, 72, 116, 117, 188, 189, 305, 305, 494, 493, 799, 798, 1292, 1292, 2090, 2091, 3382, 3383, 5473, 5473, 8856, 8855, 14329, 14328, 23184, 23184, 37512, 37513, 60696

Offset: 1

Mark Dols, Mar 23 2010

Combination a(2n)=A005252(n-1) and a(2n+1)=A024490(n). Consecutive pairs add up to A000045 and subtract to A010892. If a(1)= 1 formula gives: A103609.

			As consecutive pairs: (0,1),(0,1),(1,1),(2,1),(3,2),(4,4),...

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

Cf. A000045, A005252, A010892, A024490, A079977, A110161.

R:=PowerSeriesRing(Integers(), 70);
[0] cat Coefficients(R!( x^2*(1-x^2+x^3)/((1-x^2+x^4)*(1-x^2-x^4)) )); // G. C. Greubel, Oct 23 2024

nxt[{n_,a_,b_,c_,d_,e_}]:={n+1,b,c,d,e,If[EvenQ[n],d+c,d+a]}; NestList[nxt,{5,0,1,0,1,1},50][[All,2]] (* or *) LinearRecurrence[ {0,2,0,-1,0,0,0,1},{0,1,0,1,1,1,2,1},60] (* Harvey P. Dale, Nov 15 2019 *)

def A174618(n): return (kronecker(12,n-3) - kronecker(12,n-2) + ((n+1)%2)*fibonacci(n//2) + (n%2)*fibonacci((n+1)//2))//2
[A174618(n) for n in range(1,71)] # G. C. Greubel, Oct 23 2024

G.f.: x^2*(1-x^2+x^3) / ( (1-x^2+x^4)*(1-x^2-x^4) ). - R. J. Mathar, Jan 27 2011

a(n) = (1/2)*(A110161(n-3) - A110161(n-2) + A079977(n-2) + A079977(n-1)). - G. C. Greubel, Oct 23 2024

cp's OEIS Frontend

A289845 p-INVERT of A079977, where p(S) = 1 - S - S^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A289780 p-INVERT of the positive integers (A000027), where p(S) = 1 - S - S^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Mathematica

PARI

Formula

A053602 a(n) = a(n-1) - (-1)^n*a(n-2), a(0)=0, a(1)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A226206 Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles of area > 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A051792 a(n) = (-1)^(n-1)*(a(n-1) - a(n-2)), a(1)=1, a(2)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A096748 Expansion of (1+x)^2/(1-x^2-x^4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A099574 Diagonal sums of triangle A099573.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath