A026639 -id:A026639 - cp's OEIS frontend

A026637 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,1) = T(n,n-1) = floor((3*n-1)/2) for n >= 1, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2, n >= 4.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 5, 8, 5, 1, 1, 7, 13, 13, 7, 1, 1, 8, 20, 26, 20, 8, 1, 1, 10, 28, 46, 46, 28, 10, 1, 1, 11, 38, 74, 92, 74, 38, 11, 1, 1, 13, 49, 112, 166, 166, 112, 49, 13, 1, 1, 14, 62, 161, 278, 332, 278, 161, 62, 14, 1, 1, 16, 76, 223, 439, 610, 610, 439, 223, 76, 16, 1

Offset: 0

Clark Kimberling

T(n, k) = number of paths from (0, 0) to (n-k, k) in directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, j)-to-(i+1, j+1) for i=0, j >= 1 and odd and for j=0, i >= 1 and odd.

See A228053 for a sequence with many terms in common with this one. - T. D. Noe, Aug 07 2013

			Triangle begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  4,   1;
  1,  5,  8,   5,   1;
  1,  7, 13,  13,   7,   1;
  1,  8, 20,  26,  20,   8,   1;
  1, 10, 28,  46,  46,  28,  10,   1;
  1, 11, 38,  74,  92,  74,  38,  11,  1;
  1, 13, 49, 112, 166, 166, 112,  49, 13,  1;
  1, 14, 62, 161, 278, 332, 278, 161, 62, 14,  1;

Reinhard Zumkeller, Rows n = 0..100 of table, flattened

Cf. A026638, A026639, A026640, A026641, A026642, A026643, A026644.
Cf. A026966, A026967, A026968, A026969, A026970, A228053.
Sums include: A000007 (alternating sign row), A026644 (row), A026645, A026646, A026647 (diagonal).

a026637 n k = a026637_tabl !! n !! k
a026637_row n = a026637_tabl !! n
a026637_tabl = [1] : [1,1] : map (fst . snd)
   (iterate f (0, ([1,2,1], [0,1,1,0]))) where
   f (i, (xs, ws)) = (1 - i,
     if i == 1 then (ys, ws) else (zipWith (+) ys ws, ws'))
        where ys = zipWith (+) ([0] ++ xs) (xs ++ [0])
              ws' = [0,1,0,0] ++ drop 2 ws
-- Reinhard Zumkeller, Aug 08 2013

function T(n,k) // T = A026637
   if k eq 0 or k eq n then return 1;
   elif k eq 1 or k eq n-1 then return Floor((3*n-1)/2);
   else return T(n-1, k) + T(n-1, k-1);
   end if;
end function;
[T(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jun 28 2024

A026637 := proc(n,k)
      option remember;
      if k=0 or k=n then
        1
    elif k=1 or k=n-1 then
        floor((3*n-1)/2) ;
    elif k <0 or k > n then
        0;
    else
        procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Apr 26 2015

T[n_, k_] := T[n, k] = Which[k == 0 || k == n, 1, k == 1 || k == n-1, Floor[(3n-1)/2], k < 0 || k > n, 0, True, T[n-1, k-1] + T[n-1, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

def T(n,k): # T = A026637
    if k==0 or k==n: return 1
    elif k==1 or k==n-1: return ((3*n-1)//2)
    else: return T(n-1, k) + T(n-1, k-1)
flatten([[T(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Jun 28 2024

From G. C. Greubel, Jun 28 2024: (Start)
T(n, n-k) = T(n, k).
T(2*n-1, n-1) = A026641(n), n >= 1.
Sum_{k=0..n} T(n, k) = A026644(n).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)

A026642 a(n) = A026637(2*n-1, n-2).

Original entry on oeis.org

1, 7, 28, 112, 439, 1711, 6652, 25846, 100450, 390670, 1520764, 5925718, 23112931, 90239407, 352654084, 1379410438, 5400188206, 21157958962, 82959736504, 325514137048, 1278093308806, 5021436970822, 19740128055928

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A026637, A026638, A026639, A026640, A026641, A026643, A026644.
Cf. A026645, A026646, A026647, A026966, A026967, A026968, A026969.
Cf. A026970.

[n le 2 select 7^(n-1) else ((7*n^2+3*n+2)*Self(n-1) + 2*n*(2*n+1)*Self(n-2))/(2*(n-1)*(n+2)): n in [1..40]]; // G. C. Greubel, Jul 01 2024

a[n_]:= a[n]= If[n<4, 7^(n-2), ((7*n^2-11*n+6)*a[n-1] + 2*(n-1)*(2*n- 1)*a[n-2])/(2*(n-2)*(n+1))];
Table[a[n], {n,2,40}] (* G. C. Greubel, Jul 01 2024 *)

@CachedFunction
def a(n): # a = A026642
    if n<4: return 7^(n-2)
    else: return ((7*n^2-11*n+6)*a(n-1) + 2*(n-1)*(2*n-1)*a(n-2))/(2*(n-2)*(n+1))
[a(n) for n in range(2,41)] # G. C. Greubel, Jul 01 2024

a(n) = ( (7*n^2 - 11*n + 6)*a(n-1) + 2*(n-1)*(2*n-1)*a(n-2) )/(2*(n-2)*(n+1)), n >= 4. - G. C. Greubel, Jul 01 2024

A026638 a(n) = A026637(2*n, n).

Original entry on oeis.org

1, 2, 8, 26, 92, 332, 1220, 4538, 17036, 64412, 244928, 935684, 3588392, 13806704, 53271548, 206040506, 798600332, 3101109164, 12062148368, 46986821516, 183276382472, 715748620424, 2798274135368, 10951009023716, 42895901012792, 168167959150232, 659793819847040

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A026637, A026639, A026640, A026642, A026643, A026644, A026645.

Cf. A026646, A026647, A026966, A026967, A026968, A026969, A026970.

[1] cat [n le 2 select 2^(2*n-1) else ((7*n-4)*Self(n-1) + 2*(2*n-1)*Self(n-2))/(2*n): n in [1..40]]; // G. C. Greubel, Jul 01 2024

CoefficientList[Series[1/(2+x)+3/((2+x)*Sqrt[1-4*x])-1,{x,0,20}],x] (* Vaclav Kotesovec, Oct 21 2012 *)

my(x='x+O('x^66)); Vec( 1/(2+x)+3/((2+x)*sqrt(1-4*x))-1 ) \\ Joerg Arndt, May 04 2013

@CachedFunction
def a(n): # a = A026638
    if n<3: return 2^(n*(n+1)/2)
    else: return ((7*n-4)*a(n-1) + 2*(2*n-1)*a(n-2))/(2*n)
[a(n) for n in range(41)] # G. C. Greubel, Jul 01 2024

From Vaclav Kotesovec, Oct 21 2012: (Start)
G.f.: (3 - (x+1)*sqrt(1-4*x))/((x+2)*sqrt(1-4*x)).
Recurrence: 2*n*a(n) = (7*n-4)*a(n-1) + 2*(2*n-1)*a(n-2).
a(n) ~ 2^(2*n+2)/(3*sqrt(Pi*n)) (End)

A026640 a(n) = A026637(2*n, n-2).

Original entry on oeis.org

1, 8, 38, 161, 662, 2672, 10676, 42398, 167756, 662252, 2610758, 10283861, 40490702, 159394424, 627456188, 2470223186, 9726696572, 38308366784, 150916209308, 594704861546, 2344206594332, 9243186573248, 36456892635848

Offset: 2

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A026637, A026638, A026639, A026641, A026642, A026643, A026644.
Cf. A026645, A026646, A026647, A026966, A026967, A026968, A026969.
Cf. A026970.

[1] cat [n le 2 select 4^(n+1) -3^(n+1) +1 else ((7*n^2+24*n+24 )*Self(n-1) + 2*(2*n+3)*(n+2)*Self(n-2))/(2*n*(n+4)): n in [1..40]]; // G. C. Greubel, Jul 01 2024

a[n_]:= a[n]= If[n<5, 4^(n-1) -3^(n-1) +1 -Boole[n==2], ((7*n^2 -4*n + 4)*a[n-1] +2*n*(2*n-1)*a[n-2])/(2*(n-2)*(n+2))];
Table[a[n], {n,2,40}] (* G. C. Greubel, Jul 01 2024 *)

@CachedFunction
def a(n): # a = A026640
    if n<5: return 4^(n-1) -3^(n-1) +1 -int(n==2)
    else: return ((7*n^2-4*n+4)*a(n-1) + 2*n*(2*n-1)*a(n-2))/(2*(n-2)*(n+2))
[a(n) for n in range(2,41)] # G. C. Greubel, Jul 01 2024

a(n) = ((7*n^2 - 4*n + 4)*a(n-1) + 2*n*(2*n-1)*a(n-2))/(2*(n-2)*(n+2)), n >= 5. - G. C. Greubel, Jul 01 2024

A026643 a(n) = A026637(n, floor(n/2)).

Original entry on oeis.org

1, 1, 2, 4, 8, 13, 26, 46, 92, 166, 332, 610, 1220, 2269, 4538, 8518, 17036, 32206, 64412, 122464, 244928, 467842, 935684, 1794196, 3588392, 6903352, 13806704, 26635774, 53271548, 103020253, 206040506, 399300166, 798600332

Offset: 0

Clark Kimberling

nonn

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

Cf. A026637, A026638, A026639, A026640, A026641, A026642, A026644.

Cf. A026966, A026967, A026968, A026969, A026970.

[1] cat [n le 4 select 2^(n-1) else (4*Self(n-1) +(7*n-9)*Self(n-2) +2*Self(n-3) +4*(n-1)*Self(n-4))/(2*(n+1)): n in [1..40]]; // G. C. Greubel, Jul 01 2024

T[n_, k_]:= T[n, k] = If[k==0 || k==n, 1, If[k==1 || k==n-1, Floor[(3*n -1)/2], T[n-1,k-1] + T[n-1,k] ]]; (* A026637 *)
A026643[n_]:= T[n, Floor[n/2]];
Table[A026643[n], {n,0,40}] (* G. C. Greubel, Jul 01 2024 *)

@CachedFunction
def a(n): # a = A026643
    if n<5: return (1,1,2,4,8)[n]
    else: return (4*a(n-1) +(7*n-9)*a(n-2) +2*a(n-3) +4*(n-1)*a(n-4))/(2*(n+1))
[a(n) for n in range(41)] # G. C. Greubel, Jul 01 2024

a(n) = (4*a(n-1) + (7*n-9)*a(n-2) + 2*a(n-3) + 4*(n-1)*a(n-4))/(2*(n+1)) with a(0) = a(1) = 1, a(2) = 2, a(3) = 4, a(4) = 8. - G. C. Greubel, Jul 01 2024

A026646 a(n) = Sum_{i=0..n} Sum_{j=0..n} A026637(i,j).

Original entry on oeis.org

1, 3, 7, 17, 37, 79, 163, 333, 673, 1355, 2719, 5449, 10909, 21831, 43675, 87365, 174745, 349507, 699031, 1398081, 2796181, 5592383, 11184787, 22369597, 44739217, 89478459, 178956943, 357913913, 715827853, 1431655735

Offset: 0

Clark Kimberling

nonn

a(n) indexes the corner blocks on the Jacobsthal spiral built from blocks of unit area (using J(1) and J(2) as the sides of the first block). - Paul Barry, Mar 06 2008

Partial sums of A026644, which are the row sums of A026637. - Paul Barry, Mar 06 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Cf. A000126, A001045, A026637, A026644, A084639, A109613.
Cf. A026637, A026638, A026639, A026640, A026641, A026642, A026643.
Cf. A026644, A026966, A026967, A026968, A026969, A026970.

[(2^(n+4) -(6*n+9+(-1)^n))/6: n in [0..40]]; // G. C. Greubel, Jul 01 2024

CoefficientList[Series[(1-x^2+2x^3)/((1-x)(1-2x)(1-x^2)), {x, 0, 29}], x] (* Metin Sariyar, Sep 22 2019 *)
LinearRecurrence[{3,-1,-3,2}, {1,3,7,17}, 41] (* G. C. Greubel, Jul 01 2024 *)

[(2^(n+4) - (-1)^n -9 - 6*n)/6 for n in range(41)] # G. C. Greubel, Jul 01 2024

G.f.: (1 -x^2 +2*x^3)/((1-x)*(1-2*x)*(1-x^2)). - Ralf Stephan, Apr 30 2004
From Paul Barry, Mar 06 2008: (Start)
a(n) = A001045(n+3) - 2*floor((n+2)/2).
a(n) = -n + Sum_{k=0..n} A001045(k+2) = A084639(n+1) - n. (End)
a(n+1) = 2*a(n) + A109613(n), a(0)=1. - Paul Curtz, Sep 22 2019

A026647 a(n) = Sum_{k=0..floor(n/2)} A026637(n-k, k).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 17, 27, 45, 73, 119, 192, 312, 505, 818, 1323, 2142, 3466, 5609, 9075, 14685, 23761, 38447, 62208, 100656, 162865, 263522, 426387, 689910, 1116298, 1806209, 2922507, 4728717, 7651225, 12379943, 20031168

Offset: 0

Clark Kimberling

nonn

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

Cf. A026637, A026638, A026639, A026640, A026641, A026642, A026643.
Cf. A026644, A026645, A026646, A026966, A026967, A026968, A026969.
Cf. A026970.

[1] cat [n le 5 select Binomial(n, Floor(n/2)) else Self(n-2) +Self(n-3) +2*Self(n-4) +Self(n-5) +3: n in [1..40]]; // G. C. Greubel, Jul 01 2024

a[n_]:= a[n]= If[n<6, Binomial[n, Floor[n/2]], a[n-2] +a[n-3] +2*a[n- 4] +a[n-5] +3]; (* a = A026647 *)
Table[a[n], {n,0,40}] (* G. C. Greubel, Jul 01 2024 *)
LinearRecurrence[{1,1,0,1,-1,-1},{1,1,2,3,6,10,17},40] (* Harvey P. Dale, Jan 19 2025 *)

@CachedFunction
def a(n): # a = A026647
    if n<6: return binomial(n, n//2)
    else: return a(n-2) + a(n-3) + 2*a(n-4) + a(n-5) + 3
[a(n) for n in range(41)] # G. C. Greubel, Jul 01 2024

G.f.: (1 + x^5 + x^6)/((1-x^4)*(1-x-x^2)).
From G. C. Greubel, Jul 01 2024: (Start)
a(n) = [n=0] - (3/4) + (1/4)*(-1)^n - (1/10)*2^((1-(-1)^n)/2)*(-1)^floor((n+1)/2) + (3/5)*LucasL(n+1).
a(n) = (1/20)*( 12*LucasL(n+1) + 5*(-1)^n - 15 - 2*cos(n*Pi/2) + 4*sin(n*Pi/2) ) + [n=0].
a(n) = a(n-2) + a(n-3) + 2*a(n-4) + a(n-5) + 3, with a(0) = a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 6, a(5) = 10. (End)

A026645 a(n) = Sum_{k=0..floor(n/2)} A026637(n, k).

Original entry on oeis.org

1, 1, 3, 5, 14, 21, 55, 85, 216, 341, 848, 1365, 3340, 5461, 13191, 21845, 52208, 87381, 206968, 349525, 821514, 1398101, 3264044, 5592405, 12979006, 22369621, 51642594, 89478485, 205592744, 357913941, 818848135, 1431655765, 3262611696, 5726623061, 13003800704, 22906492245

Offset: 0

Clark Kimberling

nonn

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

Cf. A026637, A026638, A026639, A026640, A026642, A026643, A026644.

Cf. A026646, A026647, A026966, A026967, A026968, A026969, A026970.

T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k==1 || k==n-1, Floor[(3*n- 1)/2], T[n-1,k] + T[n-1,k-1] ]];
A026645[n_]:= Sum[T[n, k], {k, 0, Floor[n/2]}];
Table[A026645[n], {n,0,40}] (* G. C. Greubel, Jun 29 2024 *)

@CachedFunction
def T(n,k): # T = A026637
    if k==0 or k==n: return 1
    elif k==1 or k==n-1: return ((3*n-1)//2)
    else: return T(n-1, k) + T(n-1, k-1)
def A026645(n): return sum(T(n,k) for k in range((n//2)+1))
[A026645(n) for n in range(41)] # G. C. Greubel, Jun 29 2024

cp's OEIS Frontend

A026637 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for n >= 0, T(n,1) = T(n,n-1) = floor((3*n-1)/2) for n >= 1, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k) for 2 <= k <= n-2, n >= 4.

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A026642 a(n) = A026637(2*n-1, n-2).

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A026638 a(n) = A026637(2*n, n).

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A026640 a(n) = A026637(2*n, n-2).

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A026643 a(n) = A026637(n, floor(n/2)).

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A026646 a(n) = Sum_{i=0..n} Sum_{j=0..n} A026637(i,j).

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A026647 a(n) = Sum_{k=0..floor(n/2)} A026637(n-k, k).

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