A122950 -id:A122950 - cp's OEIS frontend

A133679 a(n) = 7a(n-1) + 56a(n-2) for n>=3, a(0)=1, a(1)=7, a(2)=98.

1, 7, 98, 1078, 13034, 151606, 1791146, 21027958, 247499882, 2910064822, 34230447146, 402576760054, 4734942360554, 55688895086902, 654979037799338, 7703431389461878, 90602845842996074, 1065612078710837686, 12533043918183643946

Offset: 0

Philippe Deléham, Dec 31 2007

Harvey P. Dale, Table of n, a(n) for n = 0..934
Index entries for linear recurrences with constant coefficients, signature (7,58)

Cf. A122950.

CoefficientList[Series[(1 - 7*x^2)/(1 - 7*x - 56*x^2), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jan 21 2017 *)
LinearRecurrence[{7,56},{1,7,98},20] (* Harvey P. Dale, Dec 27 2019 *)

G.f.: (1 - 7*x^2)/(1 - 7*x - 56*x^2).

a(n) = Sum_{k=0..n} A122950(n,k)*7^k.

a(8) and a(13) corrected by Georg Fischer, May 03 2019

A133680 a(n)=8a(n-1)+72a(n-2) for n>=3, a(0)=1, a(1)=8, a(2)=128 .

Original entry on oeis.org

1, 8, 128, 1600, 22016, 291328, 3915776, 52301824, 700350464, 9368535040, 125373513728, 1677522632704, 22447074050048, 300358221955072, 4019055107244032, 53778232838717440, 719597830431309824

Offset: 0

Philippe Deléham, Dec 31 2007

Harvey P. Dale, Table of n, a(n) for n = 0..887
Index entries for linear recurrences with constant coefficients, signature (8,72).

LinearRecurrence[{8,72},{1,8,128},20] (* Harvey P. Dale, Sep 15 2015 *)

G.f.: (1-8*x^2)/(1-8*x-72*x^2) . a(n) = Sum_{k, 0<=k<=n}A122950(n,k)*8^k .

A167704 a(0)=1, a(1)=0, a(2)=2, a(3)=1, a(n)=a(n-2)+a(n-3)+a(n-4) for n>3.

Original entry on oeis.org

1, 0, 2, 1, 3, 3, 6, 7, 12, 16, 25, 35, 53, 76, 113, 164, 242, 353, 519, 759, 1114, 1631, 2392, 3504, 5137, 7527, 11033, 16168, 23697, 34728, 50898, 74593, 109323, 160219, 234814, 344135, 504356, 739168, 1083305, 1587659, 2326829, 3410132, 4997793, 7324620

Offset: 0

Philippe Deléham, Nov 10 2009

nonn

Diagonal sums of triangle in A122950.

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

LinearRecurrence[{0,1,1,1},{1,0,2,1},50] (* Harvey P. Dale, Aug 09 2015 *)

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

a(n) = a(n-2) + a(n-3) + a(n-4). - G. C. Greubel, Jun 20 2016

Corrected and extended by Harvey P. Dale, Aug 09 2015

A133594 a(n)=3a(n-1)+12a(n-2) for n>=3, a(0)=1, a(1)=3, a(2)=18 .

Original entry on oeis.org

1, 3, 18, 90, 486, 2538, 13446, 70794, 373734, 1970730, 10396998, 54839754, 289283238, 1525926762, 8049179142, 42458658570, 223966125414, 1181402279082, 6231800342214, 32872228375626, 173398289233446, 914661608207850

Offset: 0

Philippe Deléham, Dec 31 2007

Index entries for linear recurrences with constant coefficients, signature (3,12).

LinearRecurrence[{3,12},{1,3,18},30] (* Harvey P. Dale, Dec 20 2020 *)

G.f.: (1-3*x^2)/(1-3*x-12*x^2) . a(n) = Sum_{k, 0<=k<=n}A122950(n,k)*3^k .

A133642 a(n) = 4a(n-1)+20a(n-2) for n>=3, a(0)=1, a(1)=4, a(2)=32.

Original entry on oeis.org

1, 4, 32, 208, 1472, 10048, 69632, 479488, 3310592, 22832128, 157540352, 1086803968, 7498022912, 51728171008, 356873142272, 2462055989248, 16985686802432, 117183866994688, 808449204027392, 5577474156003328, 38478880704561152, 265465005938311168

Offset: 0

Philippe Deléham, Dec 31 2007

Index entries for linear recurrences with constant coefficients, signature (4,20).

Cf. A122950.

G.f.: (1-4*x^2)/(1-4*x-20*x^2).

a(n) = Sum_{k=0..n} A122950(n,k)*4^k.

a(15) corrected by Georg Fischer, Mar 13 2020

A133678 a(n)=6a(n-1)+42a(n-2) for n>=3, a(0)=1, a(1)=6, a(2)=72 .

Original entry on oeis.org

1, 6, 72, 684, 7128, 71496, 728352, 7372944, 74828448, 758634336, 7694600832, 78030247104, 791354717568, 8025398683776, 81389290240512, 825402486161664, 8370765107071488, 84891495061218816, 860921104864315392, 8730969421757082624, 88544502934843742208

Offset: 0

Philippe Deléham, Dec 31 2007

D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989.
D. E. Flath, Introduction to Number Theory, Wiley-Interscience, 1989.

Index entries for linear recurrences with constant coefficients, signature (6,42).

Join[{1},LinearRecurrence[{6,42},{6,72},20]] (* Harvey P. Dale, Mar 25 2015 *)

G.f.: (1-6*x^2)/(1-6*x-42*x^2) . a(n) = Sum_{k, 0<=k<=n}A122950(n,k)*6^k .

a(1)=6, a(2)=72, a(n)=6*a(n-1)+42*a (n-2). - Harvey P. Dale, Mar 25 2015

Corrected and extended by Harvey P. Dale, Mar 25 2015

A133681 a(n)=9a(n-1)+90a(n-2) for n>=3, a(0)=1, a(1)=9, a(2)=162 .

Original entry on oeis.org

1, 9, 162, 2268, 34992, 519048, 7820712, 117100728, 1757770632, 26359001208, 395430367752, 5931183418488, 88969383864072, 1334530962440568, 20018023209731592, 300269995507235448, 4504052048440962312

Offset: 0

Philippe Deléham, Dec 31 2007

G.f.: (1-9*x^2)/(1-9*x-90*x^2) . a(n) = Sum_{k, 0<=k<=n}A122950(n,k)*9^k .

A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 5, 3, 0, 0, 8, 7, 1, 0, 0, 13, 15, 4, 0, 0, 0, 21, 30, 12, 1, 0, 0, 0, 34, 58, 31, 5, 0, 0, 0, 0, 55, 109, 73, 18, 1, 0, 0, 0, 0, 89, 201, 162, 54, 6, 0, 0, 0, 0, 0, 144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 10 2012

A skew version of A122075.

			Triangle begins :
  1
  2, 0
  3, 1, 0
  5, 3, 0, 0
  8, 7, 1, 0, 0
  13, 15, 4, 0, 0, 0
  21, 30, 12, 1, 0, 0, 0
  34, 58, 31, 5, 0, 0, 0, 0
  55, 109, 73, 18, 1, 0, 0, 0, 0
  89, 201, 162, 54, 6, 0, 0, 0, 0, 0
  144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A122075, A122950, A000045, A023610, A129707

T[0, 0] := 1; T[1, 0] := 2; T[1, 1] := 0; T[n_, k_] := T[n, k] = If[n<0, 0, If[k > n, 0, T[n - 1, k] + T[n - 2, k] + T[n - 2, k - 1]]]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 19 2017 *)

G.f.: (1+x)/(1-x-(1+y)*x^2).
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.

A236076 A skewed version of triangular array A122075.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 1, 7, 8, 0, 0, 0, 4, 15, 13, 0, 0, 0, 1, 12, 30, 21, 0, 0, 0, 0, 5, 31, 58, 34, 0, 0, 0, 0, 1, 18, 73, 109, 55, 0, 0, 0, 0, 0, 6, 54, 162, 201, 89, 0, 0, 0, 0, 0, 1, 25, 145, 344, 365, 144, 0, 0, 0, 0, 0, 0, 7, 85, 361

Offset: 0

Philippe Deléham, Jan 19 2014

Triangle T(n,k), 0 <= k <= n, read by rows, given by (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Subtriangle of the triangle A122950.

			Triangle begins:
  1;
  0,  2;
  0,  1,  3;
  0,  0,  3,  5;
  0,  0,  1,  7,  8;
  0,  0,  0,  4, 15, 13;
  0,  0,  0,  1, 12, 30, 21;
  0,  0,  0,  0,  5, 31, 58, 34;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
H. Fuks and J.M.G. Soto, Exponential convergence to equilibrium in cellular automata asymptotically emulating identity, arXiv:1306.1189 [nlin.CG], 2013.

Cf. variant: A055830, A122075, A122950, A208337.
Cf. A167704 (diagonal sums), A000079 (row sums).
Cf. A111006.

a236076 n k = a236076_tabl !! n !! k
a236076_row n = a236076_tabl !! n
a236076_tabl = [1] : [0, 2] : f [1] [0, 2] where
   f us vs = ws : f vs ws where
     ws = [0] ++ zipWith (+) (zipWith (+) ([0] ++ us) (us ++ [0])) vs
-- Reinhard Zumkeller, Jan 19 2014

T[n_, k_]:= If[k<0 || k>n, 0, If[n==0 && k==0, 1, If[k==0, 0, If[n==1 && k==1, 2, T[n-1, k-1] + T[n-2, k-1] + T[n-2, k-2]]]]]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 21 2019 *)

{T(n,k) = if(k<0 || k>n, 0, if(n==0 && k==0, 1, if(k==0, 0, if(n==1 && k==1, 2, T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2) ))))}; \\ G. C. Greubel, May 21 2019

def T(n, k):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    elif (k==0): return 0
    elif (n==1 and k==1): return 2
    else: return T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 21 2019

G.f.: (1+x*y)/(1 - x*y - x^2*y - x^2*y^2).
T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0)=1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k < 0 or if k > n.
Sum_{k=0..n} T(n,k) = 2^n = A000079(n).
Sum_{n>=k} T(n,k) = A078057(k) = A001333(k+1).
T(n,n) = Fibonacci(n+2) = A000045(n+2).
T(n+1,n) = A023610(n-1), n >= 1.
T(n+2,n) = A129707(n).

cp's OEIS Frontend

A133679 a(n) = 7*a(n-1) + 56*a(n-2) for n>=3, a(0)=1, a(1)=7, a(2)=98.

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A133680 a(n)=8*a(n-1)+72*a(n-2) for n>=3, a(0)=1, a(1)=8, a(2)=128 .

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A167704 a(0)=1, a(1)=0, a(2)=2, a(3)=1, a(n)=a(n-2)+a(n-3)+a(n-4) for n>3.

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A133594 a(n)=3*a(n-1)+12*a(n-2) for n>=3, a(0)=1, a(1)=3, a(2)=18 .

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A133642 a(n) = 4*a(n-1)+20*a(n-2) for n>=3, a(0)=1, a(1)=4, a(2)=32.

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A133678 a(n)=6*a(n-1)+42*a(n-2) for n>=3, a(0)=1, a(1)=6, a(2)=72 .

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A133681 a(n)=9*a(n-1)+90*a(n-2) for n>=3, a(0)=1, a(1)=9, a(2)=162 .

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A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A236076 A skewed version of triangular array A122075.

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A133679 a(n) = 7a(n-1) + 56a(n-2) for n>=3, a(0)=1, a(1)=7, a(2)=98.

A133680 a(n)=8a(n-1)+72a(n-2) for n>=3, a(0)=1, a(1)=8, a(2)=128 .

A133594 a(n)=3a(n-1)+12a(n-2) for n>=3, a(0)=1, a(1)=3, a(2)=18 .

A133642 a(n) = 4a(n-1)+20a(n-2) for n>=3, a(0)=1, a(1)=4, a(2)=32.

A133678 a(n)=6a(n-1)+42a(n-2) for n>=3, a(0)=1, a(1)=6, a(2)=72 .

A133681 a(n)=9a(n-1)+90a(n-2) for n>=3, a(0)=1, a(1)=9, a(2)=162 .