A078008 -id:A078008 - cp's OEIS frontend

A101615 Number of representations of n as a sum of the Jacobsthal numbers A078008 (2 is allowed twice as a part).

1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 2

Offset: 0

Paul Barry, Dec 09 2004

Partial sums are A101616.

Cf. A026465.

G.f.: Product{k>=0, 1+x^A078008(k)}/2

A102564 A000120(A078008(n))-A078008(A000120(n)).

Original entry on oeis.org

0, 0, 1, -1, 2, 0, 1, 1, 4, 2, 3, 3, 4, 4, 5, 1, 8, 6, 7, 7, 8, 8, 9, 5, 10, 10, 11, 7, 12, 8, 9, 5, 16, 14, 15, 15, 16, 16, 17, 13, 18, 18, 19, 15, 20, 16, 17, 13, 22, 22, 23, 19, 24, 20, 21, 17, 26, 22, 23, 19, 24, 20, 21, 9, 32, 30, 31, 31, 32, 32, 33, 29, 34, 34, 35, 31, 36, 32, 33, 29, 38, 38, 39, 35, 40, 36, 37, 33, 42, 38, 39, 35, 40, 36

Offset: 0

Paul Barry, Jan 14 2005

It would appear that a(2^(n+1))=2^n=A000079(n); a(2^(n+1)-1)+1=A078008(n); a(2^(n+1)+1)+1=2^n-1; a(4n+1)-a(2n+1)=n; a(8n+1)-a(4n+1)=2n.

Cf. A102563, 102565.

A160760 Triangle read by rows, binomial transform of an infinite lower triangular Toeplitz matrix with A078008 in every column.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 9, 5, 3, 1, 27, 14, 8, 4, 1, 81, 41, 22, 12, 5, 1, 243, 122, 63, 34, 17, 6, 1, 729, 365, 185, 97, 51, 23, 7, 1, 2187, 1094, 550, 282, 148, 74, 30, 8, 1, 6561, 3281, 1644, 832, 430, 222, 104, 38, 9, 1

Offset: 0

Gary W. Adamson, May 25 2009

Row sums = A025192: (1, 2, 6, 18, 54, 162, 486, 1458,...).

A triangle formed like Pascal's triangle, but with 3^n for n>=0 on the left border instead of 1. - Boris Putievskiy, Aug 19 2013

			First few rows of the triangle =
     1;
     1,    1;
     3,    2,    1;
     9,    5,    3,   1;
    27,   14,    8,   4,   1;
    81,   41,   22,  12,   5,   1;
   243,  122,   63,  34,  17,   6,   1;
   729,  365,  185,  97,  51,  23,   7,  1;
  2187, 1094,  550, 282, 148,  74,  30,  8, 1;
  6561, 3281, 1644, 832, 430, 222, 104, 38, 9, 1;
...

Cf. A078008, A025192.

A007318 * an infinite lower triangular Toeplitz matrix with A078008 in every column: (1, 0, 2, 2, 6, 10, 22, 42, 86,...).

Closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 19 2013

T(7,4) corrected by Georg Fischer, Oct 08 2021

A091598 Triangle read by rows: T(n,0) = A078008(n), T(n,m) = T(n-1,m-1) + T(n-1,m).

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 2, 3, 2, 1, 6, 5, 5, 3, 1, 10, 11, 10, 8, 4, 1, 22, 21, 21, 18, 12, 5, 1, 42, 43, 42, 39, 30, 17, 6, 1, 86, 85, 85, 81, 69, 47, 23, 7, 1, 170, 171, 170, 166, 150, 116, 70, 30, 8, 1, 342, 341, 341, 336, 316, 266, 186, 100, 38, 9, 1, 682, 683, 682, 677, 652, 582, 452, 286, 138, 47, 10, 1

Offset: 0

Paul Barry, Jan 23 2004

A Jacobsthal-Pascal triangle.

			Triangle starts as:
   1;
   0,  1;
   2,  1,  1;
   2,  3,  2,  1;
   6,  5,  5,  3,  1;
  10, 11, 10,  8,  4,  1;
  22, 21, 21, 18, 12,  5, 1;
  42, 43, 42, 39, 30, 17, 6, 1; ...

G. C. Greubel, Rows n = 0..20 of triangle, flattened

Columns include A078008, A001045, A000975, A011377. Row sums give A084219.

Cf. A091597.

T[n_, k_]:= If[k==0, (2^n + 2*(-1)^n)/3, If[k<0 || k>n, 0, T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 04 2019 *)

{T(n,k) = if(k==0, (2^n + 2*(-1)^n)/3, if(k<0 || k>n, 0, T(n-1,k-1) + T(n-1,k)))}; \\ G. C. Greubel, Jun 04 2019

def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return (2^n + 2*(-1)^n)/3
    else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jun 04 2019

k-th column has e.g.f. ((1-x)/(1-x-x^2))*(x/(1-x))^k.

A092900 A Jacobsthal sequence (A078008) to base 4.

Original entry on oeis.org

1, 0, 2, 2, 12, 22, 112, 222, 1112, 2222, 11112, 22222, 111112, 222222, 1111112, 2222222, 11111112, 22222222, 111111112, 222222222, 1111111112, 2222222222, 11111111112, 22222222222, 111111111112, 222222222222, 1111111111112

Offset: 0

Paul Barry, Mar 12 2004

			a(8)= 1112 because A078008(8) = 86 (in base 10) = 64 + 16 + 4 + 2 = 1*(4^3) + 1*(4^2) + 1*(4^1) + 2.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,11,0,-10).

Cf. A081857.

Vec((1-9*x^2+2*x^3)/((1-x)*(1+x)*(1-10*x^2)) + O(x^30)) \\ Colin Barker, Apr 01 2016

For n > 0, a(2*n+1) is represented as a string of n 2's and a(2*n) as a string of (n-1) 1's followed by a 2.
From Colin Barker, Apr 01 2016: (Start)
a(n) = (6+10*(-1)^n+10^(1/2*(-1+n))*(2-2*(-1)^n+sqrt(10)+(-1)^n*sqrt(10)))/18.
a(n) = (10^(n/2)+8)/9 for n even.
a(n) = (2^((n+1)/2)*5^((n-1)/2)-2)/9 for n odd.
a(n) = 11*a(n-2)-10*a(n-4) for n>3.
G.f.: (1-9*x^2+2*x^3) / ((1-x)*(1+x)*(1-10*x^2)).
(End)

A113678 Sequence array for A078008.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 2, 2, 0, 1, 6, 2, 2, 0, 1, 10, 6, 2, 2, 0, 1, 22, 10, 6, 2, 2, 0, 1, 42, 22, 10, 6, 2, 2, 0, 1, 86, 42, 22, 10, 6, 2, 2, 0, 1, 170, 86, 42, 22, 10, 6, 2, 2, 0, 1, 342, 170, 86, 42, 22, 10, 6, 2, 2, 0, 1, 682, 342, 170, 86, 42, 22, 10, 6, 2, 2, 0, 1, 1366, 682, 342, 170, 86

Offset: 0

Paul Barry, Nov 04 2005

Row sums are A001045(n+1). Diagonal sums are A053088. Inverse is A113680.

			Triangle begins
1;
0, 1;
2, 0, 1;
2, 2, 0, 1;
6, 2, 2, 0, 1;
10, 6, 2, 2, 0, 1;
22, 10, 6, 2, 2, 0, 1;

Riordan array ((1-x)/(1-x-2x^2), x); Number triangle T(n, k)=if(k<=n, (2^(n-k)+2(-1)^(n-k))/3, 0); T(n, k)=sum{i=0..n, C(n-i, k)C(k, n-i)(2^i+2(-1)^i)/3}.

A128210 Number triangle T(n,k) = (-1)^(n-k)[k<=n]Product_{i=k+1..n} Sum_{j=0..i-1} A078008(j-1).

Original entry on oeis.org

1, -1, 1, 1, -1, 1, -3, 3, -3, 1, 15, -15, 15, -5, 1, -165, 165, -165, 55, -11, 1, 3465, -3465, 3465, -1155, 231, -21, 1, -148995, 148995, -148995, 49665, -9933, 903, -43, 1, 12664575, -12664575, 12664575

Offset: 0

Paul Barry, Feb 19 2007

Inverse is A128208. Subdiagonals include -A001045(n+1) and A084175(n+1).

			Triangle begins:
        1;
       -1,      1;
        1,     -1,       1;
       -3,      3,      -3,     1;
       15,    -15,      15,    -5,     1;
     -165,    165,    -165,    55,   -11,   1;
     3465,  -3465,    3465, -1155,   231, -21,   1;
  -148995, 148995, -148995, 49665, -9933, 903, -43, 1;
  ...

A156550 a(n) = 5(-1)^nA078008(n).

Original entry on oeis.org

5, 0, 10, -10, 30, -50, 110, -210, 430, -850, 1710, -3410, 6830, -13650, 27310, -54610, 109230, -218450, 436910, -873810, 1747630, -3495250, 6990510, -13981010, 27962030, -55924050, 111848110, -223696210, 447392430, -894784850, 1789569710, -3579139410

Offset: 0

Paul Curtz, Feb 09 2009

Index entries for linear recurrences with constant coefficients, signature (-1,2).

Cf. A001045, A078008, A140966.

LinearRecurrence[{-1, 2}, {5, 0}, 32] (* Robert P. P. McKone, Jan 31 2021 *)

a(n) = A140966(n)+A140966(n+2).
a(n) = 3*A140966(n)-A140966(n+1).
a(n+1) = 10*(-1)^n*A001045(n).
G.f.: 5*(1+x)/(1+x-2*x^2). - R. J. Mathar, Feb 23 2009
a(n) = (5*(2+(-2)^n))/3. - Colin Barker, Jun 10 2012
a(n) = -a(n-1) + 2*a(n-2) for n > 1. - Klaus Purath, Jan 30 2021
E.g.f.: 5*exp(-2*x)*(1 + 2*exp(3*x))/3. - Stefano Spezia, Jan 30 2021

Edited and extended by R. J. Mathar Feb 23 2009

A160756 Triangle read by rows, infinite lower triangular Toeplitz matrix with A078008 in every column convolved with A001333.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 2, 2, 0, 3, 6, 2, 2, 0, 7, 10, 6, 2, 6, 0, 17, 22, 10, 6, 6, 14, 0, 41, 42, 22, 10, 18, 14, 34, 0, 99, 86, 42, 22, 30, 42, 34, 82, 0, 239, 170, 86, 42, 66, 70, 102, 82, 198, 0, 577

Offset: 0

Gary W. Adamson, May 25 2009

Row sums = A001333: (1, 1, 3, 7, 17, 41,...). Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
0, 1;
2, 0, 1;
2, 2, 0, 3;
6, 2, 2, 0, 7;
10, 6, 2, 6, 0, 17;
22, 10, 6, 6, 14, 0, 41;
42, 22, 10, 18, 14, 34, 0, 99;
86, 42, 22, 30, 42, 34, 82, 0, 239;
170, 86, 42, 66, 70, 102, 82, 198, 0, 577;
...
Example: row 4 = (6, 2, 2, 0, 7) = (6, 2, 2, 0, 1) * (1, 1, 1, 3, 7).

Cf. A078008, A001333

Let M = an infinite lower triangular Toeplitz matrix with A078008 (1, 0, 2, 2, 6, 10, 22, 42, 86, 170,...). Let Q = the eigensequence of that triangle prefaced with a 1: (1, 1, 1, 3, 7, 17,...) where A001333 = (1, 1, 3, 7, 17,...). The triangle = M * Q.

A091085 a(n) = mod(A078008(n),10).

Original entry on oeis.org

1, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6, 0, 2, 2, 6

Offset: 0

Paul Barry, Dec 18 2003

A078008(0), followed by A078008(1), A078008(2), A078008(3), A078008(4) repeating.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A078008.

LinearRecurrence[{0, 0, 0, 1}, {1, 0, 2, 2, 6}, 105] (* Georg Fischer, May 15 2019 *)

G.f.: (1+2x^2+2x^3+5x^4)/(1-x^4).
E.g.f.: 2cos(x)-sin(x)+3exp(-x)/2+5exp(x)/2-5.
a(n) = 2cos(Pi*n/2)-sin(Pi*n/2)+3(-1)^n/2+5/2-5*0^n.

Terms a(75) ff. corrected by Georg Fischer, May 15 2019

cp's OEIS Frontend

A101615 Number of representations of n as a sum of the Jacobsthal numbers A078008 (2 is allowed twice as a part).

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A102564 A000120(A078008(n))-A078008(A000120(n)).

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A160760 Triangle read by rows, binomial transform of an infinite lower triangular Toeplitz matrix with A078008 in every column.

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A091598 Triangle read by rows: T(n,0) = A078008(n), T(n,m) = T(n-1,m-1) + T(n-1,m).

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A092900 A Jacobsthal sequence (A078008) to base 4.

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A113678 Sequence array for A078008.

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A128210 Number triangle T(n,k) = (-1)^(n-k)*[k<=n]*Product_{i=k+1..n} Sum_{j=0..i-1} A078008(j-1).

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A156550 a(n) = 5*(-1)^n*A078008(n).

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A160756 Triangle read by rows, infinite lower triangular Toeplitz matrix with A078008 in every column convolved with A001333.

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A091085 a(n) = mod(A078008(n),10).

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A128210 Number triangle T(n,k) = (-1)^(n-k)[k<=n]Product_{i=k+1..n} Sum_{j=0..i-1} A078008(j-1).

A156550 a(n) = 5(-1)^nA078008(n).