A131269 -id:A131269 - cp's OEIS frontend

A131300 a(n) = 3a(n-1) - 2a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=7.

1, 2, 3, 7, 14, 27, 49, 86, 147, 247, 410, 675, 1105, 1802, 2931, 4759, 7718, 12507, 20257, 32798, 53091, 85927, 139058, 225027, 364129, 589202, 953379, 1542631, 2496062, 4038747, 6534865, 10573670, 17108595, 27682327, 44790986, 72473379, 117264433, 189737882

Offset: 0

Gary W. Adamson, Jun 27 2007

Row sums of A131299 and A131301.

a(n)/a(n-1) tends to phi.

			a(4) = 14 = (1 + 1 + 7 + 4 + 1).

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

Cf. A131269, A131299, A131301.

/* By first comment: */ [&+[3*Binomial(n-Floor((k+1)/2), Floor(k/2))-2: k in [0..n]]: n in [0..37]]; // Bruno Berselli, May 03 2012

seq(add(3*binomial(floor((n+k)/2),k)-2,k=0..n),n=0..50); # Nathaniel Johnston, Jun 29 2011

LinearRecurrence[{3, -2, -1, 1}, {1, 2, 3, 7}, 38] (* Bruno Berselli, May 03 2012 *)

makelist(expand(3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1)), n, 0, 37); /* Bruno Berselli, May 03 2012 */

Vec((1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2)+O(x^38)) \\ Bruno Berselli, May 03 2012

From Bruno Berselli, May 03 2012: (Start)
G.f.: (1-x-x^2+3*x^3)/((1-x-x^2)*(1-x)^2).
a(n) = (3*A131269(n)-n-1)/2.
a(n) = 3*((1+sqrt(5))^(n+2)-(1-sqrt(5))^(n+2))/(2^(n+2)*sqrt(5))-2*(n+1). (End)
a(n) = 3*A000045(n+2)-2*(n+1). - R. J. Mathar, Mar 24 2018

Terms after a(9) from Nathaniel Johnston, Jun 29 2011

New definition from Bruno Berselli, May 03 2012

A131268 Triangle read by rows: T(n,k) = 2*binomial(n-floor((k+1)/2),floor(k/2)) - 1, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 3, 1, 1, 1, 7, 5, 5, 1, 1, 1, 9, 7, 11, 5, 1, 1, 1, 11, 9, 19, 11, 7, 1, 1, 1, 13, 11, 29, 19, 19, 7, 1, 1, 1, 15, 13, 41, 29, 39, 19, 9, 1, 1, 1, 17, 15, 55, 41, 69, 39, 29, 9, 1, 1, 1, 19, 17, 71, 55, 111, 69, 69, 29, 11, 1, 1, 1, 21, 19, 89, 71, 167, 111, 139, 69, 41, 11, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums are in A131269. Reversal = triangle A131270.

			Triangle begins:
1;
1, 1;
1, 1,  1;
1, 1,  3,  1;
1, 1,  5,  3,  1;
1, 1,  7,  5,  5,  1;
1, 1,  9,  7, 11,  5,   1;
1, 1, 11,  9, 19, 11,   7,   1;
1, 1, 13, 11, 29, 19,  19,   7,   1;
1, 1, 15, 13, 41, 29,  39,  19,   9,  1;
1, 1, 17, 15, 55, 41,  69,  39,  29,  9,  1;
1, 1, 19, 17, 71, 55, 111,  69,  69, 29, 11,  1;
1, 1, 21, 19, 89, 71, 167, 111, 139, 69, 41, 11, 1;
...

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

Cf. A065941, A000012, A131269, A131270.

[2*Binomial(n-Floor((k+1)/2), Floor(k/2))-1: k in [0..n], n in [0..14]]; // Bruno Berselli, May 03 2012

T := proc (n, k) options operator, arrow; 2*binomial(n-floor((1/2)*k+1/2), floor((1/2)*k))-1 end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form. - Emeric Deutsch, Jul 15 2007

Table[2*Binomial[n -Floor[(k+1)/2], Floor[k/2]] -1, {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 10 2019 *)

T(n,k) = 2*binomial(n- (k+1)\2, k\2) -1; \\ G. C. Greubel, Jul 10 2019

[[2*binomial(n -floor((k+1)/2), floor(k/2)) -1 for k in (0..n)] for n in (0..14)] # G. C. Greubel, Jul 10 2019

Equals 2*A065941 - A000012, where A065941 = Pascal's triangle with repeated columns; and A000012 = (1; 1,1; 1,1,1;...) as an infinite lower triangular matrix.

More terms from Emeric Deutsch, Jul 15 2007

A131270 Triangle T(n,k) = 2*A046854(n,k) - 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 5, 1, 1, 1, 5, 5, 7, 1, 1, 1, 5, 11, 7, 9, 1, 1, 1, 7, 11, 19, 9, 11, 1, 1, 1, 7, 19, 19, 29, 11, 13, 1, 1, 1, 9, 19, 39, 29, 41, 13, 15, 1, 1, 1, 9, 29, 39, 69, 41, 55, 15, 17, 1, 1, 1, 11, 29, 69, 69, 111, 55, 71, 17, 19, 1, 1

Offset: 0

Gary W. Adamson, Jun 23 2007

Row sums = A131269: {1, 2, 3, 6, 11, 20, 35, 60, 101, 168, ...}.

			First few rows of the triangle:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  1,  1;
  1,  3,  5,  1,  1;
  1,  5,  5,  7,  1,  1;
  1,  5, 11,  7,  9,  1,  1;
  1,  7, 11, 19,  9, 11,  1,  1;
  ...

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

Cf. A046854, A065941, A000012, A131268, A131269.

[[2*Binomial(Floor((n+k)/2), k) -1: k in [0..n]]:n in [0..12]]; // G. C. Greubel, Jul 09 2019

Table[2*Binomial[Floor[(n+k)/2], k] - 1, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 09 2019 *)

T(n,k) = 2*binomial((n+k)\2, k)-1; \\ G. C. Greubel, Jul 09 2019

[[2*binomial(floor((n+k)/2), k) -1 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 09 2019

T(n,k) = 2*A046854(n,k) - 1.

Reversed triangle of A131268.

A210728 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 7, 14, 27, 48, 83, 140, 233, 384, 629, 1026, 1669, 2710, 4395, 7122, 11535, 18676, 30231, 48928, 79181, 128132, 207337, 335494, 542857, 878378, 1421263, 2299670, 3720963, 6020664, 9741659, 15762356, 25504049, 41266440, 66770525, 108037002, 174807565

Offset: 0

Alex Ratushnyak, May 10 2012

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

Cf. A065220: a(n)=a(n-1)+a(n-2)+n-5, a(0)=1,a(1)=2 (except first 2 terms).
Cf. A168043: a(n)=a(n-1)+a(n-2)+n-3, a(0)=1,a(1)=2 (except first 2 terms).
Cf. A131269: a(n)=a(n-1)+a(n-2)+n-2, a(0)=1,a(1)=2.
Cf. A000126: a(n)=a(n-1)+a(n-2)+n-1, a(0)=1,a(1)=2.
Cf. A104161: a(n)=a(n-1)+a(n-2)+n,   a(0)=1,a(1)=2 (except the first term).
Cf. A192969: a(n)=a(n-1)+a(n-2)+n+1, a(0)=1,a(1)=2.
Cf. A210729: a(n)=a(n-1)+a(n-2)+n+3, a(0)=1,a(1)=2.

RecurrenceTable[{a[0] == 1, a[1] == 2, a[n] == a[n - 1] + a[n - 2] + n + 2}, a, {n, 36}] (* Bruno Berselli, Jun 27 2012 *)
nxt[{n_,a_,b_}]:={n+1,b,a+b+n+3}; NestList[nxt,{1,1,2},40][[;;,2]] (* Harvey P. Dale, Aug 26 2024 *)

G.f.: (1-x+3*x^2-2*x^3)/((1-x)^2*(1-x-x^2)). - Bruno Berselli, Jun 27 2012
a(n) = ((5+sqrt(5))*(1+sqrt(5))^(n+1)-(5-sqrt(5))*(1-sqrt(5))^(n+1))/(2^(n+1)*sqrt(5))-n-5. - Bruno Berselli, Jun 27 2012
a(n) = -n-5+A022112(n+1). R. J. Mathar, Jul 03 2012

A210729 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 8, 16, 31, 55, 95, 160, 266, 438, 717, 1169, 1901, 3086, 5004, 8108, 13131, 21259, 34411, 55692, 90126, 145842, 235993, 381861, 617881, 999770, 1617680, 2617480, 4235191, 6852703, 11087927, 17940664, 29028626, 46969326, 75997989, 122967353

Offset: 0

Alex Ratushnyak, May 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Cf. A065220: a(n)=a(n-1)+a(n-2)+n-5, a(0)=1,a(1)=2 (except first 2 terms).
Cf. A168043: a(n)=a(n-1)+a(n-2)+n-3, a(0)=1,a(1)=2 (except first 2 terms).
Cf. A131269: a(n)=a(n-1)+a(n-2)+n-2, a(0)=1,a(1)=2.
Cf. A000126: a(n)=a(n-1)+a(n-2)+n-1, a(0)=1,a(1)=2.
Cf. A104161: a(n)=a(n-1)+a(n-2)+n,   a(0)=1,a(1)=2 (except the first term).
Cf. A192969: a(n)=a(n-1)+a(n-2)+n+1, a(0)=1,a(1)=2.
Cf. A210728: a(n)=a(n-1)+a(n-2)+n+2, a(0)=1,a(1)=2.
Cf. A000032, A000045.

F:=Fibonacci;; List([0..40], n-> 2*F(n+3)+3*F(n+1)-n-6); # G. C. Greubel, Jul 09 2019

[3*Fibonacci(n+1)+2*Fibonacci(n+3)-n-6: n in [0..40]]; // Vincenzo Librandi, Jul 18 2013

Table[3*Fibonacci[n+1]+2*Fibonacci[n+3]-n-6,{n,0,40}] (* Vaclav Kotesovec, May 13 2012 *)

vector(40, n, n--; f=fibonacci; 2*f(n+3)+3*f(n+1)-n-6) \\ G. C. Greubel, Jul 09 2019

prpr, prev = 1,2
for n in range(2, 99):
    current = prev+prpr+n+3
    print(prpr, end=',')
    prpr = prev
    prev = current

f=fibonacci; [2*f(n+3)+3*f(n+1)-n-6 for n in (0..40)] # G. C. Greubel, Jul 09 2019

G.f.: (1-x+4*x^2-3*x^3)/((1-x-x^2)*(1-x)^2).
a(n) = 3*Fibonacci(n+1)+2*Fibonacci(n+3)-n-6. - Vaclav Kotesovec, May 13 2012
a(n) = 2*Lucas(n+2) + Fibonacci(n+1) - (n+6). - G. C. Greubel, Jul 09 2019

A131373 A046854 + A065941 - A000012.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 10, 7, 10, 3, 1, 1, 4, 11, 14, 14, 11, 4, 1, 1, 4, 16, 15, 29, 15, 16, 4, 1, 1, 5, 17, 26, 35, 35, 26, 17, 5, 1

Offset: 0

Gary W. Adamson, Jul 03 2007

Row sums = A131269: (1, 2, 3, 6, 11, 20, 35, 60,...).

			First few rows of the triangle are:
1;
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 2, 5, 2, 1;
1, 3, 6, 6, 3, 1;
1, 3, 10, 7, 10, 3, 1;
1, 4, 11, 14, 14, 11, 4, 1;
1, 4, 16, 15, 29, 15, 16, 4, 1;
...

Cf. A056854, A065941, A000012, A131269.

A046854 + A065941 - A000012 as infinite lower triangular matrices.

cp's OEIS Frontend

A131300 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=7.

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A131268 Triangle read by rows: T(n,k) = 2*binomial(n-floor((k+1)/2),floor(k/2)) - 1, 0<=k<=n.

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A131270 Triangle T(n,k) = 2*A046854(n,k) - 1, read by rows.

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A210728 a(n) = a(n-1) + a(n-2) + n + 2 with n>1, a(0)=1, a(1)=2.

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A210729 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0)=1, a(1)=2.

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A131373 A046854 + A065941 - A000012.

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Formula

A131300 a(n) = 3a(n-1) - 2a(n-2) - a(n-3) + a(n-4) with n>3, a(0)=1, a(1)=2, a(2)=3, a(3)=7.