cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A213586 Antidiagonal sums of the convolution array A213584.

Original entry on oeis.org

1, 6, 20, 51, 112, 224, 421, 758, 1324, 2263, 3808, 6336, 10457, 17158, 28036, 45675, 74256, 120544, 195485, 316790, 513116, 830831, 1344960, 2176896, 3523057, 5701254, 9225716, 14928483, 24155824, 39086048, 63243733, 102331766
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2012

Keywords

Comments

a(n) is the number of bit strings of length n+5 with the pattern 01 at least thrice, and without the pattern 110, see example. - John M. Campbell, Jan 25 2013

Examples

			From _John M. Campbell_, Jan 25 2013: (Start)
There are a(3) = 20 bit strings of length 3+5 with the pattern 01 at least thrice, and without the pattern 110:
00010101, 00100101, 00101001, 00101010, 00101011,
01000101, 01001001, 01001010, 01001011, 01010001,
01010010, 01010011, 01010100, 01010101, 01010111,
10010101, 10100101, 10101001, 10101010, 10101011.
(End)

Links

Crossrefs

Cf. A213500, A213584, A213590.

Programs

  • GAP
    List([1..40], n-> Fibonacci(n+8) -(21+10*n+2*n^2)) # G. C. Greubel, Jul 06 2019
  • Magma
    [Fibonacci(n+8) -(21+10*n+2*n^2): n in [1..40]]; // G. C. Greubel, Jul 06 2019
  • Mathematica
    (See A213584.)
With[{F = Fibonacci}, Table[F[n+8] -(21+10*n+2*n^2), {n,40}]] (* G. C. Greubel, Jul 06 2019 *)
  • PARI
    vector(40, n, fibonacci(n+8) -(21+10*n+2*n^2)) \\ G. C. Greubel, Jul 06 2019
  • Sage
    [fibonacci(n+8) -(21+10*n+2*n^2) for n in (1..40)] # G. C. Greubel, Jul 06 2019

Formula

a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
G.f.: x*(1 + 2*x + x^2)/((1 - x - x^2)*(1 - x)^3).
a(n) = Fibonacci(n+8) - (21 + 10*n + 2*n^2). - G. C. Greubel, Jul 06 2019

A213585 Principal diagonal of the convolution array A213584.

Original entry on oeis.org

1, 7, 22, 54, 116, 232, 443, 821, 1490, 2664, 4710, 8256, 14373, 24883, 42878, 73594, 125880, 214664, 365087, 619425, 1048666, 1771852, 2988362, 5031744, 8459401, 14201887, 23811238, 39873726, 66695420, 111440104, 186016835
Offset: 1

Views

Author

Clark Kimberling, Jun 18 2012

Keywords

Links

Crossrefs

Cf. A000045, A213500, A213584.

Programs

  • GAP
    F:=Fibonacci;; List([1..40], n-> F(n+4) +n*F(n+3) -(4*n+3)) # G. C. Greubel, Jul 08 2019
  • Magma
    F:=Fibonacci; [F(n+4) +n*F(n+3) -(4*n+3): n in [1..40]]; // G. C. Greubel, Jul 08 2019
  • Mathematica
    (* First program *)
b[n_]:= Fibonacci[n+1]; c[n_]:= n;
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213584 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
d = Table[T[n, n], {n, 1, 40}] (* A213585 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A213586 *)
(* Second program *)
Table[Fibonacci[n+4] + n*Fibonacci[n+3] -4*n-3, {n, 40}] (* G. C. Greubel, Jul 08 2019 *)
  • PARI
    vector(40, n, f=fibonacci; f(n+4) +n*f(n+3) -(4*n+3)) \\ G. C. Greubel, Jul 08 2019
  • Sage
    f=fibonacci; [f(n+4) +n*f(n+3) -(4*n+3) for n in (1..40)] # G. C. Greubel, Jul 08 2019

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) + a(n-5).
G.f.: x*(1 + 3*x - 2*x^2 - 4*x^3 - 2*x^4)/(1 - 2*x + x^3)^2.
a(n) = Fibonacci(n+4) + n*Fibonacci(n+3) - (4*n + 3). - G. C. Greubel, Jul 08 2019

A213500 Rectangular array T(n,k): (row n) = b**c, where b(h) = h, c(h) = h + n - 1, n >= 1, h >= 1, and ** = convolution.

Original entry on oeis.org

1, 4, 2, 10, 7, 3, 20, 16, 10, 4, 35, 30, 22, 13, 5, 56, 50, 40, 28, 16, 6, 84, 77, 65, 50, 34, 19, 7, 120, 112, 98, 80, 60, 40, 22, 8, 165, 156, 140, 119, 95, 70, 46, 25, 9, 220, 210, 192, 168, 140, 110, 80, 52, 28, 10, 286, 275, 255, 228, 196, 161, 125, 90
Offset: 1

Views

Author

Clark Kimberling, Jun 14 2012

Keywords

Comments

Principal diagonal: A002412.
Antidiagonal sums: A002415.
Row 1: (1,2,3,...)**(1,2,3,...) = A000292.
Row 2: (1,2,3,...)**(2,3,4,...) = A005581.
Row 3: (1,2,3,...)**(3,4,5,...) = A006503.
Row 4: (1,2,3,...)**(4,5,6,...) = A060488.
Row 5: (1,2,3,...)**(5,6,7,...) = A096941.
Row 6: (1,2,3,...)**(6,7,8,...) = A096957.
...
In general, the convolution of two infinite sequences is defined from the convolution of two n-tuples: let X(n) = (x(1),...,x(n)) and Y(n)=(y(1),...,y(n)); then X(n)**Y(n) = x(1)*y(n)+x(2)*y(n-1)+...+x(n)*y(1); this sum is the n-th term in the convolution of infinite sequences:(x(1),...,x(n),...)**(y(1),...,y(n),...), for all n>=1.
...
In the following guide to related arrays and sequences, row n of each array T(n,k) is the convolution b**c of the sequences b(h) and c(h+n-1). The principal diagonal is given by T(n,n) and the n-th antidiagonal sum by S(n). In some cases, T(n,n) or S(n) differs in offset from the listed sequence.
b(h)........ c(h)........ T(n,k) .. T(n,n) .. S(n)
h .......... h .......... A213500 . A002412 . A002415
h .......... h^2 ........ A212891 . A213436 . A024166
h^2 ........ h .......... A213503 . A117066 . A033455
h^2 ........ h^2 ........ A213505 . A213546 . A213547
h .......... h*(h+1)/2 .. A213548 . A213549 . A051836
h*(h+1)/2 .. h .......... A213550 . A002418 . A005585
h*(h+1)/2 .. h*(h+1)/2 .. A213551 . A213552 . A051923
h .......... h^3 ........ A213553 . A213554 . A101089
h^3 ........ h .......... A213555 . A213556 . A213547
h^3 ........ h^3 ........ A213558 . A213559 . A213560
h^2 ........ h*(h+1)/2 .. A213561 . A213562 . A213563
h*(h+1)/2 .. h^2 ........ A213564 . A213565 . A101094
2^(h-1) .... h .......... A213568 . A213569 . A047520
2^(h-1) .... h^2 ........ A213573 . A213574 . A213575
h .......... Fibo(h) .... A213576 . A213577 . A213578
Fibo(h) .... h .......... A213579 . A213580 . A053808
Fibo(h) .... Fibo(h) .... A067418 . A027991 . A067988
Fibo(h+1) .. h .......... A213584 . A213585 . A213586
Fibo(n+1) .. Fibo(h+1) .. A213587 . A213588 . A213589
h^2 ........ Fibo(h) .... A213590 . A213504 . A213557
Fibo(h) .... h^2 ........ A213566 . A213567 . A213570
h .......... -1+2^h ..... A213571 . A213572 . A213581
-1+2^h ..... h .......... A213582 . A213583 . A156928
-1+2^h ..... -1+2^h ..... A213747 . A213748 . A213749
h .......... 2*h-1 ...... A213750 . A007585 . A002417
2*h-1 ...... h .......... A213751 . A051662 . A006325
2*h-1 ...... 2*h-1 ...... A213752 . A100157 . A071238
2*h-1 ...... -1+2^h ..... A213753 . A213754 . A213755
-1+2^h ..... 2*h-1 ...... A213756 . A213757 . A213758
2^(n-1) .... 2*h-1 ...... A213762 . A213763 . A213764
2*h-1 ...... Fibo(h) .... A213765 . A213766 . A213767
Fibo(h) .... 2*h-1 ...... A213768 . A213769 . A213770
Fibo(h+1) .. 2*h-1 ...... A213774 . A213775 . A213776
Fibo(h) .... Fibo(h+1) .. A213777 . A001870 . A152881
h .......... 1+[h/2] .... A213778 . A213779 . A213780
1+[h/2] .... h .......... A213781 . A213782 . A005712
1+[h/2] .... [(h+1)/2] .. A213783 . A213759 . A213760
h .......... 3*h-2 ...... A213761 . A172073 . A002419
3*h-2 ...... h .......... A213771 . A213772 . A132117
3*h-2 ...... 3*h-2 ...... A213773 . A214092 . A213818
h .......... 3*h-1 ...... A213819 . A213820 . A153978
3*h-1 ...... h .......... A213821 . A033431 . A176060
3*h-1 ...... 3*h-1 ...... A213822 . A213823 . A213824
3*h-1 ...... 3*h-2 ...... A213825 . A213826 . A213827
3*h-2 ...... 3*h-1 ...... A213828 . A213829 . A213830
2*h-1 ...... 3*h-2 ...... A213831 . A213832 . A212560
3*h-2 ...... 2*h-1 ...... A213833 . A130748 . A213834
h .......... 4*h-3 ...... A213835 . A172078 . A051797
4*h-3 ...... h .......... A213836 . A213837 . A071238
4*h-3 ...... 2*h-1 ...... A213838 . A213839 . A213840
2*h-1 ...... 4*h-3 ...... A213841 . A213842 . A213843
2*h-1 ...... 4*h-1 ...... A213844 . A213845 . A213846
4*h-1 ...... 2*h-1 ...... A213847 . A213848 . A180324
[(h+1)/2] .. [(h+1)/2] .. A213849 . A049778 . A213850
h .......... C(2*h-2,h-1) A213853
...
Suppose that u = (u(n)) and v = (v(n)) are sequences having generating functions U(x) and V(x), respectively. Then the convolution u**v has generating function U(x)*V(x). Accordingly, if u and v are homogeneous linear recurrence sequences, then every row of the convolution array T satisfies the same homogeneous linear recurrence equation, which can be easily obtained from the denominator of U(x)*V(x). Also, every column of T has the same homogeneous linear recurrence as v.

Examples

			Northwest corner (the array is read by southwest falling antidiagonals):
  1,  4, 10, 20,  35,  56,  84, ...
  2,  7, 16, 30,  50,  77, 112, ...
  3, 10, 22, 40,  65,  98, 140, ...
  4, 13, 28, 50,  80, 119, 168, ...
  5, 16, 34, 60,  95, 140, 196, ...
  6, 19, 40, 70, 110, 161, 224, ...
T(6,1) = (1)**(6) = 6;
T(6,2) = (1,2)**(6,7) = 1*7+2*6 = 19;
T(6,3) = (1,2,3)**(6,7,8) = 1*8+2*7+3*6 = 40.

Links

Crossrefs

Cf. A000027.

Programs

  • Mathematica
    b[n_] := n; c[n_] := n
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213500 *)
  • PARI
    t(n,k) = sum(i=0, k - 1, (k - i) * (n + i));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(t(k,n - k + 1),", ");); print(););};
tabl(12) \\ Indranil Ghosh, Mar 26 2017
  • Python
    def t(n, k): return sum((k - i) * (n + i) for i in range(k))
for n in range(1, 13):
    print([t(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

Formula

T(n,k) = 4*T(n,k-1) - 6*T(n,k-2) + 4*T(n,k-3) - T(n,k-4).
T(n,k) = 2*T(n-1,k) - T(n-2,k).
G.f. for row n: x*(n - (n - 1)*x)/(1 - x)^4.

A023554 Convolution of natural numbers >= 3 and (Fib(2), Fib(3), Fib(4), ...).

Original entry on oeis.org

3, 10, 22, 43, 78, 136, 231, 386, 638, 1047, 1710, 2784, 4523, 7338, 11894, 19267, 31198, 50504, 81743, 132290, 214078, 346415, 560542, 907008, 1467603, 2374666, 3842326, 6217051, 10059438, 16276552, 26336055, 42612674, 68948798, 111561543, 180510414
Offset: 1

Views

Author

Clark Kimberling

Keywords

Comments

a(n) is the sum of row n in the triangle T(n,k) defined by: T(n,1) = T(n,n) = 2*n+1 for n>=1 and T(n,k) = 3*T(n-1,k-1) - 2*T(n-1,k) + T(n-2,k-1) for n>2, 2<=k<=n-1. - Lechoslaw Ratajczak, Nov 07 2020
Floretion Algebra Multiplication Program, FAMP code: (a(n)) = 4jesleftforcycseq[ - .25'i + .5'k - .25i' - .5j' + .5k' - .75'ii' + .75'jj' - .25'kk' + .25'jk' - .5'ki' + .25'kj' + .25e ], apart from initial terms. 4jesrightforcycseq = A022308; 2jesforcycseq(n+2) = n+2; identity: jesleft + jesright = jes; vesforcycseq was set to the constant sequence = (-1,-1,-1,-1,-1...). (Dement)

Links

Crossrefs

Cf. A000045, A213584.

Programs

  • GAP
    F:=Fibonacci; List([1..40], n-> F(n+5)+2*F(n+3)-(2*n+9)); # G. C. Greubel, Jul 08 2019
  • Magma
    F:=Fibonacci; [F(n+5)+2*F(n+3)-(2*n+9): n in [1..40]]; // G. C. Greubel, Jul 08 2019
  • Mathematica
    Table[Fibonacci[n+5] + 2*Fibonacci[n+3] -2*n-9, {n, 40}] (* G. C. Greubel, Jul 08 2019 *)
  • PARI
    Vec(x*(1+x)*(3-2*x) / ((1-x)^2*(1-x-x^2)) + O(x^60)) \\ Colin Barker, Feb 20 2017
  • PARI
    vector(40, n, f=fibonacci; f(n+5)+2*f(n+3)-(2*n+9)) \\ G. C. Greubel, Jul 08 2019
  • SageMath
    f=fibonacci; [f(n+5)+2*f(n+3)-(2*n+9) for n in (1..40)] # G. C. Greubel, Jul 08 2019

Formula

G.f.: x*(1+x)*(3-2*x) / ((1-x)^2*(1-x-x^2)).
2*(n+5) = A022308(n+4) - a(n+1) (conjectured). Note offset of A022308 is 0. - Creighton Dement, Feb 02 2005
From Colin Barker, Feb 20 2017: (Start)
a(n) = -7 + (2^(-1-n)*((1-t)^n*(-19+9*t) + (1+t)^n*(19+9*t)))/t - 2*(1+n) where t=sqrt(5).
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4) for n>4. (End)
a(n) = Fibonacci(n+5) + 2*Fibonacci(n+3) - (2*n + 9). - G. C. Greubel, Jul 08 2019
a(n) = a(n-1) + a(n-2) + 2*n + 3 for n>2. - Lechoslaw Ratajczak, Nov 07 2020
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