A213590 -id:A213590 - cp's OEIS frontend

A213504 Principal diagonal of the convolution array A213590.

1, 6, 35, 138, 488, 1564, 4733, 13734, 38711, 106846, 290496, 781264, 2084753, 5531846, 14619811, 38527834, 101328712, 266119228, 698218525, 1830665830, 4797572551, 12568780126, 32920653120, 86214096768, 225758326273

Offset: 1

Clark Kimberling, Jun 19 2012

Clark Kimberling, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-10,-2,15,-2,-8,0,1).

Cf. A213590, A213500.

F:=Fibonacci;; List([1..40], n-> F(2*n+6) -F(n+6) -2*n*F(n+3) -n^2*F(n+1)); # G. C. Greubel, Jul 06 2019

F:=Fibonacci; [F(2*n+6) -F(n+6) -2*n*F(n+3) -n^2*F(n+1): n in [1..40]]; // G. C. Greubel, Jul 06 2019

(* First program *)
b[n_]:= n^2; c[n_]:= Fibonacci[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}]] (* A213590 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213504 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213557 *)
(* Second program *)
With[{F = Fibonacci}, Table[F[2*n+6] -F[n+6] -2*n*F[n+3] -n^2*F[n+1], {n, 40}]] (* G. C. Greubel, Jul 06 2019 *)

vector(40, n, my(f=fibonacci); f(2*n+6) - f(n+6) - 2*n*f(n+3) - n^2*f(n+1)) \\ G. C. Greubel, Jul 06 2019

f=fibonacci; [f(2*n+6) -f(n+6) -2*n*f(n+3) -n^2*f(n+1) for n in (1..40)] # G. C. Greubel, Jul 06 2019

a(n) = 6*a(n-1) - 10*a(n-2) - 2*a(n-3) + 15*a(n-4) - 2*a(n-5)- 8*a(n-6) + a(n-8).
G.f.: x*(1 + 9*x^2 - 10*x^3 + 7*x^4 - 2*x^5)/((1 - 3*x + x^2)*(1 - x - x^2)^3). [corrected by Georg Fischer, May 11 2019]
a(n) = Fibonacci(2*n+6) - Fibonacci(n+6) - 2*n*Fibonacci(n+3) - n^2*Fibonacci(n+1). - G. C. Greubel, Jul 06 2019

A213557 Antidiagonal sums of the convolution array A213590.

Original entry on oeis.org

1, 6, 23, 70, 184, 438, 971, 2042, 4125, 8076, 15424, 28876, 53189, 96670, 173747, 309362, 546456, 958690, 1672015, 2901170, 5011321, 8621976, 14781888, 25263000, 43053769, 73186038, 124119311, 210055582, 354806200, 598245006

Offset: 1

Clark Kimberling, Jun 19 2012

Clark Kimberling, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (5,-8,2,6,-4,-1,1).

Cf. A213590, A213500.

F:=Fibonacci;; List([1..40], n-> n*F(n+7) -2*F(n+9) +2*(n^2+10*n+ 34)); # G. C. Greubel, Jul 06 2019

F:=Fibonacci; [n*F(n+7) -2*F(n+9) +2*(n^2+10*n+34): n in [1..40]]; // G. C. Greubel, Jul 06 2019

(* First program *)
b[n_]:= n^2; c[n_]:= Fibonacci[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}]] (* A213590 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213504 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213557 *)
(* Second program *)
With[{F = Fibonacci}, Table[n*F[n+7] -2*F[n+9] +2*(n^2+10*n+34), {n,40}]] (* G. C. Greubel, Jul 06 2019 *)
LinearRecurrence[{5,-8,2,6,-4,-1,1},{1,6,23,70,184,438,971},30] (* Harvey P. Dale, Jun 04 2025 *)

vector(40, n, f=fibonacci; n*f(n+7) -2*f(n+9) +2*(n^2+10*n+34)) \\ G. C. Greubel, Jul 06 2019

f=fibonacci; [n*f(n+7) -2*f(n+9) +2*(n^2+10*n+34) for n in (1..40)] # G. C. Greubel, Jul 06 2019

a(n) = 5*a(n-1) - 8*a(n-2) + 2*a(n-3) + 6*a(n-4) - 4*a(n-5) - a(n-6) + a(n-7).
G.f.: f(x)/g(x), where f(x) = x*(1 + x + x^2 + x^3) and g(x) = (1 - x)^3 (1 - x - x^2)^2.
a(n) = n*Fibonacci(n+7) - 2*Fibonacci(n+9) + 2*n^2 + 20*n + 68. - G. C. Greubel, Jul 06 2019

A213500 Rectangular array T(n,k): (row n) = bc, 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

Clark Kimberling, Jun 14 2012

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.

			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.

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000027.

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 *)

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

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

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.

A053808 Partial sums of A001891.

Original entry on oeis.org

1, 5, 15, 36, 76, 148, 273, 485, 839, 1424, 2384, 3952, 6505, 10653, 17383, 28292, 45964, 74580, 120905, 195885, 317231, 513600, 831360, 1345536, 2177521, 3523733, 5701983, 9226500, 14929324, 24156724, 39087009, 63244757, 102332855, 165578768, 267912848

Offset: 0

Barry E. Williams, Mar 27 2000

Antidiagonal sums of the convolution array A213579 and row 1 of the convolution array A213590. - Clark Kimberling, Jun 18 2012

Also number CG(n,2) of complete games with n players of 2 types. - N. J. A. Sloane, Dec 29 2012

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. Freixas and S. Kurz, The golden number and Fibonacci sequences in the design of voting structures, 2012. - From _N. J. A. Sloane_, Dec 29 2012
W. Lang, Problem B-858, Fibonacci Quarterly, 36,3 (1998) 373-374; Solution, ibid. 37,2 (1999) 183-184.
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Convolution of A000290 (squares) with A000045, n >= 1. (Fibonacci) - Wolfdieter Lang, Apr 10 2000

Right-hand column 7 of triangle A011794.

List([0..40], n-> Fibonacci(n+8) - (n^2 +8*n+20)); # G. C. Greubel, Jul 06 2019

[Fibonacci(n+8) - (n^2+8*n+20): n in [0..40]]; // G. C. Greubel, Jul 06 2019

Table[Fibonacci[n+8] -(n^2 +8*n+20), {n,0,40}] (* G. C. Greubel, Jul 06 2019 *)
LinearRecurrence[{4,-5,1,2,-1},{1,5,15,36,76},40] (* Harvey P. Dale, Apr 14 2022 *)

vector(40, n, n--; fibonacci(n+8) - (n^2 +8*n+20)) \\ G. C. Greubel, Jul 06 2019

[fibonacci(n+8) - (n^2 +8*n+20) for n in (0..20)] # G. C. Greubel, Jul 06 2019

a(n) = a(n-1) + a(n-2) + (n+1)^2, a(-n)=0.
G.f.: (1+x)/((1-x-x^2)*(1-x)^3).
a(n) = Fibonacci(n+6) - (n^2 + 4*n + 8), n >= 2 (see p. 184 of FQ reference).
a(n-2) = Sum_{i=0..n} Fibonacci(i)*(n-i)^2. - Benoit Cloitre, Mar 06 2004

A280154 a(n) = 5*Lucas(n).

Original entry on oeis.org

10, 5, 15, 20, 35, 55, 90, 145, 235, 380, 615, 995, 1610, 2605, 4215, 6820, 11035, 17855, 28890, 46745, 75635, 122380, 198015, 320395, 518410, 838805, 1357215, 2196020, 3553235, 5749255, 9302490, 15051745, 24354235, 39405980, 63760215, 103166195, 166926410, 270092605, 437019015

Offset: 0

Bruno Berselli, Dec 27 2016

Fibonacci sequence beginning 10, 5.
After 5, the sequence provides the 3rd column of the rectangular array in A213590.
After 5, all terms belong to A191921 because a(n) = Lucas(n+4) - 3*Lucas(n-1).
From G. C. Greubel, Dec 27 2016: (Start)
{a(n) mod 3} yields (1,2,0,2,2,1,0,1), repeated, and is given as A082115.
{a(n) mod 6} yields (4,5,3,2,5,1,0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1) and is given as A082117. (End)

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).

Subsequence of A084176.
Cf. A022088: 5*Fibonacci(n).
Cf. A022359: Lucas(n+5) + Lucas(n-5).
Cf. A000032, A000045, A191921, A213590.
Cf. sequences with formula Fibonacci(n+k) + Fibonacci(n-k): A006355 (k=0, without the initial 1), A000032 (k=1), A022086 (k=2), A022112 (k=3, with an initial 4), A022090 (k=4), this sequence (k=5), A022352 (k=6).

Magma
```
[5*Lucas(n): n in [0..40]];
```

F := n -> combinat:-fibonacci(n):
seq(F(n+5) + F(n-5), n=0..38); # Peter Luschny, Dec 29 2016

Mathematica
```
Table[5 LucasL[n], {n, 0, 40}]
```

vector(40, n, n--; fibonacci(n+5)+fibonacci(n-5))

def A280154():
    x, y = 10, 5
    while True:
        yield x
        x, y = y, x + y
a = A280154(); print([next(a) for  in range(39)]) # _Peter Luschny, Dec 29 2016

G.f.: 5*(2 - x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2) for n>1.
a(n) = Fibonacci(n+5) + Fibonacci(n-5), with Fibonacci(-k) = -(-1)^k*Fibonacci(k) for the negative indices.

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

Clark Kimberling, Jun 18 2012

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

			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)

Clark Kimberling, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Cf. A213500, A213584, A213590.

List([1..40], n-> Fibonacci(n+8) -(21+10*n+2*n^2)) # G. C. Greubel, Jul 06 2019

[Fibonacci(n+8) -(21+10*n+2*n^2): n in [1..40]]; // G. C. Greubel, Jul 06 2019

(See A213584.)
With[{F = Fibonacci}, Table[F[n+8] -(21+10*n+2*n^2), {n,40}]] (* G. C. Greubel, Jul 06 2019 *)

vector(40, n, fibonacci(n+8) -(21+10*n+2*n^2)) \\ G. C. Greubel, Jul 06 2019

[fibonacci(n+8) -(21+10*n+2*n^2) for n in (1..40)] # G. C. Greubel, Jul 06 2019

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

cp's OEIS Frontend

A213504 Principal diagonal of the convolution array A213590.

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A213557 Antidiagonal sums of the convolution array A213590.

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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.

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A053808 Partial sums of A001891.

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A280154 a(n) = 5*Lucas(n).

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A213586 Antidiagonal sums of the convolution array A213584.

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