A054602 -id:A054602 - cp's OEIS frontend

A292022 a(n) = 4n(n^2 + 2).

12, 48, 132, 288, 540, 912, 1428, 2112, 2988, 4080, 5412, 7008, 8892, 11088, 13620, 16512, 19788, 23472, 27588, 32160, 37212, 42768, 48852, 55488, 62700, 70512, 78948, 88032, 97788, 108240, 119412, 131328, 144012, 157488, 171780, 186912, 202908, 219792, 237588

Offset: 1

Eric W. Weisstein, Sep 07 2017

For n > 1, Wiener index of the 2n-crossed prism graph.

Eric Weisstein's World of Mathematics, Crossed Prism Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006527, A054602, A217873.

Table[4 n (n^2 + 2), {n, 50}]
LinearRecurrence[{4, -6, 4, -1}, {12, 48, 132, 288}, 20]
CoefficientList[Series[(12 (1 + x^2))/(-1 + x)^4, {x, 0, 20}], x]

a(n) = 4*n*(n^2 + 2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: 12*x*(1 + x^2)/(-1 + x)^4.
From Elmo R. Oliveira, Aug 09 2025: (Start)
E.g.f.: 4*x*(3 + 3*x + x^2)*exp(x).
a(n) = 12*A006527(n) = 4*A054602(n) = 3*A217873(n). (End)

A173044 Product plus sum of five consecutive nonnegative numbers.

Original entry on oeis.org

10, 135, 740, 2545, 6750, 15155, 30280, 55485, 95090, 154495, 240300, 360425, 524230, 742635, 1028240, 1395445, 1860570, 2441975, 3160180, 4037985, 5100590, 6375715, 7893720, 9687725, 11793730, 14250735, 17100860, 20389465, 24165270, 28480475, 33390880, 38956005

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 21 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A028387, A054602, A166941.

[(n+2)*(n^4 +8*n^3 +19*n^2 +12*n +5): n in [0..40]]; // G. C. Greubel, Feb 19 2021

A173044:= n-> (n+2)*(n^4 +8*n^3 +19*n^2 +12*n +5); seq(A173044(n), n=0..40) # G. C. Greubel, Feb 19 2021

a[n_]:= n*(n+1)*(n+2)*(n+3)*(n+4) + n + (n+1)+(n+2)+(n+3)+(n+4);
Table[a[n],{n,0,5!}]

[(n+2)*(n^4 +8*n^3 +19*n^2 +12*n +5) for n in (0..40)] # G. C. Greubel, Feb 19 2021

a(n) = n*(n+1)*(n+2)*(n+3)*(n+4) +n +(n+1) +(n+2) +(n+3) +(n+4).
G.f.: 5*(2 +15*x +16*x^2 -14*x^3 +6*x^4 -x^5)/(1-x)^6. - Colin Barker, Jun 25 2012
a(n) = (n+2)*(n^4 +8*n^3 +19*n^2 +12*n +5) = n^5 +10*n^4 +35*n^3 +50*n^2 +29*n +10. - Bruno Berselli, Jun 25 2012
E.g.f.: (10 +125*x +240*x^2 +120*x^3 +20*x^4 +x^5)*exp(x). - G. C. Greubel, Feb 19 2021

Offset corrected by G. C. Greubel, Feb 19 2021

A277625 Nontrivial values of Fibonacci polynomials.

Original entry on oeis.org

2, 3, 5, 8, 10, 12, 13, 17, 21, 26, 29, 33, 34, 37, 50, 55, 65, 70, 72, 82, 89, 101, 109, 122, 135, 144, 145, 169, 170, 197, 226, 228, 233, 257, 290, 305, 325, 357, 360, 362, 377, 401, 408, 442, 485, 528, 530, 577, 610, 626, 677, 701, 730, 747, 785, 842, 901, 962, 985, 987

Offset: 1

Bobby Jacobs, Oct 24 2016

nonn

The polynomial FibonacciPolynomial(x, y) satisfies the recurrence FibonacciPolynomial(0, y) = 0, FibonacciPolynomial(1, y) = 1, and FibonacciPolynomial(x, y) = y*FibonacciPolynomial(x-1, y) + FibonacciPolynomial(x-2, y).
Nontrivial means a value FibonacciPolynomial(x, y) with x>=3 and y>=1. For FibonacciPolynomial(0, y) = 0 and FibonacciPolynomial(1, y) = 1 for all y, and any number y can be represented trivially as FibonacciPolynomial(2, y).
5 = FibonacciPolynomial(5, 1) = FibonacciPolynomial(3, 2) is the only known number that can be represented as a nontrivial Fibonacci polynomial in more than one way.
Numbers obtained as A104244(n,A206296(k)), where n >= 1 and k >= 3 (all terms from array A073133 except its two leftmost columns) and then sorted into ascending order, with any possible duplicate (5) removed. - Antti Karttunen, Oct 29 2016

			12 is in this sequence because FibonacciPolynomial(4, 2) = 12.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Wikipedia, Fibonacci polynomials.

Cf. A000045, A000129, A001076, A006190, A052918 (FibonacciPolynomial(x, y) for different values of y).
Cf. A002522, A054602, A085151 (FibonacciPolynomial(x, y) for different values of x).
Cf. A073133, A104244, A206296.

Take[Union[Flatten[Table[Fibonacci[x, y], {x, 3, 20}, {y, 50}]]], 60] (* Robert G. Wilson v, Oct 24 2016 *)

list(lim)=my(v=List()); for(y=1,sqrtint(lim\1-1), my(a=y,b=y^2+1); while(b<=lim, listput(v,b); [a,b]=[b,a+y*b])); Set(v) \\ Charles R Greathouse IV, Oct 30 2016

FibonacciPolynomial(x, y) with x>=3 and y>=1.

a(n) = n^2 - 2*n^(5/3) - O(n^(3/2)). - Charles R Greathouse IV, Nov 03 2016

More terms from Robert G. Wilson v, Oct 24 2016

A325173 Perfect squares of the form a + b^2 + c^3, where a,b,c are consecutive numbers.

Original entry on oeis.org

9, 144, 1089, 5184, 18225, 51984, 127449, 278784, 558009, 1040400, 1830609, 3069504, 4941729, 7683984, 11594025, 17040384, 24472809, 34433424, 47568609, 64641600, 86545809, 114318864, 149157369, 192432384, 245705625, 310746384, 389549169, 484352064, 597655809, 732243600

Offset: 1

Philip Mizzi, Sep 05 2019

			9 = 0 + 1^2 + 2^3. 0,1,2 are consecutive numbers and 9 is a perfect square. Hence, 9 is a member of the sequence.
18225 = 24 + 25^2 + 26^3. 24,25,26 are consecutive numbers and 18225 is a perfect square. Hence 18225 is a member of the sequence.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Intersection of A000290 and A027620.

Cf. A005563 (the indices n that give these squares), A054602.

a(n) = n^2*(2 + n^2)^2 \\ David A. Corneth, Sep 11 2019

Vec(9*x*(1 + x)*(1 + 8*x + 22*x^2 + 8*x^3 + x^4) / (1 - x)^7 + O(x^40)) \\ Colin Barker, Sep 11 2019

a(n) = A000290(A054602(n)). - Michel Marcus, Sep 05 2019
From Colin Barker, Sep 05 2019: (Start)
G.f.: 9*x*(1 + x)*(1 + 8*x + 22*x^2 + 8*x^3 + x^4) / (1 - x)^7.
a(n) = n^2*(2 + n^2)^2.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)
E.g.f.: exp(x)*x*(9 + 63*x + 114*x^2 + 69*x^3 + 15*x^4 + x^5). - Conjectured by Stefano Spezia, Sep 05 2019 after Colin Barker
From Chai Wah Wu, Sep 10 2019: (Start)
Above conjectures are true. Proof: k + (k+1)^2 + (k+2)^3 = (k+1)*(k+3)^2 and thus is a perfect square if and only if k+1 = n^2 is a perfect square. This implies that (k+1)*(k+3)^2 = n^2*(n^2+2)^2.
(End)

A329655 Square array read by antidiagonals: T(n,k) is the number of relations between set A with n elements and set B with k elements that are both right unique and left unique.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 33, 20, 5, 6, 30, 72, 72, 30, 6, 7, 42, 135, 208, 135, 42, 7, 8, 56, 228, 500, 500, 228, 56, 8, 9, 72, 357, 1044, 1545, 1044, 357, 72, 9, 10, 90, 528, 1960, 4050, 4050, 1960, 528, 90, 10, 11, 110, 747, 3392, 9275, 13326, 9275, 3392, 747, 110, 11

Offset: 1

Roy S. Freedman, Nov 18 2019

A relation R between set A with n elements and set B with k elements is a subset of the Cartesian product A x B. A relation R is right unique if (a, b1) in R and (a,b2) in R implies b1=b2. A relation R is left unique if (a1,b) in R and (a2,b) in R implies a1=a2.

			The symmetric array T(n,k) begins:
  1,   2,    3,    4,     5,      6,       7,       8,        9, ...
  2,   6,   12,   20,    30,     42,      56,      72,       90, ...
  3,  12,   33,   72,   135,    228,     357,     528,      747, ...
  4,  20,   72,  208,   500,   1044,    1960,    3392,     5508, ...
  5,  30,  135,  500,  1545,   4050,    9275,   19080,    36045, ...
  6,  42,  228, 1044,  4050,  13326,   37632,   93288,   207774, ...
  7,  56,  357, 1960,  9275,  37632,  130921,  394352,  1047375, ...
  8,  72,  528, 3392, 19080,  93288,  394352, 1441728,  4596552, ...
  9,  90,  747, 5508, 36045, 207774, 1047375, 4596552, 17572113, ...

Roy S. Freedman, Some New Results on Binary Relations, arXiv:1501.01914 [cs.DM], 2015.

The diagonal T(n,n) is A097662. T(1,k)=A000027; T(2,k)=A002378; T(3,k)=A054602.

T:= (n,k)-> value(Sum(binomial(n,j)*binomial(k, j)*j!, j=1..k)):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

T[n_, k_] := Sum[Binomial[n, j] * Binomial[k, j] * j!, {j, 1, k}]; Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)

T:=(n,k)->_plus (binomial(n,j)*binomial(k, j)* j! $ j=1..k):

T(n,k) = Sum_{j=1..k} binomial(n,j)*binomial(k,j)*j!.

T(n,k) = A088699(n,k)-1.

A120566 G.f. satisfies: A(x) = A(A(x)) - x*A(A(A(x))), with A(0)=0.

Original entry on oeis.org

1, 1, 1, 3, 7, 33, 109, 643, 2623, 17929, 85349, 652395, 3517911, 29484193, 176844781, 1605009651, 10575269935, 103033059513, 738834271605, 7676696689275, 59466011617671, 655467253898577, 5451048833933693

Offset: 1

Paul D. Hanna, Jun 14 2006

nonn

If A(0, x) = x, A(n+1, x) = A( A(n, x)) = A(n, A(x)). Then A(n, x) = x + n*x^2 + n^2*x^3 + (n^3 + 2*n)*x^4 + (n^4 + 6*n^2)*x^5 + ... where [x^4] A(n, x) = A054602(n). - Michael Somos, Jan 22 2012

			A(x) = x + x^2 + x^3 + 3x^4 + 7x^5 + 33x^6 + 109x^7 + 643x^8 +...
A(A(x)) = x + 2x^2 + 4x^3 + 12x^4 + 40x^5 + 168x^6 + 736x^7 + 3784x^8+..
x*A(A(A(x))) = x^2 + 3x^3 + 9x^4 + 33x^5 + 135x^6 + 627x^7 + 3141x^8+...

{a(n)=local(A=x+x^2+x*O(x^n));if(n<1,0, for(i=1,n,A=x-subst(A,x,-x)*subst(A,x,A));polcoeff(A,n))}

G.f. satisfies: A(-A(-x)) = x ; Also: A(x) = x + A(A(x))*series_reversion(A(x)).

Since g.f. satisfies: A(A(x)) = ( x - A(x) )/A(-x), then higher order self-compositions of A(x) reduce into expressions involving A(x) and A(-x). - Paul D. Hanna, Jul 22 2006

cp's OEIS Frontend

A292022 a(n) = 4*n*(n^2 + 2).

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A173044 Product plus sum of five consecutive nonnegative numbers.

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A277625 Nontrivial values of Fibonacci polynomials.

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A325173 Perfect squares of the form a + b^2 + c^3, where a,b,c are consecutive numbers.

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A329655 Square array read by antidiagonals: T(n,k) is the number of relations between set A with n elements and set B with k elements that are both right unique and left unique.

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A120566 G.f. satisfies: A(x) = A(A(x)) - x*A(A(A(x))), with A(0)=0.

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A292022 a(n) = 4n(n^2 + 2).