A183862 a(n) = n + floor(sqrt(5*n/2)); complement of A183863.

2, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94

Offset

1,1

Comments

Also n + floor(sqrt(5*n/2 + 1/2)) since 5*n/2 + 1/2 is never a square. - Charles R Greathouse IV, Feb 11 2011

Links

Table of n, a(n) for n=1..80.

Mathematica

a=5/2; b=1/2;
Table[n+Floor[(a*n+b)^(1/2)], {n, 80}]
Table[n-1+Ceiling[(n*n-b)/a], {n, 70}]

Prog

(PARI) a(n)=n+sqrtint(5*n/2)

Crossrefs

Cf. A183863.

Keyword

nonn,easy

Author

Clark Kimberling, Jan 07 2011

Status

approved

A183875 Triangle T(n,k) for A(x)^k=sum(n>=k T(n,k)*x^n), where o.g.f. A(x) satisfies A(x)=(a+b*x*A(x))/(c-d*x*A(x)), a=1,b=2,c=1,d=2.

1, 4, 1, 24, 8, 1, 176, 64, 12, 1, 1440, 544, 120, 16, 1, 12608, 4864, 1168, 192, 20, 1, 115584, 45184, 11424, 2112, 280, 24, 1, 1095424, 432128, 113088, 22528, 3440, 384, 28, 1, 10646016, 4227584, 1133952, 237824, 39840, 5216, 504, 32, 1, 105522176, 42115072, 11506944, 2505728, 448064, 65280, 7504, 640, 36, 1

Offset

1,2

Comments

For o.g.f G(x), G(A(x,a,b,c,d))=g(0)+sum(n>0, sum(k=1..n, T(n,k,a,b,c,d)*g(k))x^n).

T(n,k,1,1,1,1)=A080247(n,k),

T(n,k,2,-1,1,1)=A108891(n,k),

T(n,k,1,-2,1,1)=A125692(n,k),

T(n,k,1,-3,1,1)=A125694(n,k),

T(n,k,-2,1,1,1)=A085403(n,k).

Links

Table of n, a(n) for n=1..55.

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

T(n,k,a,b,c,d):=k/n*sum(i=0..n-k, binomial(n,n-k-i)*a^(k+i)*b^(n-k-i)*binomial(i+n-1,n-1)*c^(-i-n)*d^i), a,b,c,d !=0, n>0.

T(n,k,1,2,1,2):=k/n*2^(n-k)*sum(i=0..n-k, binomial(n,n-k-i)*binomial(i+n-1,n-1)), n>0.

Conjecture: T(n,1) = A156017(n-1). - R. J. Mathar, Nov 14 2011

Example

1,
4,1,
24,8,1,
176,64,12,1,
1440,544,120,16,1,
12608,4864,1168,192,20,1,
115584,45184,11424,2112,280,24,1,
1095424,432128,113088,22528,3440,384,28,1,
10646016,4227584,1133952,237824,39840,5216,504,32,1,
105522176,42115072,11506944,2505728,448064,65280,7504,640,36,1

Mathematica

T[n_, k_, a_, b_, c_, d_] := k/n Sum[Binomial[n, n - k - i] a^(k + i) b^(n - k - i) Binomial[i + n - 1, n - 1] c^(-i - n) d^i, {i, 0, n - k}];
T[n_, k_] := T[n, k, 1, 2, 1, 2];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 08 2018, from formula *)

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Feb 12 2011

Status

approved