A101103 Partial sums of A101104. First differences of A005914.

1, 13, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 756, 780, 804, 828, 852, 876, 900, 924, 948, 972, 996, 1020, 1044, 1068, 1092, 1116, 1140, 1164, 1188, 1212, 1236, 1260

Offset

1,2

Comments

For more information, cross-references etc., see A101104.

For n >= 3, a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that Im(f) contains 3 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Mar 08 2007

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.

Claudio de J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq., Vol. 16 (2013) , Article 13.5.7.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = 2*a(n-1) - a(n-2), n > 4.

G.f.: x*(1+x)*(1 + 10*x + x^2)/(1-x)^2.

a(n) = 24*n - 36, n >= 3.

a(n) = Sum_{j=0..n} (-1)^j*binomial(3, j)*(n - j)^4. [Indices shifted, Nov 01 2010]

a(n) = Sum_{i=1..4} A008292(4,i)*binomial(n-i+1,1). [Indices shifted, Nov 01 2010]

Sum_{n>=1} (-1)^(n+1)/a(n) = 157/156 - Pi/48. - Amiram Eldar, Jan 26 2022

Maple

seq(coeff(series(x*(1+x)*(1+10*x+x^2)/(1-x)^2, x, n+1), x, n), n = 1 .. 60); # Muniru A Asiru, Dec 02 2018

Mathematica

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}]; Table[MagicNKZ, {n, 4, 4}, {z, 2, 2}, {k, 0, 34}] OR SeriesAtLevelR = Sum[Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; Table[SeriesAtLevelR, {n, 4, 4}, {r, -3, -3}, {x, 3, 35}]
Join[{1, 13}, LinearRecurrence[{2, -1}, {36, 60}, 33]] (* Ray Chandler, Sep 23 2015 *)

Prog

(PARI) my(x='x+O('x^60)); Vec(x*(1+x)*(1+10*x+x^2)/(1-x)^2) \\ G. C. Greubel, Dec 01 2018
(Magma) I:=[36, 60]; [1, 13] cat [n le 2 select I[n] else 2*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 01 2018
(Sage) s=(x*(1+x)*(1+10*x+x^2)/(1-x)^2).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 01 2018
(GAP) Concatenation([1, 13], List([3..60], n->24*n-36)); # Muniru A Asiru, Dec 02 2018

Crossrefs

Cf. A073762.

Sequence in context: A272108 A034119 A054285 * A051865 A081928 A034129

Adjacent sequences: A101100 A101101 A101102 * A101104 A101105 A101106

Keyword

easy,nonn

Author

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Extensions

Removed redundant information already in A101104. Reduced formulas by expansion of constants - R. J. Mathar, Nov 01 2010

Status

approved