A048777 First partial sums of A005409; second partial sums of A001333.

1, 5, 16, 44, 113, 281, 688, 1672, 4049, 9789, 23648, 57108, 137889, 332913, 803744, 1940432, 4684641, 11309749, 27304176, 65918140, 159140497, 384199177, 927538896, 2239277016, 5406092977, 13051463021, 31509019072, 76069501220, 183648021569, 443365544417

Offset: 0

Author

Barry E. Williams

Keywords

• easy
• nonn

Comments

Form an array having the first column all 1's and the first row the squares 1, 4, 9, ..., so m(n,1) = 1 and m(1,n) = n^2 for n = 1, 2, 3, ..., and let the interior terms be m(i,j) = m(i,j-1) + m(i-1,j-1) + m(i-1,j). Then the sums of the terms in the antidiagonals are the terms of this sequence. - J. M. Bergot, Nov 16 2012

Define a triangle with T(n,n)=n+1 and T(n,0)=n*(n+1)+1 for n >= 0. Define the interior terms via T(r,c) = T(r-2,c-1) + T(r-1,c-1) + T(r-1,c). Then the row sums are a(n) = Sum_{k=0..n} T(n,k). - J. M. Bergot, Feb 27 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Crossrefs

Cf. A000129, A001333, A002203, A005409, A048776.

Programs

Magma

I:=[1,5,16,44]; [n le 4 select I[n] else 4*Self(n-1) -4*Self(n-2) +Self(n-4): n in [1..36]]; // G. C. Greubel, Apr 23 2021

Mathematica

LinearRecurrence[{4,-4,0,1},{1,5,16,44},40] (* Harvey P. Dale, Nov 12 2017 *)
Table[(LucasL[n+3, 2] -2*(2n+5))/4, {n,0,35}] (* G. C. Greubel, Apr 23 2021 *)

Sage

[(lucas_number2(n+3,2,-1) -2*(2*n+5))/4 for n in (0..35)] # G. C. Greubel, Apr 23 2021

Formula

a(n) = 2*a(n-1) + a(n-2) + 2*n+1 with a(0)=1, a(1)=5.

a(n) = ( {(5+(7/2)*sqrt(2))*(1+sqrt(2))^n - (5-(7/2)*sqrt(2))*(1-sqrt(2))^n}/2*sqrt(2) ) - (2*n+5)/2.

a(n) = (1/2)*( Pell(n+3) + Pell(n+2) -2*n -5 ), with Pell(n) = A000129(n). - Ralf Stephan, May 15 2007

From Colin Barker, Sep 20 2012: (Start)

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).

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

a(n) = A048776(n-1) + A048776(n). - R. J. Mathar, Feb 28 2013

a(n) = (A002203(n+3) - 2*(2*n+5))/4. - G. C. Greubel, Apr 23 2021

E.g.f.: exp(x)*(7*cosh(sqrt(2)*x) + 5*sqrt(2)*sinh(sqrt(2)*x) - 2*x - 5)/2. - Stefano Spezia, May 13 2023

Extensions

More terms from Harvey P. Dale, Nov 12 2017