A122931 Row sums of triangular array A122930.

1, 2, 7, 18, 50, 132, 351, 924, 2431, 6380, 16732, 43848, 114869, 300846, 787815, 2062830, 5401054, 14140940, 37022755, 96928920, 253766591, 664375032, 1739365272, 4553731728, 11921847625, 31211839802, 81713718151, 213929389674

Offset: 1

Author

Alford Arnold, Sep 20 2006

Keywords

• easy
• nonn

Comments

Also sums of the natural numbers with A000045 entries per row: for example, 1 2 3+4 5+6+7 8+9+10+11+12.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Crossrefs

Cf. A000027, A000045, A001654, A003714, A010049, A035514, A122930.

Cf. A000217, A191797, A301809.

Programs

Maple

A000045 := proc(n) if n <= 1 then RETURN(n) ; else RETURN( A000045(n-1)+A000045(n-2)) ; fi ; end: A000071 := proc(n) RETURN(A000045(n)-1) ; end: A122931 := proc(n) local a45 ; a45 := A000045(n) ; RETURN (a45*(A000071(n+1)+(a45+1)/2)) ; end: for n from 1 to 30 do printf("%d,",A122931(n)) ; od ; # R. J. Mathar, Oct 07 2006

Mathematica

(#[[2]]^2-#[[1]]^2-#[[2]]+#[[1]])/2&/@Partition[Fibonacci[ Range[ 2,30]],2,1] (* or *) Module[{nn=30,fib},fib=Fibonacci[Range[nn]];Total/@ TakeList[ Range[Total[ fib]], fib]](* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Nov 19 2018 *)

Formula

From R. J. Mathar, Oct 07 2006: (Start)

a(n) = Sum_{i=A000071(n+1)+1..A000071(n+2)} i.

a(n) = A000045(n)*floor(A000071(n+1) + (A000045(n)+1)/2). (End)

a(n) = Sum_{k=1..n} A000045(k)^2*A000045(n-k+1). - Gerald McGarvey, Nov 08 2007

a(n) = (F(n+2)^2 - F(n+1)^2 - F(n+2) + F(n+1))/2 where F(n)=Fibonacci(n). - Gary Detlefs, Mar 10 2011

G.f.: x*(1-x)/((1+x)*(1-3*x+x^2)*(1-x-x^2)). - Colin Barker, Mar 12 2012

a(n) = F(n)*(F(n+3)-1)/2. - J. M. Bergot, Mar 16 2013

a(n) = (F(n+1) - 1)*(F(n+2) + 1)/2 + (n mod 2). - Greg Dresden, Sep 25 2021

a(n) = A301809(n+1) - A000045(n) = A191797(n+2) - A191797(n+1). - J.S. Seneschal, Jul 07 2025

Extensions

More terms from R. J. Mathar, Oct 07 2006