1,2

Partial sum of main diagonal of array defined in A125127 = Array L(k,n) read by antidiagonals: k-step Lucas numbers. The many formulae of A000295 give as many formulae for this sequence.

a(n) = SUM[i=1..n]L(i,i) where L(k,n) = L(k,n-1) + L(k,n-2) + ... + L(k,n-k); L(k,1) = 1 and for n<=0, L(k,n) = 0. a(n) = SUM[i=1..n]((2^i)-1) = (2^(n+1)) - n - 2 = A000295(n+1).

a(1) = 1 because "1-step Lucas number"(1) = 1.

a(2) = 4 = a(1) + [2-step] Lucas number(2) = 1 + 3.

a(3) = 11 = a(2) + 3-step Lucas number(3) = 1 + 3 + 7.

a(4) = 26 = a(3) + 4-step Lucas number(4) = 1 + 3 + 7 + 15.

a(5) = 57 = a(4) + 5-step Lucas number(5) = 1 + 3 + 7 + 15 + 31.

a(6) = 120 = a(5) + 6-step Lucas number(6) = 1 + 3 + 7 + 15 + 31 + 63.

a(7) = 247 = a(6) + 7-step Lucas number(7) = 1 + 3 + 7 + 15 + 31 + 63 + 127.

a(8) = 502 = a(7) + 8-step Lucas number(8) = 1 + 3 + 7 + 15 + 31 + 63 + 127 + 255.

a(9) = 1013 = a(8) + 9-step Lucas number(9) = 1 + 3 + 7 + 15 + 31 + 63 + 127 + 255 + 511.

seq(sum(binomial(n-1, k), k=2..n), n=3..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 23 2007

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+n od: seq(a[n], n=1..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008

Cf. A000012, A000032, A000204, A001644, A001648, A048887, A048888, A074048, A074584, A092921, A104621, A105754, A105755, A125127, A125129.

Row sums of A143291. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jun 01 2009]

Sequence in context: A002940 A030196 A000295 this_sequence A130103 A034334 A036891

Adjacent sequences: A125125 A125126 A125127 this_sequence A125129 A125130 A125131

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 22 2006